UNIT-1 ALGEBRA
SEE SPECIFICATION GRID K U A HA Total
21=2 32=6 24=8 15=5 2+6+8+5=21
Contents
1. Function
2. Polynomial
3. Sequence and Series
4. Linear Programming
5. Quadratic equations and Graphs
1. FUNCTION
A. Algebraic Function
1. (a) KNOWLEDGE (K) BASED PROBLEMS
Define algebraic function with an example.
(b) What do you mean linear function? Give an example.
(c) Define constant function with an example.
(d) State identity function with an example.
(e) What is quadratic function? Give an example.
(f) Define cubic function with an example.
2. (a) What type of function is defined by f (x) = mx +c?
(b) State the type of is defined by f (x) = x.
(c) Write the type of the function will be f (x) = mx +c when m = 0?
(d) Write the type of the function will be f (x) = mx + c when m = 1 and c = 0?
3. (a) What value of m in the function f (x) = mx +c makes it a constant function?
(b) What values of m and c in the function f (x) = mx +c makes it an identity function?
4. (a) If f (x) = 3, find the value of f (5)
(b) If f (x) = 10, find the value of f (100).
(c) If f (x) = x then what is the value of f (2)?
(d) If f (x) = x then what is the value of f (10)?
5. (a) Write the nature of the straight line given by a constant function in the graph.
(b) Find the angle in degree inclined by the line representing an identity function with x-axis in positive
direction.
6. (a) Identify and name he functions from the following graphs.
(i) (ii) (iii)
(iv)
(iv) (v) (vi)
(b) Classify the following functions. (iii) y = x (iv) y = 7 (v) y = x3 – 1
(i) y = 2x + 3 (ii) y = x2
BY: TARA BDR MAGAR
B. Composite Function
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 Page 1
KNOWLEDGE (K) BASED PROBLEMS
1. (a) A function f is defined from set A to set B and another function g is defined from set B to set C. What is
the special name for a function so defined from set A to C?
(b) Define composite function.
2. (a) If f: AB and g: BC are two functions then denote the composite function defined from AC.
(b) If g: YZ and h: XYare two functions then denote the composite function defined from XZ.
3. (a) Write the name of the function which has the domain same as that of fog.
(b) Which function has the range same as that of fog?
4. (a) If f (x) = 2x and g (x) = 5, find fog (x).
(b) If f (x) = 5x and g (x) = x + 1, find gof (x).
(c) If f (x) = 4x and g (x) = x2, find gof(x).
(d) If f (x) = 2x and g (x) = x3, find fog(x).
UNDERSTANDING (U) BASED PROBLEMS
1. (a) If f = {(a, 1), (b, 2), (c, 3)} and g = {(1, p), (2, q), (3, r)}, show the function gof in an arrow diagram and
write it in ordered pair form. Ans: {(a, p), (b, q), (c, r)}
(b) If f = {(1, 1), (2, 4), (3, 9)} and g = {(-1, 1), (0, 2), (1, 3)}, show the function fog in an arrow diagram
and write it in ordered pair form. Ans:{(-1, 1), (0, 4), (1, 9)}
(c) If f = {(1, 3), (0, 0), (-1, -3)} & g = {(0, 2), (-3, -1), (3, 5)}, write gof in ordered pair form by
representing in a mapping diagram. Ans:{(1, 5), (0, 2), (-1, -1)}
(d) If g = {(3, 2), (4, 1), (5, 0)} and h = {(1, 3), (2, 4), (3, 4)}, find the function goh by the help of arrow
diagram. Also, find the values of goh (2) and hog (4).Ans: goh = {(1, 2), (2, 1), (3, 1)}; 1, 3
2. (a) If f = {(-1, 1), (0, 2), (1, 3)} & gof = {(-1, 1), (0, 4), (1, 9)}, find the function g in ordered pair form by
using an arrow diagram. Ans: g = {(1, 1), (2, 4), (3, 9)}
(b) If g = {(p, 1), (q, 2), (r, 3)} and fog = {(p, 4), (q, 4), (r, 4)}, find the function f in ordered pair form by
the help of mapping diagram. Also, mention the type of the function.
Ans: f = {(1, 4), (2, 4), (3, 4)}, constant function
(c) If g = {(3, 2), (4, 3), (5, 4)} & fog = {(3, 9), (4, 9), (5, 9)}, find the function f in ordered pair form by the
help of mapping diagram. Also, mention the type of the function.
Ans: f = {(1, 4), (2, 9), (3, 9)}, , constant function
(d) If f = {(0, 1), (1, 2), (2, 3)} and gf = {(0, 1), (1, 2), (2, 3)}, find the function g in ordered pair form by
the help of balloon diagram. Also, mention the type of the function.
Ans: g = {(1, 1), (2, 2), (3, 3)}, identity function
3. (a) If f(x) = 2x – 1 and g (x) = 4x, find the value of gof (x). [SEE MODEL-2076] Ans:8x - 4
(b) If f(x) = 3x + 2 and g (x) = 2x + 1, find fog (x). Ans:6x + 5
(c) If f: R R: f(x) = 3x + 1 and g: R R: g (x) = 5x – 3, find gof (x). Ans:15x + 2
(d) If f: N N: f(x) = x + 2 & g: R R: g (x) = 4 – x2, find (gof) (x). Ans:-x2 - 4x
4. (a) If f(x) = x + 2, x R and g (x) = 3x – 1, x R; find (gof ) (5). Ans: 20
(b) If f(x) = 5 – x, x R and g (x) = 11- 3x, x R; find (gof ) (2). Ans: 2
(c) If f(x) = 6x + 7, calculate the value of ff (-1). Ans:13
Ans:3
(d) If g: R R: g (x) = 2x 1 , find the value of g2(8)
Ans: 4
3 Ans: 2
5. (a) If f (x) = 9x + 1 and f [g(x) – 3] = x + 3, find g (7).
(b) If f (x) = 2x – 1 and g [f(x) + 4] = 6x + 5, find g (2).
6. (a) If g(x) = x + 1 and h(x) = 2x – 1are two functions, prove that hog (x) = x.
2
(b) If f(x) = m am xm , show that: ff (x) = x
7. (a) If f (x) = 3x and g (x) = x + 2, find fog(x) = 18, find the value of x. Ans: 4
(b) If f (x) = 2x + 1, g (x) = 3x + 1 and fog (x) = 15, find the value of x. Ans: 2
(c) If f (x) = 3x + 2, g (x) = 4x – 5 and gof (x) = 9, find the value of x. Ans: 1/2
(d) If f: x 3x + b and ff (2) = 12, find the value of b. Ans:-3/2
(e) If g: R R: g: x 4x – p and gg (4) + 1 = 0, find the value of p. Ans:13
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8. (a) If f (x) = 3x - 2 and fog (x) = 6x - 2, find g(x). Ans: 2x
(b) If f (x) = 5x - 6 and fog (x) = 10x – 1, find g(x). Ans:2x + 1
Ans:2x + 2
(c) If g (x) = 4x + 5 and gof(x) = 8x + 13, find f(x) Ans:2x - 1
(d) If g (x) = 2x + 1 and gof(x) = 4x – 1 , find f(x) Ans4x + 5
Ans: 2x + 3
9. (a) If g(x) = 2x + 2 and (fog)(x) = 8x + 13, find f (x) Ans: x – 1
(b) Let g: R R: g (x) = 4 - x and (fog) (x) = 11 – 2x then find f (x). Ans: x+ 1
(c) If g: N R: g (x) = x2 and (fog)(x) = x2 – 2x + 1 then find f (x).
(d) If g: N R: g (x) = x2 – 1 and (fog) (x) = x2 then find f (x).
APPLICATION (A) BASED PROBLEMS
1. (a) If f: NN: f(x) = 2x and g: NR: g (x) = 3x + 4, find the values of (fog) (4) and (gof) (3). Can (gof) (-
1) be defined? Ans: 32, 22, No
(b) If f: NN: f(x) = 3x+1 and g: NR: g (x) = 4x -3, find the values of (fog) (6) and (gof) (9). Can (gof)
(-3) be defined? Ans: 64, 109, No
2. (a) If f: NN: f(x) = 4x + 5 and (fog)(x) = 8x + 13, find the value of x such that (gof)(x) = 28.Ans:2
(b) If f: RR: f(x) = 3x + 2, (fog) (x) = 6x +17 and (gof) (x) = 3 find the value of x. Ans:-1
(c) If f (x) = 4x + 1 and (gof) (x) = 12x + 13, then find the value of x such that (fog) (x) = 65. Ans:2
3. (a) If f (x) = 3x + 4 and g(x) = 2(x + 1), prove that fog = gof
(b) If g (x) = 2x – 1 and h (x) = 3x - 2, then prove that (goh)(x) = (hog) (x)
(c) If g (x) = x + 2and h (x) = x – 2, then prove that (goh)(x) = (hog) (x)
4. (a) If f (x) = 3x + a and fof (6) = 10, find the value of ‘a’. Also, find the value of ff (10). Ans:–11, 19
(b) If f (x) = 2x + p, fof (2) = 2f (2) + 1, find the value of ‘p’ and ff (3). Ans:1, 15
HIGHER ABILITY (HA) BASED PROBLEMS
1. The number of bacteria in the food kept in a refrigerator is N (T) = 20T2 – 80T + 500 (2 T 14), where
T denotes the temperature and T (t) = 4t + 2 (0 t 3), where t represents the time in hour.
(i) Find (NoT)(t) Ans: 320T2 + 420
(ii) How many bacteria may be in the food afer 2 hours? Ans: 1700
(iii) After how long time, the number of bacterias may be 3300? Ans: 3 hrs
1.3 Inverse Function
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define inverse function.
(b) Under what condition, the inverse of a function is possible?
2. (a) If f and g are two functions such that fog (x) = fog(x) = x, what is the relation between the functions
f and g?
(b) If f is a function then what is the value of (fof – 1) (x)?
3. (a) If f = {(1, 3), (2, 4)} is a function, find f – 1.
(b) Find the inverse of the function g = {(a, p), (b, q)}.
(c) If f – 1 = {(1, 10), (2, 20)}, find the function f.
(d) If h– 1 = {(-2, 4), (-3, 9)}, find the function h.
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Find f – 1(x) if f (x) = 4x + 5. [SEE MODEL-2076] Ans (x -5) /4
(b) If f (x) = 8x – 3 is a one to one onto function, find f – 1(x).
Ans: x 3 / 8
(c) Find the inverse of the function f = {(x, x + 2): x}
(d) Find the inverse of the function f: x 2x + 1, x R. Ans: x – 2
Ans: f 1 : x x 1 / 2
(e) If g:x 3 ,x -2 , find g – 1 Ans: g-1(x)= 3-2x
x+2
x
2. (a) If f (x) = 2x – 3 is a one to one onto function, find the value of f – 1 (5). Ans:4
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(b) If g(x) = 4x – 3 is a one to one onto function, find the value of g 1 (7). Ans: 5/2
Ans: 8
(c) If g(x) = 3x 4 is a one to one onto function, find the value of h – 1(4).
Ans: 2
5 Ans: 11
3. (a) If f – 1(x) = 4x – 3, find the value of f (5). Ans: 7
Ans: -1, 2
(b) If g – 1(x) = 2x 1 , find the value of g (7). Ans: (x + 19)/5
Ans: (x – 7) /3
3 Ans: x / 3
Ans: (x + 1) /10
(c) If h – 1(x) = 3x + 1 and h (m) =2, find the value of m.
(d) If g – 1(x) = kx - 2 and g (k) = k, find k.
4. (a) If f (x+ 2) = 5x – 9, find f – 1 (x).
(b) If f (3x – 1) = 9x + 4, find f – 1 (x).
5. (a) If f (x) = 3x, g (x) = x, find (gof) – 1(x).
(b) If f (x) = 2x, g (x) = 5x – 1, find (gof) – 1(x).
APPLICATION (A) BASED PROBLEMS
1. (a) If f (x) = 2x + 7, find f (x + 2) and f – 1 (x + 2). Ans: 2x + 11, (x – 5) / 2
(b) If g(x) = 3x + 4, find g (2x – 1) and g – 1 (2x – 1). Ans: 6x + 1, (2x – 5) / 3
2. (a) If f (x) = 2x – 1, g(x) = 4x 3 and fog – 1 (x) = 5, find the value of x. Ans: 3
5
(b) If f (x) = x 1 , g(x) = x - 5 and fog – 1 (x) = 6, find the value of x. Ans: 3
22
(c) If f (x) = 3x – 7, g(x) = 5x 2 and g – 1f (x) = 8, find the value of x. Ans: 7
3 Ans: 1
(d) If f (x) = 4x + 7, g (x) = 3x – 5 and fog – 1 (x) = 15, find the value of x.
(e) If f (x) = 3x + 4, g (x) = x + 1 and f – 1g (x) = 2, find the value of x. Ans: 9
3. (a) If f (x) = 4x – 17, g (x) = 2x + 8 and ff(x) = g– 1 (x), find the value of x. Ans: 6
5
(b) f: RR: f(x) = 2x + 1 and g: RR: g(x) = 3x 1 are two given functions. If ff(x) = g– 1 (x), find the
2
value of x. Ans: -1
(c) If f(x) = 3x 10 and g (x) =3x – 5 are two given functions such that gg (x) = f – 1 (x), find the value
2
of x. Ans: 2
(d) If f = (x, 3x – 11), g = x, 2x - 3 and ff (x) = g – 1 (x), find the value of x. Ans:7
5
(e) If f = (x, 2x + 5), g = x, 3x -1 and f g – 1 (x) = f (x), find the value of x. Ans:1
2
4. (a) If f (x) = 2x + 1, g (x) = px – 1 and f – 1 (3) = g – 1(4), find the value of p. Ans : 5
(b) If f (x) = ax + 9, g (x) = 3x + 8 and f – 1 (10) = g – 1(11), find the value of a. Ans: 1
(c) If f (x) = 5x– 9, g (x) = 2x + k and f – 1 (6) = g – 1(9), find the value of k. Ans: 3
(d) If f (x) = 3x – m, g (x) = 5x – 3 and f – 1 (11) = g – 1(22), find the value of m. Ans: 4
HIGHER ABILITY (HA) BASED PROBLEMS
1. (a) Functions f (x) = 3x 11 , x 3and g (x) = x - 3 are given. If f – 1(x) = go f(x), find the value of x.
x-3 2
Ans: -2, 5
(b) If f (x) = x - 2 , x 1 and g (x) = 1 , x 0 and f (x) = g – 1 (x), find the value of x. Ans: 1
2x 1
2 x
(c) If f (x) = x2 – 2x, g (x) = 2x + 3 and fog – 1 (x) = 3 then find the value of x. Ans: 1, 9
2. (a) If g(x) = x and g (x) = g– 1 (x), find the value of x. Ans: 0, 2
2x 3
(b) If h (x) = x and h (x) = h– 1 (x), find the value of x. Ans:-1, 0
3 2x
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3. (a) If f (x) = 2 ; x 3 and g(x) = a2 + 3ax + x2 and gof – 1(1) = 5, find the possible values of a.
x+3
Ans:-1, 4
(b) If f(x) = 3x + 8, g(x) = x2 – 5x + 3 and g(x) = f – 1 (5), find the values of a.Ans:1, 4
4. (a) If f (x) = 3x – 4 and f –1g(x) = 3x 2 are given functions, find g(x) and f – 1 1 Ans: 3x – 2, 3/2
3 2
(b) If f (x) = 2x + 3 and f –1g(x) = x – 5 , find g(x) and f – 1 1 Ans: (2x + 1)/2, 0
4 2
5. (a) Ans:x/6
(b) If f (x) = 3x – 1 and fog(x) = 6x + 5, find (gof) – 1 Ans: (x-5)/10
If f (x) = 5x + 2 and fog(x) = 10x + 7, find (gof) – 1
6. (a) Let f (x) = 2x 1 , g (x) = 3x 1 , and gof – 1 (x) is an identity function, find the value of x. Ans: 1
34
(b) Let g (x) = 6x -1 , h (x) = 5x - 2 and g – 1oh– 1 (x) is an identity function, find the value of x. Ans: 1
53
1.4 Trigonomrtic Function
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define trigonometric function. [SEE MODEL-2076]
(b) What is period of trigonometric function?
2. (a) Write the domain of sine function.
(b) Write the range of the function f (x) = sinx.
(c) Write the y- intercept of the function f (x) = sinx.
(d) Find the maximum and minimum value of sine function.
(e) What is the period of sine function?
3. (a) Write the domain of cosine function.
(b) Write the range of the function f (x) = cosx.
(c) Write the y- intercept of cosine function.
(d) Find the maximum and minimum value of cosine function.
(e) What is the period of cosine function?
4. (a) Write the domain of tangent function.
(b) Write the range of the function f (x) = tanx.
(c) Write the y- intercept of tangent function.
(d) Find the maximum and minimum value of tangent function.
(e) What is the period of tangent function?
5. (a) Identify and name the functions from the following figures.
(i)
(ii)
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(iii)
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Draw the graph of trigonometric function y = sinx (-2 x2)
(b) Draw the graph of trigonometric function y = cosx (-2 x2)
(c) Draw the graph of trigonometric function y = tanx (-2 x 2)
2. POLYNOMIAL
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define polynomial.
(b) Define degree of polynomial.
2. (a) If f (x) a polynomial, d (x) the divisor, q (x) the quotient and r (x) the remainder, then express the
polynomial f (x) in terms of d(x), q (x) and r (x).
(b) What do q (x) and d (x) denote in division algorithm f (x) = q (x) × d (x) + r(x)?
3. (a) What is the degree of the polynomial x3 – 4x2 + x + 5?
(b) What is the degree of the polynomial 5x3 + 2x2 – 3x + 10?
4. (a) In a division of two polynomials, which has higher degree, quotient or remainder?
(b) In a division of two polynomials, which has higher degree, divisor or dividend?
(c) In a division of two polynomials, which has higher degree, quotient or dividend?
5. (a) If the polynomial f (x) of degree 3 is divided by (x – a), what is the degree of quotient Q (x)?
(b) While dividing a polynomial f (x) divided by (x – a) the degree of quotient Q (x) is 4, what was the
degree of f (x)?
6. (a) State remainder theorem.
(b) What do you mean by factor theorem?
(c) Define converse of factor theorem.
7. (a) If the polynomial f (x) is divided by (x – a), what is the remainder?
(b) What is the remainder of polynomial f (x) when it is divided by (x – 3)?
(c) If the polynomial f (x) is divided by (x + p), what is the remainder?
(d) What is the remainder of polynomial f (x) when it is divided by (x + 2)?
(e) If a polynomial f (x) is divided by (ax – b), find the remainder.
(f) What is the remainder of polynomial f (x) when it is divided by (ax + b)?
8. (a) If (x – a) is the factor of a polynomial f (x), find the value of f (a).
(b) If (x + p) is the factor of a polynomial f (x), what is the value of f (-p)?
(c) If (x –3) is the factor of a polynomial f (x) , what is the value of f (3)?
(d) For what value of f (c), will (x – c) be a factor of the polynomial f (x)?
9. (a) In a polynomial f (x), if f (1) = 0, find one of the linear factors of f (x).
(b) In a polynomial f (x), if f (- 2) = 0, find one of the linear factors of f (x).
10. (a) Is (x – 1) a factor of x3 – 5x2 – x + 5?
(b) Is (x + 1) a factor of x3 – x2 + x + 1?
11. (a) What do you mean by root or zero of a polynomial?
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(b) What are the zeros of the polynomial f (x) = x2– 9?
(c) What are the roots of the polynomial f (x) = x2– x?
(d) What are the roots of the polynomial f (x) = x2 + 3x?
UNDERSTANDING (U) BASED PROBLEMS
1. Find the polynomial f (x) when quotient q(x), remainder r (x) and divisor d (x) are given.
(a) d (x) = (x + 2), q (x) = 3x + 4 & r (x) = 5 Ans: 3x2 + 10x + 8
(b) d (x) = (7– x), q (x) = x2 + x + 1 & r (x) = 7 Ans: - x3 + 6x2 + 6x + 14
2. (a) If divisor d (x) is x – 1, the quotient is x2 – 8x – 1and the remainder is – 4. What is the original
polynomial? Ans: x3 – 9x2+ 7x - 3
(b) A polynomial p(x) is divided by x – 1 and the quotient is x2 + x + 1 with the remainder 5, what is the
original polynomial p(x)? Ans: x3 + 4
3. Find the quotient and remainder by using synthetic division method.
(a) (2x3 - 3x2 + x + 2) (x – 2) Ans: 2x2 + x + 3, 8
(b) (3x3 - 2x2 + x + 3) (x – 3) Ans: 3x2 + 4x + 9, 21
(c) (5x3 + x2 – 2x + 7) (x + 1) Ans: 5x2 - 4x + 2, 5
(d) (x3 + 5x2 + x - 1) (x + 2) Ans: x2 + 3x - 5, 4
4. Find the quotient and remainder by using synthetic division method.
(a) (4x3 + 2x2 – 4x + 9) (2x – 1) Ans: 2x2 + 2x – 1 , 8
(b) (3x3 + x2 + x + 1) (3x – 2) Ans: x2 + x + 1, 3
(c) (2x3 + x2 – 2x + 10) (2x + 1) Ans: x2 – 1, 11
(d) (6x3 – x2 + 2x + 11) (3x + 1) Ans: 2x2 – x + 1, 10
5. (a) If 2x3 – 7x2 + x + 10 = (x – 1) Q (x) + R, find Q (x) and R. Ans: 2x2 – 5x – 4, 6
(b) If 3x3 + 4x2 + 5x - 6 = (x + 2) Q (x) + R, find Q (x) and R. Ans: 3x2 - 2x + 9, -24
(c) If 6x3 + x2 + 2x + 1 = (2x–1). Q (x) + R, find Q (x) and R. Ans: 3x2 + 2x + 2, 3
(d) If 4x3 – 25 = (2x–5). Q (x) + R, find Q (x) and R. Ans: 2x + 5, 0
6. (a) State remainder theorem. Find the remainder when a polynomial x6 –1 is divided (x – 1). Ans: 0
(b) If p(x) = 4x3 + 5x2 - 6x + 7 is a dividend and (x + 2) a divisor, what is the remainder? Ans: 7
(c) Find the remainder when f (x) = 2x3 + 3x2 – 4x + 10 is divided (2x – 1). Ans: 9
(d) Calculate the remainder if 3x3 –2x2 + 5x + 8 is divided (3x + 1) Ans: 6
(e) Find the remainder when the polynomial (x + 2) (x – 3) (x + 4) (x – 5) + 75 is divided by (x – 2)
Ans: 3
7. (a) The polynomial x3 + 3x2 + ax + 4 leaves the remainder 4 when divided by x - 2, find the value of a.
Ans:10
(b) If x3 – (p + 2) x2 – 3px + 28 leaves a remainder 1 when divided by (x – 3), find the value of p. Ans: 2
(c) If 4x3 – 3mx + 15 leaves the remainder 10 when divided by x - 1, find the value of m. Ans: 3
(d) If 2x3 + 3x2 + px + 4 leaves a remainder 2(4 – p) when divided by (x – 2), find the value of p. Ans: -6
8. (a) Given that the polynomial f(x) = 2x4 – 3x3 + 6x + k. If f (1) = 0, find the value of k. Ans: –5
(b) Given that f(x) = x3 + 5x2 – 7mx + 4m + 2. If f (2) = 0, find the value of m. Ans: 3
(c) For a polynomial f(x) = x3 – 9x2 + (k + 1) x – 7 so that f (7) = 0 what is the value of k? Ans: 14
9. (a) If ax3 + 2x2 – 3 and x2 – ax + 4 leaves the same remainder when divided by (x + 1), find the value of a.
Ans:-3
(b) If 2x2 -5x+ b and x3 - x2 + bx + 9 leaves same remainder when divided by x + 2, find the value of b.
Ans:-7
(c) If kx3 + 3x2-3 and 2x3-5x + k leaves the same remainder when divided by x - 4, find the value of k.
Ans: 1
(d) If 2x3 - 5nx2 + 7nx +13 and 6x3 – 3nx2 + 8x + 11 leaves the equal remainder when divided by (x – 1),
what will be the value of n? Ans: 2
10. (a) If the polynomial p (x) = kx3 + 3x2 – 3x– k leaves the same remainder when it is divided by both (x – 1)
and (x + 2), find the value of k. Ans: 2
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 7
(b) If the polynomial p (x) =3x3 – 2mx2 – (m + 3) x + 2 leaves the same remainder when it is divided by
both (x + 1) and (x – 2), find m. Ans: 2
11. (a) Show that (x + 2) is a factor of the polynomial x3 – 3x2 – 4x + 12.
(b) State factor theorem. Show that (x + 1) is a factor of 6x3 + 7x2 – x – 2.
(c) Prove that (x – a) is a factor of (xn – an) for any integer n.
(d) Show that (x + a) is a factor of (xn + an) when n is an odd integer.
12. (a) Test whether (x – 3) is a factor of x3 + 7x2 – 2x – 27 or not. Ans: Yes
(b) Decide whether (x + 1) is a factor of 3x3 –4x2 – 5x + 2 or not. Ans: Yes
(c) Is (x + 2) a factor of the polynomial 3x3 + 13x2 –16? Ans: No
(d) Is (x + 1) a factor of the polynomial 5x3 – 3x2 + x + 10? Ans: No
13. (a) If (x – 2) is a factor of x3 –2ax2 + ax + 10, find the value of k. Ans:3
(b) If 4x3 –3x2 + 2bx + b – 11 has a factor (x – 3), find the value of b. Ans:-10
(c) If (x + 3) is a factor of x3 – (k – 1) x2 + kx + 54, find the value of k. Ans: 3
(d) If (x – p) is a factor of x3 – px2 – 4x + p + 12, find the value of p. Ans: 4
14. (a) If (x – 1) is a factor of polynomial x2 + kx + 1, show that (x – k) is a factor of x2 + 3x + 2.
(b) If (x – 2) is a factor of polynomial kx3 + x2 – 8x – 4, show that (x + k) is a factor of 2x3 + 3x2 – 3x – 2.
15. (a) If (x – 1) is a factor of a polynomial x3 + ax2 + bx – 6, prove that a + b = 5.
(b) If (x – p) is a factor of polynomial x3 – px2 + px + x – 10, prove that p2 + p = 10.
16. (a) What should be added to x3 – 5x2 + x – 6 to make a polynomial having a factor x – 2? Ans: 16
(b) What should be added to f(x) =2x3 + x2 – 4 x + 5 to make a polynomial having a factor x + 3? Ans: 28
(c) What should be subtracted from g (x) = 3x3 + x2 + 2x + 7 so that x + 1 would be a factor of g (x)?Ans: 3
(d) What should be subtracted from f (x) = x3 + 140 to make a polynomial having a factor x + 5? Ans: 15
APPLICATION (A) BASED PROBLEMS
1. (a) Factorize: x3 – 4x2 – 7x + 10 Ans: (x- 1)(x + 2) (x -5)
(b) Factorize: 2x3 + 13x2 -36 [SEE MODEL -2076] Ans: (x+2)(x+6) (2x-3)
(c) Factorize: x3 – 19x – 30
2. (a) Solve: x3 – 7x2 + 7x + 15 = 0 Ans: (x+3)(x+2)(x-5)
(b) Solve: x3 – 3x2 – 4x + 12 = 0 Ans:– 1, 3, 5
(c) Solve: x3 – 6x2 + 3x + 10 = 0 Ans:– 2, 2, 3
(d) Solve: 8x3 – 2x2 – 5x – 1 = 0 Ans:– 1, 2, 5
Ans:1, -1/4, -1/2
3. (a) Solve: 2x3 + 6 = 3x2 + 11 Ans: -2, 3, 1/2
(b) Solve: 3y3 + 10 = 7y (2y + 1) Ans: 1, 5, 2 /3
(c) Solve: 6 (y3 + 1) = 7y (y + 1) Ans: -1, 2/3, 3/2
4. (a) Solve: 3x3 – 13x2 + 16 = 0 Ans:-1, 4, 4/3
Ans:-1, -2, 2/3
(b) Solve: 3t3 + 7t2 – 4 = 0 Ans:-2, 3, 6
(c) Solve: y3 = 7y2 – 36 Ans:1, 2, -2
(d) Solve: 3x3 = 7x2 – 4 Ans:-2, -2/3, ½
(e) Solve: 6y3 = 4 – 13y2 Ans: – 3, – 2, 5
5. (a) Solve: y3 – 19y – 30 = 0 Ans: 2, 3, - 5
(b) Solve: z3 – 19z+ 30 = 0
(c) Solve: x3 + 84 = 37x Ans: -7, 3, 4
6. (a) Solve: (x+1)(x2 – 5x + 10) – 12= 0 Ans: 1, 1, 2
Ans:1, 3, 4
(b) Solve: (x-2) (x2 - 6x + 7) – 2= 0 Ans: -2, -6, 3/2
(c) Solve: (x-1) (2x2 +15x + 15) – 21 = 0 Ans: – 1, 2, 3
7. (a) Solve: y = x3 – 4x2 + x + 8 and y = 2 Ans:-3, 2,4
(b) Solve: y = x3 – 3x2 – 10x + 30 and y = 6 Ans:1, 3,4
(c) Solve: y = z3 – 8z2 + 19z – 9 and y = 3
HIGH ABILITY (HA) BASED PROBLEMS
1. (a) If f (x) = x3 + 24x and g (x) = 9x2 + 20 are two given polynomials, solve for x when f (x) = g(x).
(b) If f (x) = x3 + 10 and g (x) = 4x2 + 7x are two given polynomials, solve for x when f (x) = g(x).
(c) If f (x) = x3 – 3 and g (x) = x (3x -11) are given polynomials, solve for x when 2f (x) + g(x) = 0.
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(d) If f (x) = x3 – x2 and g (x) = x2 + 2x - 3 are given polynomials, solve for x when 2f (x) - 3g(x) = 0.
Ans: (a) 1, 3, -3/2 (b) 2, 2, 5 (c) 1, -2, 5 (d) 2, -3, -1/2
2. (a) A polynomial x3 + px2 + qx + 6 has a factor x – 2 and leaves a remainder 3 when it is divided by x – 3,
find the values of p and q. Ans:- 3,-1
(b) The polynomial 3x3 + 2x2 – bx + a is exactly divisible by (x – 1) and leaves a remainder 10 when
divided by (x + 4). Find the values of a and b. Ans: 30, 35
(c) If (x + 2) is a factor of px3 – qx2 – 7x + 10 and leaves a remainder 12 when divided by (x + 1), find the
values of a and b. Ans: p = 1, q = 4
3. (a) If (x – 1) and (x + 2) are the factors of the polynomial x3 – 2mx2 – 7x + n, find the values of m and n
(b) If a polynomial x3 + ax2 – bx – 6 has the factors (x + 1) and (x – 2), find the values of a and b.
(c) If (x2 + 2x – 3) is a factor of the polynomial px3 + 5x2 – qx – 3, find the values of p and q.
(d) If (x2 + 8x + 12) is a factor of the polynomial mx3 + 13x2 + nx – 36, find the values of m and n.
Ans: (a) 2, 10 (b) 2, 5 (c) 2, 4 (d) 2, 0
3. SEQUENCE AND SERIES
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define arithmetic sequence. (b) Define arithmetic series.
(c) Define common difference. (d) What do you mean by arithmetic mean?
2. (a) Write down the formula for finding the nth term (tn) when first term (a), common difference (d) and
number of terms (n) of an arithmetic sequence are given.
(b) What is the 6th term of the sequence having n terms?
(c) If 4 and 7 are the first and the second terms of an AP, what is its common difference?
3. (a) What is the arithmetic mean between the numbers ‘a’ and ‘b’? [SEE MODEL-2076]
(b) What is the arithmetic mean between 7 and 13?
(c) What is the arithmetic mean between 10 and 50?
4. (a) What does ‘n’ denote in the formula d = b−a ?
n+1
(b) If there are ‘n’ arithmetic means between the numbers ‘a’ and ‘b’, what will be the common difference?
(c) If there are 3 arithmetic means between 2 and 18, what will be the common difference (d)?
5. (a) Write the formula for finding the last mean (mn) of an arithmetic progression.
(b) If an AP has the first term 2, common difference 3 and 4 means, what is the last mean?
6. (a) If an AS has n terms, first term (a) and last term (l), write the formula of calculating the sum of the
series.
(b) If an AS with n terms has the first term (a) and common difference (d), what is the sum of the series?
(c) What does ‘a’ denote in the formula Sn = n [2a (n 1)d] ?
2
(d) What does ‘l’ denote in the formula of finding the sum of terms of an arithmetic series Sn = n (a l) ?
2
7. (a) Write down the formula to find the sum of the first n natural numbers.
(b) Find the sum of the first 20 odd numbers.
(c) Write down the formula to find the sum of the first n odd numbers.
(d) Find the sum of the first 10 odd numbers.
(e) Write down the formula to find the sum of the first n even numbers.
8. (a) Define geometric sequence. (b) Define geometric series. (c) Define geometric mean.
9. (a) Write down the formula for finding the nth term (tn) when first term (a), common ratio (r) and number
of terms (n) of a geometric sequence are given.
(b) Find the 10th term of a GP having ‘n’ terms.
(c) What is the 3rd term of a GS whose first term is 2 and common ratio is 3.
(d) What is the common ratio of a GP if its first term is 6 and second term is 18?
10. (a) What is the geometric mean between the numbers ‘a’ and ‘b’?
(b) If G is the geometric mean between p and q, find the value of G.
(c) What is the geometric mean between 2 and 8?
(d) What is the geometric mean between 3 and 12?
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 9
b 1/ n1
a
(a) What does n denote in the formula r = ?
11.
(b) If there are ‘n’ geometric means between the two numbers ‘a’ and ‘b’, find the formula to find the
common ratio(r).
(c) In a GP, there are 2 means between 3 and 24, find the common ratio.
12. (a) Find the formula for finding the last mean (mn) of a GP.
(b) What is the 3rd mean of a GS having first term 4 and the common ratio 2?
13. (a) If a geometric series has n terms, first term (a) and last term (l), write the formula of calculating the
sum of the series.
(b) If a geometric series has n terms, first term (a) and common ratio (r), write the formula to find the sum
of the series.
a(rn – 1)
(c) What does r denote in the formula Sn= r – 1 ?
(d) lr a
14. (a) What does l denote in the formula Sn= r 1 ?
If ‘A’ and ‘G’ are the arithmetic and geometric means between any two unequal real numbers, write
the relation between A and G.
(b) Write the relation between the arithmetic and geometric means between two numbers aand b.
UNDERSTANDING (U) BASED PROBLEMS
A. ARITHMETIC SEQUENCE/SERIES
1. (a) Examine whether 7, 10, 13, 16, .., is an arithmetic sequence or not. If so, find the tenth term.
(b) Examine whether 2, 8, 14, 20, , .., is an arithmetic sequence or not. If so, find the sixth term.
(c) Check whether 16, 12, 8, 4, …is an arithmetic sequence or not. If so, find the eleventh term.
(d) Test whether 2, 4, 8, 16, .., is an arithmetic sequence or not. Give your reason. If so, find the eighth
term. Ans: (a) 34 (b) 32 (c) -24 (d) No
2. (a) If the common difference of an AP is 3, find the difference between the 20th term and 15th term.
(b) If the common difference of an AP is 5, find the difference between the 25th and 12nd terms.
(c) If the common difference of an AP is10, find the difference between the 35h and 20nd terms.
Ans: (a) 15 (b) 65 (c) 150
3. (a) In an AS, if the difference between 15th and 10th terms is 55, find the common difference. Ans:11
(b) In an AS, if the difference between 25th and 13th terms is 84, find the common difference. Ans:7
4. (a) Find the 7th term of an AP whose first term is 3 & common difference 4. Ans:27
(b) Find the 6th term of an AP whose first term is 6 and second term is 9. Ans:19
(c) Find the 10th term of an arithmetic sequence 7, 11, 15, … Ans: 43
(d) Find the ninth term of the sequence 7, 3, -1, -5, .... Ans: -25
5. (a) If the fifth term of an AP with first term 2 is 14, find the common difference. Ans: 3
(b) The firstand 8th terms of an AP are -5 and 23, find the common difference. Ans: 4
(c) The common difference of an arithmetic series is 5 and its tenth term is 40, find its first term. Ans: -5
(d) If 3rd term of an AS with common difference -2 is 6, find its first term. Ans:10
6. (a) The first term of an AP is 5, common difference is 3 & last term is 80, find the number of terms
(b) The first term of an AP is 14, common difference is 7 and last term is 98, find the number of terms.
(c) The first term of an AP is 10, common difference is 5 and last term is 505, find the number of terms.
(d) If the nth term of the series 84 + 78 + 72 + ….is 0, find the value of n.
(e) The nth term of the series 2+ 6 + 10 + ….is 98, find the value of n.
Ans: (a) 26 (b) 13 (c) 100 (d) 15 (e) 25
Ans: 13
7. (a) How many terms are there in AP: 7 + 12 + 17 + …+ 67?
(b) How many terms are there in AP: 3, 6, 9, ..., 111? Ans: 37
(c) How many terms are there in AP:25, 50, 75, …, 1000? Ans: 40
(d) How many terms are there in the series – 3 – 1 + 1 + …+ 25? Ans: 15
8. (a) Which term of the series 2 + 5 + 8 + .. is 56? Ans: 19th
(b) Which term of the sequence 6, 9, 12, 15… is 66? Ans :21st
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 10
(c) Which term of the AP: 21, 18, 15, …, is -78? Ans :14th
(d) Which term of the series 1/2 + 2 + 7/2 + 5 is 29? Ans: 20th
9. (a) Is 23 a term of the series 3 + 7 + 9 + …? Ans: Yes
(b) Does -10 belong to series 25 22 1 20 ... ? Ans: Yes
2
Ans : No
(c) Is 35 a term of an AS 6 + 10 + 14 + …? Ans: No
(d) Is -303 a term of an AP 10 + 7 + 4 + …?
10. (a) In an AS: 2, 5, 8, 11, .., which term is 30 more than its 15st term? Ans: 25th
(b) In an AS; 3, 8, 13, 18, .., which term is 130 more than its 31st term? Ans: 57th
(c) In an AP; 5, 7, 9, 11, .., which term is 14 less than its twenty-third term? Ans: 16th
(d) In an AP; 26, 20, 14, .., which term is 60 less than its 10st term? Ans: 20th
11. (a) Find the numbers between 1 and 100 which are divisible by 3. Ans: 33
(b) Find the numbers between 100 and 200 which are divisible by 5. Ans: 19
(c) Find the numbers between 100 and 300 which are divisible by 7. Ans:28
(d) Find the two digit numbers which are divisible by 3. Ans: 30
(e) Find the three digit numbers which are divisible by 50. Ans: 18
12. (a) If x + 2, 4x + 3 and x+4 are in an A.P., find the value of x. Ans: 0
(b) If k, 4k – 9, 3k – 2 is an A.P., find the value of k. Ans :4
(c) If k + 9, 2k – 1 and 2k +7 are in an A.P., find the value of k. Ans: 18
(d) If p – 1, p + 3 and 3p – 1 are in an A.P., find the value of p. Ans: 4
(e) If x – 3, 2x + 1 and 4x – 3 are in A.P., find the value of x. Ans: 8
(f) If x – 1, x + 3 and 2x +1 are in A.P., find the 2nd term. Ans: 9
(g) If x, x2 + 1 and x+ 6 are in A.P., find the value of x. Ans: -1, 2
(h) Find the value of k, if k2 + 4k + 8, 2k2 + 3k + 6 and 3k2 + 4k + 4 are in AP. Ans: 0
13. (a) If 6, p, q, 18 are in an AP, find the values of p and q. Ans: 10, 14
(b) If 5, x, y, z, 21 are in an AP, find the values of x, y and z. Ans: 9, 13, 17
(c) If -3, a, b, c, 5 are in an AP, find the values of a, b and c. Ans: -1, 1, 3
(d) If 3, a, b, c, 11 are in an AP, find the values of a, b and c. Ans: 5, 7, 9
14. (a) Find the arithmetic mean between the numbers12 and 34. Ans: 23
(b) If (a – b) 2 and (a + b) 2 are the 3rd and 5thterms of an arithmetic sequence, find the mean. Ans: a2 + b2
(c) Two numbers are in the ratio 9:7 and the AM between them is 56, find the numbers. Ans: 63, 49
(d) If the AM between ‘a’ and ‘b’ is m, prove that: m = a + b
2
15. (a) If the arithmetic mean of the numbers 22 & 5x is 31, find the value of x. Ans: 8
(b) Find the AM between the numbers x + 10 and 50 is 55, find the value of x. Ans: 50
(c) If 10 is the arithmetic mean between x2 and 11, find the value of x. Ans: 3
(d) For what value of x, 5x - 4 is AM between 2x+ 1and 55? Ans: 20
(e) For what value of x , 4x + 1 is AM between 3x+ 1and 61? Ans: 12
16. (a) The arithmetic mean of two numbers which are reciprocal of each other is 5/4 , find the numbers.
(b) The arithmetic mean of two numbers which are reciprocal of each other is 13/5 , find the numbers.
Ans: (a) 2, 1/2 (b) 5, 1/5
17. (a) Find the sum of the series 2 + 4 + 6 + ….20 terms. Ans: 420
Ans: 210
(b) Find the sum of the series: 3 + 7 + 11 + 15 + ….10 terms. Ans: 400
Ans: 550
(c) Find the sum of the series: 1 + 3 + 5 + 7 + ….20 terms. Ans: 2496
Ans: 645
(d) Find the sum of the series: -2 + 0 + 2 + 4+ ….25 terms. Ans: 35.05
Ans: 691.2
18. (a) Find the sum of the series 5 + 7 + …+ 99
(b) Find the sum of the series 8 + 13 + 18 + ….+ 78
(c) Find the sum of the series 3.01 + 3.02 + 3.04 + ….+ 4.00.
(d) Find the sum of the series 20.4 + 20.8 + 21.2 + ….+ 30.8
19. (a) The first and second terms of an AP are 5 and 8 respectively; find the sum of its first 10 terms.
(b) The first and second terms of an AP are 6 and 11 respectively; find the sum of its first 20 terms
(c) The first and second terms of an AP are 2 and 22 respectively; find the sum of its first 50 terms.
Ans: (a) 185 (b) 1070 (c) 24600
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20. (a) An arithmetic series has 20 terms and the last term is 50. If the first term is 4, find the sum.Ans: 540
(b) An arithmetic series has 30 terms and the 30th term is 65. If the first term is 5, find the sum.Ans: 1050
(c) An AP has 50 terms and the 50th term is 120. If the first term is 20, find the sum. Ans: 3500
21. (a) Find the sum of first 20 natural numbers. Ans: 210
(b) Find the sum of first 35 natural numbers. Ans: 630
(c) Find the sum of first 30 odd numbers. Ans: 900
(d) Find the sum of first 50 odd numbers. Ans: 2500
(e) Find the sum of first 40 even numbers. Ans: 1640
(f) Find the sum of first 60 even numbers. Ans: 3660
22. (a) The sum of the first 11 terms of an AP is 1320; find the 6th term. Ans: 120
b) The sum of the first 23 terms of an AP is 1035; find the twelfth term. Ans: 45
c) If the sum of first 17 terms of an AP adds up to 1666; find the ninth term. Ans: 98
d) If the sum of first 25 terms of an AP adds up to 5000; find the 13th term. Ans: 200
e) If the sum of first 55 terms of an AP adds up to 6600; find the ninth term. Ans: 120
23. (a) If the 3rd term of an arithmetic series is 13, find the sum of first 5 terms. Ans: 65
(b) The 11th term of an AS is 10; find the sum of first 21terms. Ans: 210
(c) The 23rd term of an AS is 54; find the sum of first 45terms. Ans: 2430
(d) The 35th term of an AS is 65; find the sum of first 69 terms. Ans: 4485
24. (a) The sum of first 10 terms of an AP is 190 and common difference is 4, find its first term. Ans: 2
(b) The sum of first 20 terms of an AP is 820 and common difference is 4, find its first term. Ans: 3
(c) The sum of an arithmetic series having six terms is 129 and its last term is 39, find its first term. Ans: 4
(d) An arithmetic series with 10 terms has the sum 275 and the last term is 50, find the first term. Ans: 5
25. (a) Find the number of terms of an arithmetic series whose first term is 6, last term is 54 and the sum of
the terms is 300. Ans:10
(b) Find the number of terms of an arithmetic series whose first term is 13, last term is 87 and the sum of
the terms is 400. Ans:8
(c) How many terms in the AP 54, 51, 48, .. must be added so that the sum may be 513? Ans:18, 19
(c) Solve for x: 1+ 4+ 7 + …. + x = 715 Ans: 64
(d) Solve for x: 2+ 8+ 14 + …. + x = 2852 Ans: 182
B. GEOMETRIC SEQUENCE/SERIES
1. (a) Is 3, 6, 12, .. a geometric sequence? Give reason. Also, find the sixth term of the sequence. Ans::96
(b) Is 2, 6, 18, a geometric sequence? Give reason. Also, find the sixth erm. Ans:486
(c) Is 3 + 9 + 27 + … a geometric series? If it is then find its seventh term. Ans:2187
(d) Is 2 2 2 44 2 ...a geometric series? If it is then find its ninth term. Ans: 32
(e) Is 1 1 1 1 ... a geometric series? If it is then find its eighth term. Ans::16
842
2. (a) The first and the second term of a geometric sequence are 9 and 18 respectively. What is the fifth
term? Ans:144
(b) The first and the second term of a geometric sequence are 32 and 8 respectively. What is the sixth
term? Ans:1/32
(c) The first and the second term of a GP are 2 and -2 respectively. What is the 8th term? Ans: -16
(d) The first and the second term of a GS are 1/9 and 1/3 respectively. What is the 6th term? Ans: -27
3. (a) In a GP, the common ratio is 2 and the 8th term is 256, find the first term. Ans:2
(b) If the sixth term of a GS with common ratio 3 is 810, find the first term. Ans:5
(c) If the 7th term of a GS with common ratio 1/ 2 is 2, find the first term. Ans: 126
(d) The ninth term of a GP with common ratio -2/3 is 216/243, find the first term. Ans:27
4. (a) A geometric sequence has common ratio 3, what will be the quotient when its fifth term is divided by
the third term? Ans: 9
(b) The common ratio of a geometric sequence 5, what will be the quotient when its eighth term is divided
by the fifth term? Ans: 125
(c) In a GP, the quotient is 64when its sixth term is divided by the third term, what the common ratio?
Ans: 4
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(d) In a GP, the quotient is 243when its ninth term is divided by the fourth term, what the common ratio?
Ans: 3
5. (a) If the fourth term of a GP with first term 2 is 54, find the common ratio. Ans:3
(b) The first term of a GS is 5. If its fifth term is 80, find the common ratio. Ans: 2
(c) A geometric sequence has the first term 8. If its fourth term is 1, what is its common ratio? Ans: 1/2
(d) The 6th term of a GS with first term 2 is 8 2 , find the common ratio. Ans: 2
(e) The 5th term of a GP with first term 3 is 16/27, find the common ratio. Ans:2/3
6. (a) How many terms are there in the sequence 2 + 4 + 8 +.. + 256? Ans:8
(b) Find the number of terms in the series 3+ 6 + 12+…+ 192. Ans:7
(c) How many terms are there in the geometric series 16 + 24 + 36 + …+ 81? Ans:5
(d) How many terms are there in the sequence 96 + 48 + 24 +…+ 3? Ans:6
(e) How many terms are there in the geometric series 1 1 2 1 2 ... 64 Ans:9
42
Ans:Yes
7. (a) Is 486 a term of a GP 2, 6, 18, 54, …,? Ans:Yes
(b) Is 768 a term of a GP 3, 6, 12, 24, …? Ans:Yes
(c) Is 1/81a term of a GP 27 + 9 + 3 + 1 + …?
(d) Is 243 a term of a GP: 3,3,3 3,... ? Ans:Yes
8. (a) Which term of GP 2, 4, 8, 18, .. is 256? Ans:8th
(b) Which term of GP 5, -10, 20, -40, …. is 320? Ans:7th
(c) Which term of GP 1 + 1/2 + 1/4 +….. is 1/128? Ans:8th
(d) Which term of the series 27 + 9 + 3 + ….is 1/81 Ans:: 8th
(e) Which term of a GP 2 2 2 4 4 2 ...is 64? Ans:11th
9. (a) If x - 1, x + 2, x + 8 are in a GS, find the value of x. Ans:4
(b) If x – 1, x + 1, 3x – 1 are in a GS, find the value of x. Ans:3
(c) If a, a – 2, a + 1 are in a GP, find the value of a. Ans:4/5
(d) If 4p + 1, 2(p + 7), p + 34 are in a GP, find the value of a. Ans:2
10. (a) If the third term of a G.P. is 2, find the product of first five terms. Ans: 32
(b) If the third term of a G.P. is 3, find the product of first five terms. Ans: 243
(c) If the product of first five terms of a GP is 243, find the third term. Ans:3
(d) If the product of first five terms of a GP is 1024, find the third term. Ans:4
11. (a) If 7, x, y, 56 form a geometric sequence, find the values of x and y. Ans: 14, 28
(b) If 20, x, y and 2500 are in a GP, find the values of x and y. Ans: 100, 500
(c) For what value of m and n: 32, m, n, 1/16 forms a geometric sequence. Ans:4, 1/2
(d) Find the values of x, y and z from the given GP: 1/8, x, y, 2 Ans:1/4, 1/2, 1
(e) If 9, x, y, z, 1/9 are in GP, find the values of x, y and z. Ans: 3,1,1/3
12. (a) Insert 2 geometric means between 2 and 54. Ans:6, 18
(b) Insert 2 geometric means between 3 and 24. Ans: 6, 12
(c) Insert 2 geometric means between 16 and 2. Ans:8, 4
(d) Insert 3 geometric means between 81 and 3. Ans:27, 9
13. (a) If the arithmetic mean between 2 and x is 5, find the geometric mean. Ans:4
(b) If the arithmetic mean between 16 and x is 10, find the geometric mean. Ans:8
(c) If the geometric mean between 3 and x is 9, find the arithmetic mean. Ans:15
(d) If the geometric mean between x and 18 is 24, find the arithmetic mean. Ans:25
14. (a) If x is geometric mean of (x – 5) and (x + 10), find the value of x. Ans:-10
(b) If (x + 5) is geometric mean of (x + 1) and (x + 7), find the value of x. Ans:-9
(c) If (x + 4) is geometric mean of (x + 2) and (x + 8), find the value of x. Ans: 0
(d) If (x + 2) is GM of (2x + 4) and (x - 4), find the value of x. Ans:10, -2
15. (a) Find the sum of the series 1 + 3 + 9 + …up to 6 terms Ans: 364
Ans:510
(b) Find the sum of the series 256 + 128 + 64+ …up to 8 terms. Ans:-85
(c) Find the sum of the series 1 – 2 + 4 – 8 + … upto 8 terms
(d) Find the sum of the series 1 1 1 ... up to 6 terms. Ans: 63/32
24
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 13
(d) Find the sum of the series 2 2 2 2 ...up to 7 terms. Ans: 15 2 14
16. (a) Find the sum of the series 2 + 6 + 18 + …..+ 486. Ans:728
Ans: 1865
(b) Find the sum of the series 5 + 15 + 45+ …..+ 1245 Ans: 511
(c) Find the sum of the series 256 + 128 + 64 + …..+ 1. Ans: 364/243
(d) Find the sum of the series1 1 1 ... 1 . Ans: 255/128
3 9 243
Ans: 3280/81
(e) Find the sum of the series1 1 1 .... 1
24 128
(f) Find the sum of the series 27 9 3 ... 1
81
APPLICATION (A) BASED PROBLEMS
1. (a) If the nth rterems of an AP 7, 12, 17, … is same as the ntrh terem of an AP 27, 30, 33 …, finds the
value of n. Ans: 11
(b) If the nth rterems of an AP 63, 65, 67, … is same as the ntrh terem of an AP 3, 10, 17, …, finds the
value of n. Ans: 13
2. (a) If 6 times the 6 tems of an arithmetic sequence is equal to 9 times the 9th term, show that the its 15th
term is zero.
(b) If 7 times the 7 tems of an arithmetic sequence is equal to 11 times the 11th term, show that the its 18th
term is zero.
3. (a) If fifth and tenth terms of arithmetic sequence are 14 and 29 respectively.Find the first term and the
common diiferece. Also, find the 17th term. Ans: 2, 3, 50
(b) If third and nineth terms of arithmetic sequence are 20 and 5 respectively. Find the first term and the
common diiferece. Also, find the 15th term. Ans: 25, -2/5, -10
(c) If 3rd and 13th terms of an arithmetic sequence are -40 and 0 respectively, find its 28th term. Ans:60
(d) If 5th and 12th terms of an arithmetic series are -16 and 5 respectively, find its 20th term. Ans:29
4. (a) If the fifth and tenth terms of an arithmetic sequence are 17 and 42 respectively, find the sequence.
(b) The fourth and sixth terms of an arithmetic sequence are 24 and 16 respectively, find the sequence.
Ans: (a) -3, 2, 7, 12, … (b) 36, 32, 28 , …
5. (a) If the third and eleventh terms of an arithmetic series are 8 and -8 respectively, find the sum of the first
seven terms of the series. Ans:42
(b) If the 4th and 15th terms of an arithmetic series are 11 and 44 respectively, find the sum of its first 20
terms. Ans:610
6. (a) If the sixth and nineth terms of an arithmetic sequence are 52 and 76 respectively, which term is 100?
Find it. Ans:12th term
(b) If the 5th and 10th terms of an arithmetic sequence are 30 and 55 respectively, which term is 80? Find
it. Ans: 15th term
(c) If the 3rd and 42nd terms of arithmetic sequence are 10 and 88 respectively, which term is 24? Find it.
Ans: 10th term
(d) If the sixth and seventeenth terms of arithmetic sequence are 19 and 41 respectively, which term is
107? Find it. Ans: 50th term
7. (a) The sum of first eight terms of an arithmetic series is 180 and its fifth term is five times of the first
term, find the sum of the first 10 terms. Ans:275
(b) The sum of first ten terms of an arithmetic progression is 50 and its fifth term is treble of the second
term. Calculate the sum of the first twenty terms. Ans:200
(c) The fourth terem of sn srithmetic series is three times the firdt term and seven term exceeds twice the
third term by 1. Find the sum of first ten terms. Ans:120
8. (a) If the sum of first seven terms of an arithmetic series is 14 and the sum of the first ten terms is 125
then find the ninth term of the series. Ans:37
(b) If the sum of first four terms of an arithmetic series is 26 and the sum of the first eight terms is 100
then find the eleventh term of the series. Ans:32
9. (a) The sum of first 4 terms of arithmetic series is 26 and the sum of thirst 6 terms is 57, find the sum of
first 10 terms. Ans:155
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 14
(b) The sum of first 9 terms of arithmetic series is 72 and the sum of thirst 17 terms is 289, find the sum of
first 25 terms. Ans:650
10. (a) In an arithmetic series, if the sum of first 10 terms is equal to that of first 20 tems, find the sum of first
30 terms. Ans:0
(b) In an arithmetic series, if the sum of first 25 terms is equal to that of first 35 tems, find the sum of first
60 terms. Ans:0
(c) In an arithmetic series, if the sum of first 30 terms is equal to that of first 47 tems, find the sum of first
77 terms. Ans:0
11. (a) Three terms in an arithmetic progression have sum 21 and product 315. Find the terms.
Ans:9, 7, 5 or, 5, 7, 9
(b) The sum of three consecutive terms in arithmetic series is 30 and their product is 750, find these terms.
Ans:5, 10, 15 or, 15, 10, 5
(c) The sum of three consecutive terms in arithmetic series is 18 and their product is 750, find these terms.
Ans:1, 6, 11 or 11, 6, 1
12. (a) The first term and the last term of an arithmetic series are 8 and 128 respectively. If the sum of all
terms of the series is 1700, find the number of terms and the common difference of the seriesAns:25,5
(b) The first term and the last term of an arithmetic series are -24 and 72 respectively. If the sum of all
terms of the series is 600, find the number of terms and the common difference of the series.Ans:25, 4
13. (a) There are n arithmetic means between 7 and 77. If the ratio of first mean to the last mean is 1:6, then
find the number of arithmetic means. Ans::13
(b) There are n arithmetic means between 4 and 24. If the ratio of third mean to the last mean is 4:5, then
find the number of terms of the series. Ans:4
14. (a) The sum of three consecutive terms in GP is 62 and their product is 1000, find the terms.
(b) The sum of three consecutive terms in GP is 7 and their product is 8, find the terms.
(c) The sum of three consecutive terms in GP is 21 and their product is 216, find the terms.
Ans: (a) 10, 50 or 50, 10, 2 (b) 1, 2, 4 or 4, 2, 1 (c) 3, 6, 12 or 12, 6, 3
15. (a) Insert 4 geometric means between 2/3and 162. Ans:2, 6, 18, 54
(b) Insert 3 geometric means between 1/9 and 9. Ans:1/3, 1, 3
16. (a) There are some geometric means between 1/2and 16. If the third mean be 4, find the numbers of
means. Ans:4
(b) Find the numbers of geometric means inserted between 1 and 64 in which the ratio of first mean to the
last mean is 1:16. Ans:5
17. (a) The second, fourth and ninth terms of an arithmetic progression are in geometric progression.
Calculate the common ratio of the geometric progression. Ans:5/2
(b) In a geometric series, if the sixth term is 16 times the second term and the sum of the first seven terms
is 127/4, find positive common ratio and the first term. Ans:2, 1/4
18. (a) The sum of first four terms is 40 and the sum of the first two terms is 4 of a geometric series whose
common ratio is positive, find the sum of first 8 terms the series. Ans:3280
(b) The sum of first four terms is 45 and the sum of the first eight terms is 765 of a geometric series whose
common ratio is positive, find the sum of first 6 terms the series. Ans:189
(c) The sum of first 3 terms of a GP is 14 and that of first 6 terms is 126, find the sum of first 12 terms.
Ans: 8190
19. (a) If the arithmetic mean and geometric mean of two numbers are 5 and 4 respectively, find the numbers.
(b) If the arithmetic mean of two numbers is 25 and their geometric mesn is 24, find the numbers.
(c) Find two numbers whose arithmetic mean is 10 and geometric mean is 8.
(d) If the arithmetic mean of two numbers is 75 and gemtric mean is 60, find the numbers.
Ans: (a) Ans:2 and 8 or 8 and 2 (b) 18 and 32 or 32 and 18 (c) 4, 16 or 16, 4 (d) 30, 120 or 120, 30
HIGH ABILITY (HA) BASED PROBLEMS
1. (a) If pth , qth and rth terems of an arithmetic sequence are a, b and c respectively, then prove that:
p (b – c) + q (c – a) + r (a – b) = 0
(b) If pth , qth and rth terems of an arithmetic sequence are a, b and c respectively, then prove that:
a (q – r) + b (r – p) + c (p – q) = 0
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 15
(c) If a, b , c are in an A.P. and x, y, z are in G.P., show that: xbc .yca .zab =1
(d) The sum of first four terms of a GP is 30 and the sum of the last four terms is 960. If the first and the
last terms are 2 and 512 respectively, find the common ratio. Ans:2
2. (a) Three numbers whose sum is 21 are in AP. If 1, 3 and 10 are added to them respectively; then the
numbers are in G.P. Fid the numbers. Ans:4, 7, 10 or 19, 7, -5
(b) Three numbers whose sum is 24 are in AP. If 1, 6 and 18 are added to them respectively; then the
numbers are in G.P. Fid the numbers. Ans:6, 8, 10 or 27, 8, -11
(c) Three numbers whose sum is 18 are in AP. If 2, 4 and 11 are added to them respectively; then the
numbers are in G.P. Fid the numbers. Ans:3, 6, 9 or 18, 6, -6
3. (a) Three numbers whose sum is 7 are in GP. If 1, 3 and 4 are added to them respectively; then the
numbers are in A.P. Fid the numbers. Ans:1, 2, 4 or 4, 2, 1
(b) Three numbers whose sum is 26are in GP. If 5, 9 and 5 are added to them respectively; then the
numbers are in A.P. Fid the numbers. Ans: 2, 6, 18 or 18, 6, 2
4. (a) The product of first five terems of a geometric series is 32. If the third term of the geometric series is
equal to 10th term of an arithmetic series, find the sum of the first 19 terms of the arithmetic series.
(b) The product of first five terems of a geometric series is 243. If the third term of the geometric series is
equal to 8th term of an arithmetic series, find the sum of the first 15 terms of the arithmetic series.
Ans: (a) 38 (b) 45
5. (a) A taxi-meter shows Rs 14 at the time of stating and then runs up by Rs 36 for each additional
kilometer travelled during 6:00 am to 9:00 pm. If Mr Chaudhary travelled a journey of 7 km from
Tripureshwor to Bhaktapur, how much should be paid for journey? Also, how long can be travelled by
taxi for Rs. 554? Ans: Rs 266, 15km
(b) A taxi-meter shows Rs 21 at the time of stating and then runs up by Rs 43 for each additional
kilometer travelled from 9:00 pm to 6:00 am. If Mr Sherpa travelled for a jopurney of 5 km at 11:00
pm, how much should he pay for a jopurney? Also, how long can be travelled by taxi for Rs. 345?
Ans: Rs 236, 9km
6. (a) A firm produced 2500 pair of shoes in its first year. If it increased its production by a constant number
every year and produced 17500 pair of shoes at the end of the fifth yerar, find the increased number of
pair of shoes in each year. Ans: 500
(b) A firm produced 4400 spectacles in its first year. If it increased its production by a constant number
every year and produced 51800 spectacles at the end of the seventh yerar, find the increased number of
spectacles in each year. Ans: 500
7. (a) A contractor on construction job specifies a penalty for delay of completion beyond a certain date as
Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day and so on. The penalty for
each successing day being Rs 50 more than that of the preceeding day. How much money the
contractor has to pay as penalty, if he has delayed the work by 30 days?[SEE MODEL-2076]
(b) A contractor on construction job specifies a penalty for delay of completion beyond a certain date as
Rs 400 for the first day, Rs 700 for the second day, Rs 1000 for the third day and so on. How much
money the contractor has to pay as penalty, if he has delayed the work by 3 weeks?
Ans: (a) Rs 27750 (b) Rs 71400
8. (a) A person pays a loan of Rs 975 in monthly installments, each installment being less than a former by
Rs 5. The amount of first installment is Rs 100. In how many installments will the entire amount be
paid? Given reason. Ans:25
(b) A person pays a loan of Rs 720 in monthly installments, each installment being less than a former by
Rs 40. The amount of first installment is Rs 240. In how many installments will the entire amount be
paid? Given reason Ans:4
9. (a) Mr Gurung saves Rs. 1000 in a bank account on the occation of his daughter’s first birthday, Rs. 1500
on the occation of his daughter’s second birthday, Rs. 2000 in her third birthday and so on.
(i) How much amount will be saved in her fifth birthday? Ans:Rs.3000
(ii) On which birthday of his daughter will he save Rs.4500? Ans:7th birthday
(iii) How much amount will be saved till her 10th birday? Ans:Rs32500
(b) Mrs. Rai saves Rs. 2000 in January, Rs. 2200 in February, Rs. 2400 in March and son on in her
account of a bank. Then
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 16
(i) How much amount will she save in the month June? Ans:Rs 3000
(ii) In which month will she save Rs.3400? Ans:August
(iii) How much amount will she save in the year? Ans:Rs 37200
(c) A man was appointed as a clerk. He began his job with monthly salary Rs 15000 and contracted his
salary would be increased every year by 20% on the previus year, find how much did he receive in
fifth year? Ans: Rs 324000
10. (a) A club consists of members whose ages are in AP. The common difference of their ages is 3 months.
The youngest member of the club is 7 years old and the sum of their ages is 250 years. Find the number
of members and the age of the eldest member. Ans: 25, 13 years
(b) A communityconsists of members whose ages are in AP. The common difference of their ages is 4
months. The youngest member of the club is 8 years old and the sum of their ages is 250 years. Find
the number of members and the age of the eldest member. Ans: 16, 13 years
11. (a) A man borrows Rs 3465 without interest and repays the loan in 6 monthly installments, each
installment being double the preceeding one. Find the first and last installments. Ans: Rs 55, Rs 1760
(b) A man borrows Rs 24200 without interest and repays the loan in 5 installments, each installment being
treble the preceeding one. Find the first and last installments. Ans: Rs 200, Rs 16200
4. LINEAR PROGRAMMING AND GRAPH OF QUADRATIC EQUATION
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define linear programming. (b) What is meant by half plane of an inequality?
(c) What do you mean by objective function? (d) Define constraints. (e) Define feasible region.
2. (a) Write the meaning of x 2. (b) Write the meaning of x ≤ 3. (c) Write the meaning of x ≤ 0.
3. (a) Which quadrant does x 0, y 0 represent in the graph?
(b) Which quadrant does x 0, y 0 represent in the graph?
4. (a) Write the equation of boundary line represented by the inequality ax + by c.
(b) Write the equation of boundary line represented by the inequality 2x + 3y 6.
5. (a) Test whether the point (1, 2) lies in the solution set of inequality 3x + 4y 12 or not.
(b) Does the point (3, 5) lies in the half plane of the inequality 5x – 2y 6?
(c) Does the point (0, 2) lies in the half plane of the inequality 3x + 8y 14?
6. (a) Write the inequality of the shaded region in the given figure.
Ans: x 2
(b) Write the inequality of the shaded region in the given figure.
Ans: 3x + 2y 6
7. (a) Write the standard form of quadratic equation. (b) Write the standard form of cubic equation.
8. (a) Define parabola. (b) What do you mean by the vertex of the parabola?
9. (a) What is the coordinates of vertex of parabola y= ax2 + bx + c, a 0?
(b) What is the coordinates of vertex of parabola with equation y = ax2+ bx, a 0?
(c) What is the coordinates of vertex of parabola with equation y = ax2+ c, a 0?
(d) What is the coordinates of vertex of parabola whose equation is y = ax2, a 0?
(e) What is the vertex of parabola whose equation is y = a(x – h) 2 + k, a 0?
10. (a) For what value of ‘a’, the mouth of the graph of the quadratic equation y = ax2, a 0 faces upwards?
(b) For what value of ‘a’, the graph of the quadratic equation y = ax2, a 0 opens downwards?
(c) In which side, the mouth of the graph of the quadratic equation y = ax2, a 0 faces when a > 0?
(d) In which side, the mouth of the graph of the quadratic equation y = ax2, a 0 faces when a < 0?
11. (a) Write the nature of parabola y = ax2 + bx + c, a 0 when the positive value of ‘a’ increases.
(b) Write the nature of parabola y = ax2 + bx + c, a 0 when the positive value of ‘a’ is decreases.
(c) For what value of ‘a’, the graph of the quadratic equation y = ax2 + bx + c, a 0 becomes wider?
(d) For what value of ‘a’, the graph of the quadratic equation y = ax2 + bx + c, a 0 becomes narrower?
12. (a) What does c denote in the parabola y = ax2 + c, a 0?
(b) Write the position of the vertex of the parabola y = ax2 + c, a 0 lie when c > 0?
(c) Write the position of the vertex of the parabola y = ax2 + c, a 0 lie when c = 0?
(d) Write the position of the vertex of the parabola y = ax2 + c, a 0 lie when c < 0?
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 17
13. (a) What is the line of symmetry of the parabola y = ax2 + bx + c, a 0?
(b) What is the line of symmetry of the parabola y = ax2 + bx, a 0?
(c) What is the line of symmetry of the parabola y = ax2 + c, a 0?
(d) What is the line of symmetry of the parabola whose equation is y = ax2, a 0?
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Show the inequality x + y 2 in a graph. (b) Show the inequality 2x + y 3 in a graph.
2. (a) What will be the points of intersection of the curve f (x) = x2 – 1 and f (x) = 3? [SEE MODEL-2076]
(b) Ans: (-2, 3), (2, 3)
(c)
(d) What will be the points of intersection of the curve f (x) = x2 – 4 and f (x) = 5? Ans: (-3, 5) and (3, 5)
3. (a)
(b) What will be the points of intersection of the curve f (x) = 6 – x2 and f (x) = 2? Ans: (-2, 2) and (2, 2)
(c)
(d) Find the points of intersection of the curve y = 23 – 5x2 and y = 3. Ans (-2, 3) &(2, 3)
4. (a)
(b) Find the vertex of the parabola y = x2 – 2x – 3. Ans: (1, -4)
5. (a)
(b) Find the vertex of the parabola y = x2 – 4x – 5. Ans: (2, -9)
(c)
(d) Find the vertex of the parabola y = x2 + 2x. Ans: (-1, 1)
6. (a)
Find the vertex of the parabola y = x2 + 3. Ans: (0, 3)
(b)
Find the y-intercept of the parabola y = x2 + x – 2 . Ans: -2
(c)
Find the y-intercept of the parabola y = x2 – 2. Ans: -2
(d)
Find the x-intercept of the curve y = x2 – 3x + 2 Ans: 1, 2
7. (a)
Find the x-intercept of the curve y = x2 – 2x – 3 Ans: -1, 3
(b)
Find the x-intercept of the curve y = x2 – x. Ans: 0, 1
(c)
(d) Find the x-intercept of the curve y = x2 – 4. Ans: -2, 2
(e)
8. (a) If the graph of y = x2 – 3x cuts off x-axis at P and Q, write the coordinates of P and Q.
(b) Ans: P (0, 0) and Q (3,0)
If the graph of y = x2 – 9 cuts off x-axis at A and B, write the coordinates
9. (a)
(b) of A and B. Ans: A (-3, 0) & B (3, 0)
(c)
(d) If the graph of y = x2 –3 x – 4 cuts off x-axis at A and B, write the coordinates
of A and B. Ans: A (-3, 0) & B (4, 0)
If the graph of y = x2 – x – 2 cuts off x-axis at A and B, write the coordinates of A and B.
Ans: A (-1, 0) & B (2, 0)
If the graph of y = ax2 + bx + c is symmetrical at a line AB, find the equation of line AB.
Ans: 2y + b = 0
If the graph of y = x2 – 2x – 3 is symmetrical at a line PQ, find the equation of line PQ.
Ans: y = 1
Find the equation of line of symmetry of the parabola y = x2 – 4x – 5.Ans: y = 2
Find the equation of line of symmetry of the parabola y = x2+ x – 2.Ans: 2y + 1 = 0
Find the equation of line of symmetry of the parabola y = 2x2 + 3x.Ans: 4y + 3 = 0
Find the equation of parabola with vertex (1, -1) and passing through the point
(2, 3) as given in the figure. Ans: y = 4 (x- 1) 2 – 1
Find the equation of parabola with vertex (2, 1) and passing through the point
(0, 5) Ans: y = (x- 2) 2 + 1
Find the x-intesept and y- intercept of the parabola y = x2 – 4. Ans: 2, -4
Find the x-intesept and y- intercept of the parabola x2 – 25. Ans: 5, -25
Find the x-intesept and y- intercept of the parabola y = x2 – 3x. Ans: 0, 3; 0
Find the x-intesept and y- intercept of the parabola y = x2 + 2x. Ans: 0, -2; 0
APPLICATION (A) OR HIGHER ABILITY (HA) BASED PROBLEMS
1. Maximize the given objective functions subject to the following constraints:
SN Constraints Objective function Answer
a. x + y ≤ 6, x – y ≤ 4, x ≥ 0, y ≥ 0 P = 3x + 5y PMax = 30 at (0, 6)
b. x + y ≤ 7, x – y ≥ –1, x ≥ 0, y ≥ 0 F = 3x + 4y F Max = 25 at (3, 4)
c. x +y ≤ 5, x + 2y ≥ 1, x ≥ 0, y ≥ 0 F = 3x + 2y + 5 F Max = 20 at (5, 0)
d. x + y ≤ 7, x + 2y ≤ 10, x ≥ 0, y ≥ 0 P = 8x + 3y P Max= 56 at (7, 0)
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 18
e. x +2 y ≤ 10, 2x + y ≤ 14, x ≥ 0, y ≥ 0 P = 2x + 3y P Max= 18 at (6, 2)
f. 2y x- 1 , x + y ≤ 4, x ≥ 0, y ≥ 0 P = 3x +y P Max=10 at (3, 1)
g. x+2y 1 , x + y ≤ 5, x ≥ 0, y ≥ 0 P = 3x + 2y + 5 P Max=20 at (5, 0)
h. 2y x – 1 , x + y ≤ 4, x ≥ 0, y 2 Z = 3x + 2y ZMax=11 at (3, 1)
i. 2x + 3y 6 , 2x – 3y ≤ 4, x ≥ 0, y 2 P = 4x +y P Max=26 at (6, 2)
j. x + 2y 6 , x – 2y ≤ 2, x ≥ -2 P = 5x +3y P Max=23 at (4, 1)
k. x + y 0, x – y ≤ 0, x ≥ -1, y 2 P = 3x + 2y PMax = 10 at (2, 2)
l. 7x – y 35 , x + 6y ≤ 48, 6x + 5y ≥ 30 F = 2x + y F Max = 19 at (6, 7)
2. Minimize the given objective functions subject to the following constraints:
SN Constraints Objective function Answer
a. 2x + y ≤ 20, 2x + 3y ≥ 24, x ≥ 0, y ≥ 0
b. x+y 6 , 3x + 2y ≤ 24, x ≥ 0, y ≥ 0 Z = 5x + 3y ZMin = 0 at (0, 0)
F = 10x + y ZMin = 60 at (6, 0)
c. x+y 6 , x – y 4, x 6 P = 2x + y P Min=11 at (5, 1)
d. x + y 0, x – y ≤ 0, x ≥ -1, y 2 Z = 3x + 2y ZMin = -8 at (-1, 2)
e. 4x + 6y 16, 5x + 3y ≤ 11, x ≥ 0, y ≥ 0 Z = 40x + 30y ZMin = 100 at (1, 2)
3. Optimize the given objective functions subject to the following constraints:
OR,
Find the extreme values of the given objective functions under the following constraints:
OR,
Find the maximum and minimum values of the given objective functions under the following constraints:
SN Constraints Objective function Answer
a. x – 2y ≤ 1, x + y ≤ 4, x ≥ 0, y ≥ 0 P = 5x +4 y PMax = 19at (3, 1)
PMin = 0 at (0, 0)
[SEE MODEL – 2076] F = 2x + 3y FMax = 15 at (0, 5)
FMin = 0 at (0, 0)
b. x - 2y ≤ 2, x + y ≤ 5, x ≥ 0, y ≥ 0 F = 3x + y FMax = 10 at (3, 1)
FMin = 0 at (0, 0)
c. 2y ≥ x – 1, x + y ≤ 4, x ≥ 0, y ≥ 0 P= 10x + 8y PMax = 50at (5, 1)
PMin = 0 at (0, 0)
d. x + y ≤ 5, x +4 y ≤ 8, x ≥ 0, y ≥ 0
4. Solve graphically:
(a) x2 + 2x – 3 = 0 Ans: x = -3, 1 (b) x2 – 2x – 3 = 0 Ans: x = -1, 3
(c) x2 – 3x – 4 = 0 Ans: x = -1, 4 (d) x2 – 3x = 10 Ans: x = -2, 5
(e) x2 – 2x = 15 Ans: x = -3, 5 (f) x2 –x – 12 = 0 Ans: x = 4, -3
5. Solve graphically the following quadratic equations. Also, find the intrersecting points of parabola and
staraight line.
(a) y = x2 and y = 2 – x Ans: x = 1, -2, (1,1), (-2, 4)
(b) y = x2 and y = 3 – 2x Ans: x = 1, -3, (1, 1), (-3, 9)
(c) y = x2 and y = 2x + 8 Ans: x = 4, -2, (4, 16), (-2, 4)
(d) x2 – 3x – 10 = 0 Ans: x = -2, 5, (-2, 0), (5, 0) or, (-2, 4), (5, 25)
UNIT-2 CONTINUITY
SEE SPECIFICATION GRID
Contents K U A HA Total Marks
14=4 1+4=5
Continuity 11=1
1. (a) KNOWLEDGE (K) BASED PROBLEMS
(b)
What is meant by limit of a function?
What do you mean by functional value of function?
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 19
(c) Define continuity of a function.
(d)
2. (a) What does the discontinuity of a curve represent?
(b)
(c) Write the interval [3, 7] in a statement form.
(d)
3. (a) What does the interval (1, 3] denote in words?
(b)
4. (a) How do you state the interval [-2, 2) in words?
(b)
5. (a) What does the interval (-3, 3) state in a sentence?
(b) Express “-1 x 2 in interval form.
(c)
(d) Express “2 x 5 in interval form.
6. (a)
7. (a) Under what condition the limit of a function f (x) exists at x = a?
(c) Write the necessary condition for the continuity of a function f (x) at x = a.
8. (a) Write the left hand limit of a function f (x) at x = a in notational form.
(b)
(c) Write the left hand limit of f (x) at x = 3 in notation.
(d)
(e) Write the notational representation of right hand limit of function f(x) at x = a.
(f)
(g) Write the right hand limit of f (x) at x = 2 in notation.
(h)
What does x 4 + denote? (b) What does x 5- mean? (c) What does x 1 + mean?
9. (a)
(b) Express lim f (x) in sentence. (b) Write lim f (x) in sentence.
xa x2
10. (a)
(b) Write lim f (x) in sentence. (d) Write lim f (x) in sentence.
x2 xa
State the continuity of number line of set of natural numbers.
Write the continuity of number line of set of whole numbers.
Write the continuity of number line of set of rational numbers.
Write the continuity of set of irrational numbers.
Write the continuity of number line of set of real numbers.
Does the continuity exist in the growth of a plant?
Does continuity exist in the crawling of snake?
Does the continuity of integers exist in the number line?
Write the set of numbers which is continuous in number line. [SEE MODEL-2076]
Write a set of numbers which is discontinuous in number line.
Is the function continuous at x = 3? Give reason.
Discuss the continuity or discontinuity of the function at x = 2.
(c) Is the function continuous at x = 1? Give reason.
(d) Discuss the continuity or discontinuity of the function at x = 4.
APPLICATION (A) BASED PROBLEMS
1. (a) For a real valued function f (x) = x +3.
(b)
(i) Find the values of f (x) at x = 1.9, 1.99, 1.999, 1.9999, 2.1, 2.01, 2.001, 2.0001
(c)
(ii) Find the value of f (2).
(d)
(iii) Is this function continuous at x = 2?
For a real valued function f (x) = 2x +1.
(i) Find the values of f (x) at x = 3.9, 3.99, 3.999, 3.9999, 3.1, 3.01, 3.001, 3.0001
(ii) Find the value of f (3).
(iii) Is this function continuous at x = 3?
If f (x) = 2x – 3
(i) Find f (1.9), f (1.99), f (1.999), f (1.9999)
(ii) Find f (2.01), f (2.001), f (2.0001), f (2.00001)
(iii) Find lim f(x), lim f(x) and f(2)
x2 x2
(iv) What conclusion can be drawn?
If f (x) = 5 – x
(i) Find f (0.9), f (0.99), f (0.999), f (0.9999)
(ii) Find f (1.01), f (1.001), f (1.0001), f (1.00001)
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 20
(iii) Find lim f(x), lim f(x) and f(1)
x 1 x 1
(iv) What conclusion can be drawn?
2. (a) Given that f (x) = x2 is a real valued function.
(b) (i) Find the left hand limit at x = 4
3. (a) (ii) Find the right hand limit at x = 4
(b)
(iii) Find the value of f (4).
4. (a)
(b) (iv) Does the limit of f (x) exist at x = 4?
(c)
(v) Is this function continuous at x = 4?
5. (a) Given that f (x) = x3 is a real valued function.
(b)
(c) (i) Find the left hand limit at x = 2
6. (a) (ii) Find the right hand limit at x = 2
(b)
(iii) Find the value of f (2).
7. (a)
(iv) Does the limit of f (x) exist at x = 2?
(b)
(v) Is this function continuous at x = 2?
(c)
For a real valued function f(x) = 2x + 3
(i) Find the values of f(2.95), f(2.99), f(3.01), f(3.05) and f(3).
(ii) Is this function continuous at x = 3? [SEE MODEL-2076]
For a real valued function f(x) = 6x + 1
(i) Find the values of f(1.9), f(1.99), f(2.01), f(2.05) and f(2).
(ii) Is this function continuous at x = 2?
If f (x) = 2x + 3 then establish the condition of continuity of f (x) at x = 1. Write the conclusion.
If f (x) = 5x + 1 then establish the condition of continuity of f (x) at x = 1. Write the conclusion.
If f (x) = x2 – 1 then establish the condition of continuity of f (x) at x = 1. Write the conclusion.
Show that the fuction f (x) = x + 5 is continuous at x = 1.
Show that the fuction f (x) = 5x + 4 is continuous at x = 3.
Show that the fuction f (x) = 8x – 1 is continuous at x = 5.
Prove that the function f (x) = x 2 for x 4 is continuous at x = 2.
2x -1 for x 4
3x 1 for x 1
Prove that the function f (x) = 4 for x 1 is continuous at x = 1.
5x 1 for x 1
2x-1 when x<3
Find the value of k if the function given by f (x) = 5 when x = 3 is continuous at x = 3. Ans: 1
kx+2 when x > 3
px2 +1 when x<2
Find the value of p if the function given by f (x) = 9 when x = 2 is continuous at x = 2. Ans: 2
4x+1 when x > 2
3x2 when x<1
If the function defined by f (x) = 3 when x = 1 is continuous at x = 1, find the value of p. Ans: 2
x3 +p when x > 1
UNIT-3 MATRIX
SEE SPECIFICATION GRID K U A HA Total Marks
11=1 22=4 14=4 1+4+4 = 9
Contents
Determinant and inverse of matrix
Solvingf equations by matrix method
Cramer’s rule
KNOWLEDGE (K) BASED PROBLEMS
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 21
1. (a) Express the equations ax + by = c and px + qy = r in matrix form.
(b)
(c) Write the equations a1x + b1y = c1 and a2x + b2y = c2 in matrix form.
2. (a) Write the equations 2x + y = 3 and 5x + 2y = 8 in matrix form.
(c)
Define singular matrix. (b) What do you mean non-singular matrix?
3. (a)
(b) State inverse of a matrix. (d) If A is a matrix such that A= 0, mention the type of matrix A?
(c)
(d) What type of matrix has its determinant possible?
4. (a) Under what condition the matrix becomes singular?
(c)
Write the necessary condition for the possibility of inverse of a matrix.
In what condition the system of linear equation has not unique solution?
If A = [5], what is the value of A? (b) If A = [-3], what is the value of A?
If A = [p], what is the value of A? (d) If A = [-k], what is the value of A?
5. (a) If A = p q , what is the value of A? [SEE MODEL-2076]
r s
(b) If M = b p , what is the value of M?
p d
(c) If P = sin A cos AA , find the value of P. (d) If Y = se1c α 1 α , find the value ofY.
cos A sin sec
6. (a) Is [0] a singular matrix? (b) Is 2 3 a singular matrix?
4 6
(c) Is 13 24 a singular matrix? (d) Is 85 3 a singular matrix?
7. (a) If 2 x 0 , what is the value of x? (b) 0
69 If 3 4 7 , what is the value of p?
p5
(c) If k 2 0 , what is the value of k? (d) If 2 x 0 , what is the value of x?
8k x 18
8. (a) If AB = I = BA, where A and B are square matrices of same order and I the identity matrix chich has
the order as A and B then write the relation of matrices A and B.
(b) If P and Q are inverse to each other then what is the type of matrix PQ?
(c) If A is non-singular matrix and AX = B then what is X in terms of A– 1 and B.
(d) What is (A – 1) – 1?
9. (a) Write the adjoint matrix of A = a b . (b) Write the adjoint matrix of A = 13 42 .
c d
(c) What is the adjoint matrix of an identity matrix of order 2?
10. (a) Name the method for solving a pair of simultaneous linear equations by using determinant.
(b) If D, D1 and D2 are given then what is formula to find the value of x by using Cramer’s rule?
(c) If D, D1 and D2 are given then what is formula to find the value of y by using Cramer’s rule?
(d) To Cramer’s rule, if D = 5, D1= 15 & D2 = 10, what is the value of x?
According to Cramer’s rule for the system a1x + b1y = c1 and a2x + b2y = c2, what is D?
11. (a) According to Cramer’s rule for the system a1x + b1y = c1 and a2x + b2y = c2, what is D1?
(b) According to Cramer’s rule for the system a1x + b1y = c1 and a2x + b2y = c2, what is D2?
(c) For what value of D, is the solution of equations undefined in Cramer’s rule?
(d)
UNDERSTANDING (U) BASED PROBLEMS
1. (a) If A = 2 3 , find A. What type of matrix is it with respect to its determinant?
(b) 10
15
Ans:0, singular matrix
If P = 3 4 , find P. What type of matrix is it with respect to its determinant?
6 8
Ans:0, singular matrix
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 22
(c) If A = sinθ 1 , find A. What type of matrix is it with respect to its determinant?
(d) cosθ cotθ
2. (a) Ans:0, singular matrix
(b)
If B = 2 1 , find B. What type of matrix is it with respect to its determinant?
3. (a) 4 5
(b)
(c) Ans:6, non-singular matrix
(d)
Is 4 8 a singular matrix? Write with your reason. Ans:Yes
4. (a) 3
(b) 6
Examine whether the matrix 1 0 is a singular or not. Ans: No
0 1
If the matrix 2 3 is singular, find the value of x. Ans: 1
4
x+5
If the matrix 2m 2 is singular, find the value of m. Ans: 2
10 5
For what value of x, the matrix x 2 is singular, find the value of x. Ans: 4
8
x
For what value of x, the matrix 9 p is singular, find the value of x. Ans: 6
p
4
If x 3 = 0 , find the values of x. Ans: 6
42
x6 Ans: 7
If = 7 , find the values of x.
7x
(c) If x x = 0 , find the values of x Ans: 0, 2
6 3x Ans: 3, 4
Ans: 2, 3
(d) If x 2 = 12, find the values of x
xx
(e) If p 2p 3 , find the values of p.
=
3p 45
(f) If x 23 x , find the values of x. Ans: 2, -3
5. (a) = Ans: 19
34 xx Ans:10
(b) Ans:24
(c) If A = 3 1 and B = 1 2 , find 2A + B Ans:20
(d)
4 2 3 5 Ans:-36
(e) Ans:32
6. (a)
2 5 4 6 , MN.
If M = 1 3 and N = find
3 2
If A = 1 0 , find the determinant of AAT.
2
4
2 1
1 2 1 and Q = 14
If P = 4 3 2 , find PQ.
3 3
3 2
If A= 0 1 2 and B = 0 4 , find AB
2 1 2 1
3
If the determinant of the matrix 2 4 is 40, find the value of x.
6
x
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 23
(b) If the determinant of the matrix 3 p 2 is 6, find the value of p. Ans: 3
p
5
(c) If A = 1 2 and determinant of A2 + A is 0, find the value of x. Ans: 1
x
2
(d) If the determinant of x sin A cos A is 1, find the value of x. Ans: 1
sin A
cos A
7. (a) If A = 3 1 and B = 1 2 , verify that: AB=A B
0
0 2 4
(b) If A = 4 3 and B = 1 5 , verify that: AB=A B
3
7 2 4
8. (a) If the matrix k 3 has no inverse, find the value of k. Ans: 6
8
4
(b) For what value of p, the inverse of matrix 12 p is not defined? Ans: 9
32 24
(c) For what value of ‘a’, the inverse of 2 a 42 is not defined? Ans:21
63 3a a
9. (a) Show that the matrices 3 1 and 2 1 are inverse to each other.
5 2 5 3
(b) Show that the matrices 5 2 and 3 2 are inverse to each other.
7
7 3 5
10. (a) If the inverse of matrix 1 2 is the matrix 3 2 , find the values of x and y.
3 1 y
x
Ans:x = 1, y = 1
(b) If the matrices 2x 7 and 9 y are inverse to each other, find the values of x and y.
5 9 5 4
Ans: x = 2, y = -7
(c) If the matrices m 2m 9 and 3 5 are inverse to each other, find the values of m and n.
n 3 n m
Ans: m = 0,2; -1/5, 1
8. (a) If A = 2 1 , findA and write A – 1 is defined or not. [SEE MODEL-2076]
3
1
(b) If A = 4 2 , findA and write A – 1 is defined or not.
1
3
(c) Test whether the matrix 1 1 is invertible or not.
6
9
(d) Test whether the matrix 4 8 is invertible or not.
1 2
9. Find the inverse of the matrix:
(a) A = 4 7 Ans: 1 8 7 (b) A = 2 3 Ans: 4 3
10 3 3
6 8 6 4 4 2
(c) A= 0 2 Ans: 1 9 2 (d) A= 5 8 Ans: 19 8
10. (a) 1 6
9 2 1 0 9 3 6 5
Using Cramer’s rule, find the value of D1 and D2 for ax + by = c and px + qy = r. [SEE MODEL-2076]
Ans: D1 c ba c
r q cq br, D2 p ar cq
r
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 24
(b) According to Cramer’s rule, find the value of D and D1 for a1x + b1y = c1 and a2x + b2y = c2
Ans: D a1 b1 a1b2 a2b1, D1 c1 b1 c1b2 c2b1
a2 b2 c2 b2
11. (a) According to Cramer’s rule, find the value of D1 and D2 for 3x + 2y = 8 and 4x – y = 7. Ans: -22, -11
(b)
(c) According to Cramer’s rule, find the value of D and D1 for 2x – 3y = 5 and 3x + y = 2. Ans: 11, -11
12. (a) According to Cramer’s rule, find the value of D and D2 for 5x – 2y = 1 and 6x + 5y = 16 Ans: 37, 74
By using Cramer’s rule, find the value of ‘a’ in the system ax + 2y = 7 and 2x + 4y = 10 having the
value of D = 8. Ans: 3
(b) By using Cramer’s rule, find the value of ‘p’ in the system 2x – y = 17 and 3x + py = 6 having the
value of D1 = 91. Ans: 5
(c) By using Cramer’s rule, find the value of ‘a’ in the system 3x – y = 5 and x + 2y = k having the value
of D2 = 7. Ans: 4
1. Solve by matrix method: APPLICATION (A) BASED PROBLEMS
(a) 3x + 5y = 11, 2x – 3y = 1
[SEE MODEL-2076] Ans:x = 2, y = 1
(b) 3x + 2y = 10, 2x + 5y = 14 Ans:x = 2, y = 2
(c) 4x – 3y = 11, 3x + 7y = –1 Ans:x = 2, y = -1
(d) 9x – 8y = 12, 2x + 3y = 17 Ans:x = 4, y = 3
(e) 2x + 3y = 3, x + 2y = 1 Ans:x = 3, y = -1
(f) 5x - y = 7, 7x + 2y = 3 Ans:x = 1, y = -2
(g) 3x + 2y = -15, 2x – y = - 3 Ans:x = y = -3
2. Solve by matrix method: Ans: x = 2, y = -1
(a) 2x – 3y – 7 = 0, 4y – 3x + 10 = 0 Ans: x = 3, y = 4
(b) 2x + 3y – 18= 0, 3x – 2y – 1 = 0 Ans: x = 1, y = 0
(c) 3x = 5y + 3, 4x – 4 = – 3y Ans: x = 4, y = -3
(d) 5x = 9y + 47, 2y = 6 – 3x Ans: x = 1, y = 1
(e) 2x + 5 = 4(y+1) – 1, 3x + 4 = 5(y+1) – 3 Ans: x = 1, y = 0
(f) 4 (x – 1) + 5 (y + 2) = 10, 5 (x – 1) – 3 (y + 2) + 6 = 0
3. Solve by matrix method:
(a) x = 2 y, 4x – 3y = 1 Ans: x = - 2, y = -3
3
(b) 3 x + 2y =1 , x – y = 1 Ans:x = 2, y = -1
2 33
(c) x – y = 1, x – y + 1 = 0 Ans:x = 9, y = 4
32 2 2
(d) 5 x – y = 5, 4 x – y + 2 = 0 Ans:x = 6, y = 10
23
(e) 1 x – 2 y + 8 = 0, 3 x + 4 y = 15 Ans:x = 4, y = 15
23 45
4. Solve by matrix method:
(a) 3x 5y 5x - 2y 3 Ans:x = y = 3
83 Ans:x = y = 2
(b) 3x 5y 7x 3y 4
45
5. Solve by matrix method:
(a) 3x + 5 = 7, 7x - 10 = 12 Ans:x = 2, y = 5
yy
(b) 3x + 4 = 10, x + 1 = 3 Ans:x = 2, y = 1
yy
(c) 4 + 3y = 5, 6 - y = 2 Ans:x = 2, y = 1
xx
(d) 3 - 2y = 1, 2 - y = 1 Ans:x = 1, y = 1
xx
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 25
(e) 6 2 1, 3 2 2 Ans:x = 2, y = 1
xy yx
(f) 2 6 3, 10 9 2 Ans:x = 2, y = 3
xy x y
(g) 4 1 3, 2y – 5x + 4xy = 0 Ans:x = 2, y = 1
xy
6. (a) If A = 1 52 and B = 4 5 verify that (AB) – 1 = B – 1 A – 1
3 3 4
(b) If P = 0 - 31 and Q = 2 1 verify that (PQ) – 1 = Q – 1 P – 1
2 - 4 5
7. Solve the following system of equations by Cramer’s rule.
(a) 9x – 8y = 12, 2x + 3y = 17 Ans:x = 4, y = 3
(b) 2x + 3y = 3, x + 2y = 1 Ans:x = 3, y = -1
(c) 5x - y = 7, 7x + 2y = 3 Ans:x = 1, y = -2
(d) 3x + 2y = -15, 2x – y = - 3 Ans:x = y = -3
8. Solve the following system of equations by Cramer’s rule.
(a) 2x + 3y – 18= 0, 3x – 2y – 1 = 0 Ans: x = 3, y = 4
(b) 3x = 5y + 3, 4x – 4 = – 3y Ans: x = 1, y = 0
(c) 5x = 9y + 47, 2y = 6 – 3x Ans: x = 4, y = -3
(d) 2x + 5 = 4(y+1) – 1, 3x + 4 = 5(y+1) – 3 Ans: x = 1, y = 1
9. Solve the following system of equations by Cramer’s rule.
(a) x = 2 y, 4x – 3y = 1 Ans: x = - 2, y = -3
3
(b) 3 x + 2y =1 , x – y = 1 Ans:x = 2, y = -1
2 33
(c) x – y = 1, x – y + 1 = 0 Ans:x = 9, y = 4
32 2 2
10. Solve the following system of equations by Cramer’s rule.
(a) 3 - 2y = 1, 2 - y = 1 Ans:x = 1, y = 1
xx
(b) 6 2 1, 3 2 2 Ans:x = 2, y = 1
xy yx
UNIT-4 COORDINATE GEOMETRY
SEE SPECIFICATION GRID K U A HA Total Marks
21=2 22= 4 14=1 15=5 2+4 + 4 + 5= 15
Contents
1. Angle between two lines
2. Pair of straight lines
3. Conic section
4. Circle
A. ANGLE BETWEEN TWO STRIAGHT LINES
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Write the formula to find the angle between the t lines y = m1x + c1 and y = m2x + c2.
(b)
(c) Write the formula to find the angle between the lines y = m1x and y = m2x.
2. (a) If the slopes of two straight lines are m1 and m2 respectively and be the angle between them, write
the formula for tan. [SEE MODEL-2076]
If two straight lines y = m1x + c1 and y = m2x + c2are parallel to each other, write the relation between
m1 and m2.
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 26
(b) If two straight lines y = m1x + c1 and y = m2x + c2are perpendicular (orthogonal) to each other, write
the relation between m1 and m2.
3. (a) In the straight lines l1: y = m1x + c1 and l2: y = m2x + c2, if m1 = m2, write the relation between the lines
l1 and l2.
(b) In the straight lines l1: y = m1x + c1 and l2: y = m2x + c2, if m1. m2= -1, what kind of lines l1 and l2?
4. (a) Find the slope of the straight line passing through the points (x1, y1) and (x2, y2).
Find the slope of the straight line passing through the points (2, - 1) and (-6, 3).
(b) Find the slope of the line having equation ax + by + c = 0.
5. (a) Find the slope of the line having equation 2x + 3y = 5.
Find the slope of the line having equation y = 5.
(b) Find the slope of the line having equation x = 1.
(c) Find the slope of the line parallel to the line having equation 4x – y = 7.
(d) Find the slope of the line parallel to the line having equation x + y = 2.
6. (a) Find the slope of the line perpendicular to the line having slope 5.
(b) Find the slope of the line perpendicular to the line having equation y = 2x + 1.
7. (a) If the slope of a side AB of a rectangle ABCD is 1/2 what is the slope of its opposite side CD?
(b) If the slope of a side of a square is 3/4, what is the slope of its opposite side?
8. (a) If the slope of a diagonal AC of a square ABCD is 3 what is the slope of its diagonal BD?
(b)
9. (a) If the slope of a diagonal of a rhombus is 3 , what is the slope of its another diagonal?
(b)
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Write the condition for the straight lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 to be perpendicular
(b) and parallel to each other. Ans: a1a2 + b1b2 = 0, a1b2 = b1a2
(c) Write down the conditions of being perpendicularity and parallelism of the lines
p1x + q1y+r1 = 0& p1x + q1y+r1= 0. Ans: p1q2 – p2q1 = 0, p1p2 + q1q2 = 0.
2. (a)
(b) Write down the conditions of being perpendicularity and parallelism of the lines
(c) l1x + m1y+n1 = 0& l1x + m1y+n1= 0. Ans: l1m2 – l2m1 = 0, l1l2 + m1m2 = 0.
(d) Show that the lines 2x + 3y + 4 = 0 and 3x – 2y + 1 = 0 are perpendicular to each other.
(e) Show that the lines 6x - 8y = 7 and 4x + 3y + 8 = 0 are perpendicular to each other.
3. (a)
Show that the lines 3 x - 4y = 7 and 4 3 x + 3y + 5 = 0 are perpendicular to each other.
(b)
Show that the line 3x - 2y = 5 and the line joining the points (2, 6) and (8, 2) are perpendicular to each
4. (a)
other.
(b)
Prove that the lines y (2 3)x=7 and y (2 3)x=4 are orthogonal to each other.
(c)
If two straight lines px + qy + r = 0 and lx + my + n = 0 are perpendicular to each other, show that: pl+
(d)
qm = 0.
5. (a)
(b) If two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are orthogonal to each other, show that: a1a2 +
(c)
(d) b1b2 = 0.
Find the slopes of two straight lines 2x + 3y + 4 = 0 and 3x – 2y + 1 = 0 and write the relation between
6. (a)
(b) them. Ans: perpendicular
(c)
Find the slopes of two straight lines 7x – 5y + 13 = 0 and 5x + 7y + 31 = 0 and write the relation
between them. Ans: perpendicular
Find the slopes of two straight line 3x - y + 11 = 0 and the line passing through the points (1, 1) and (-
8, 4) and write the relation between them. Ans: perpendicular
Find the slopes of two straight line x + 3y + 6 = 0 and the line passing through the points (-2, -3) and (-
6, 9) and write the relation between them. Ans: perpendicular
Find the value of k if the straight lines 3x – 4y + 1 = 0 and 8x +ky = 7 are perpendicular. Ans:6
For what value of p, the lines px + 7y = 1 and 7x – 5y = 6 are perpendicular to each other? Ans:5
For what value of p, line px – 2y + 1 = 0 is perpendicular to the line joining (1, 3) and (p, 2)? Ans:2
If the line joining (1, 5) & (- 4, 2) is perpendicular to the line px + 3y = 13, find the value of p.Ans:5
Find the slope of the line perpendicular to the line 3x – 8y = 10. Ans:8/3
Find the slope of the line perpendicular to the line 5x + 3y + 1 = 0. Ans:-3/5
If the equation of its side BC of a square ABCD is x + 3y + 7= 0, find the slope of a side AB. Ans:-3
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 27
(d) If the equation of its side PQ of a rectangle PQRS is x - 5y + 1= 0, find the slope of a side QR.Ans:5
7. (a) Prove that the straight lines with equations 2x + 5y – 1 = 0 and 2x + 5y + 2 = 0 are parallel to each
(b) other.
Show that the line 2x + y – 7= 0 and the line passing through the points (1, 5) and (3, 1) are parallel to
(c)
each other.
(d)
If two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are perpendicular to each other, prove that: a1a2 +
8. (a)
b1b2=0
(b)
If two lines p1x + q1y + r1 = 0 and p2x + q2y + r2 = 0 are perpendicular to each other, prove that: p1p2 +
(c)
q1q2=0
(d)
Find the slopes of two straight lines 3x + 4y + 5 = 0 and 6x + 8y + 7 = 0 and write the relation between
9. (a)
them. [SEE MODEL-2076] Ans: parallel
(b)
Find the slopes of two straight lines 5x + 3y + 1 = 0 and 25x + 15y + 10 = 0 and write the relation
(c)
between them. Ans: parallel
(d)
Find the slopes of two straight line x + 3y + 8 = 0 and the line passing through the points (2, 1) and (-
10. (a)
7, 4) and write the relation between them. Ans: parallel
(b)
Find the slopes of two straight line 5x + y + 2 = 0 and the line passing through the points (-1, 5) and
(c)
(0, 0) and write the relation between them. Ans: parallel
11. (a)
(b) Find the value of k if the pair of straight lines 2x + ky = 3 and x + 3y – 2 = 0 are parallel to each other.
(c)
(d) Ans: 6
Find the value of p if the pair of straight lines px - 6y = 5 and 4x + 3y – 2 = 0 are parallel to each other.
12. (a)
(b) Ans: -8
(c) If a line passing through the points (4, – p) and (– 2, 6) is parallel to the line 2y + 3x = 4, find the value
13. (a) of p. Ans: 3
(b)
If a line passing through the points (a, 3) and (2, 9) is parallel to the straight line having equation 4y +
(c)
14. (a) ax = 4, find the value of a. Ans: 6, -4
(b) Write down the formula of the angle between the straight lines y = m1x + c1 and y = m2x + c2. Also,
(c)
(d) write the conditions of the lines of perpendicularity and parallelism.
(e)
Write down the formula of the angle between the straight lines y = ax + b and
y = px + q. Also, write the conditions of the lines of perpendicularity and parallelism.
Write down the formula of the angle between the straight lines y = mx + h and y = nx + k. Also, write
the conditions of the lines of perpendicularity and parallelism.
Find the acute angle between the lines with slopes 3 and 3 . Ans:600
Find the acute angle between the lines 4x – y + 7 = 0 and 3x – 5y = 1. Ans: 450
Find the acute angle between the lines y = 2x + 3 and x = 3y – 5 Ans: 450
Find the acute angle between the lines 7x – 4y + 2= 0 and 3x – 11y = 5 Ans: 450
Find the obtuse angle between the lines y =2 and x – 3 y + 6 = 0 Ans: 1500
Find the obyuse angle between the lines x + 2y = 5 and 2x – y = 1 Ans: 1500
Find the obtuse angle between the lines y– (2 – 3 ) x = 1 and y– (2+ 3 ) x +3 = 0.Ans: 1200
Find the angles between the lines 2y = 3 and x – 3 y + 6 = 0 Ans: 300, 1500
Find the acute angle between the lines having equations 2x – y + 1 = 0 and x – 3y + 11= 0.
Ans: 450, 1350
Find BAC if A (1, 2), B (2, 1) and C (4, 5) are any three points. Ans: 450, 1350
Find the value of m if the angle between the lines 3x – my =19 and 3x + 5y = 7 is 450 Ans:12 or – 3 / 4
For what value of a, the angle between the lines ax – y = 2 and x + 2y = 3 is 450? Ans:-3 or – 1 / 3
For what value of p, the angle between the lines 8x – y = 7 and x = py is 900? Ans:8
For what value of m, the angle between the lines mx = 2 and x + 2y = 3 is 900? Ans:-2
If the angle between two straight lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is 450, establish the
relation a1: b1 = (a1 – b1): (a2 + b2).
APPLICATION (A) BASED PROBLEMS
1. (a) Find the equation of a straight line passing through the point (4, 1) and parallel to the line 2x + 5y = 3.
Ans: 2x + 5y - 13 = 0
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 28
(b) Find the equation of a line passing through the point (2, 3) and parallel to the line 7x – 3y + 1 = 0.
(c) Ans: 7x – 3y + 5 = 0
2. (a)
(b) Find the equation of a line passing through the point (5, 1) and parallel to the line 4x + 5y + 9 = 0.
(c)
3. (a) Ans: 4x + 5y = 26
(b)
(c) Find the equation of a straight line which is parallel to the line 2x + y – 4 = 0 and making an intercept
4. (a)
(b) of length 2 units along y –axis. Ans: 2x + y = 2
(c)
5. (a) Find the equation of a straight line which is parallel to the line 5x + 3y = 9 and making an intercept – 5
(b)
(c) on the x-axis. Ans: 5x + 3y = 25
6. (a)
(b) Find the equation of a straight line which is parallel to the line 11x – 2y = 9 and making an intercept 3
(c)
7. (a) on they-axis. Ans: 11x – 2y + 6 = 0
(b)
(c) A (0, 3), B (1, -1) and C (5, -5) are the vertices of ABC; find the equation of line passing through the
8. (a)
(b) centroid of ABC and parallel to the side BC. Ans: x + y = 3
(c)
9. (a) Find the equation of a line passing through the centroid of PQR with vertices P (3, 3), Q (-2, -6) and
(b)
(c) R (5, -3) amd parallel to side QR. Ans: 3x – 7y = 20
Find the equation of a line passing through the centroid of BOY with vertices B (-4, 1), O (0, 0) and
Y (7, 5) amd parallel to side BY. Ans: 4x – 11y = -3
Find the equation of a line parallel to the line 7x – 2y + 1 = 0 and passing through the mid-point of the
line joining the points (1, 4) and (3, 0). Ans: 7x – 2y = 10
Find the equation of a line parallel to the line x – 8y + 7 = 0 and passing through the mid-point of the
line joining the points (2, 1) and (-4, 1). Ans: x – 8y + 10 = 0
Find the equation of a line parallel to the line 5x + y + 6 = 0 and passing through the mid-point of the
line joining the points (0, 0) and (-4, 6). Ans: 5x + y + 7 = 0
Find the eqwustioj of a line passinfg through (3, 2) and perpendicular to the line 4x – 3y – 10 = 0.
Ans: 3x + 4y = 17
Find the eqwustioj of a line passinfg through (4, 6) and perpendicular to the line x – 2y = 2.
Ans: 2x + y = 14
Find the eqwustioj of a line passinfg through (7, 10) and perpendicular to the line 5x + 7y + 12 = 0.
Ans: 7x – 5y = 1
Find the equation of a straight line which is perpendicular to the line x – 3y + 4 = 0 and making an
intercept of length 5 units along x –axis. Ans: 3x + y = 15
Find the equation of a straight line which is perpendicular to the line 8x + 3y = 11 and making an
intercept 4 on the x-axis. Ans: 3x – 8y = 12
Find the equation of a straight line which is perpendicular to the line 4x – 3y + 7 = 0 and making an
intercept - 2 on they-axis. Ans: 3x + 4y + 8 = 0
Find the equation of line passing through the centroid of triangle ABCwith vertices A (0, 3), B (5, 1)
and C (1, 2) and perpendicular to the side BC. Ans: 4x + y = 6
Find the equation of line passing through the centroid of triangle XYZ with vertices X (-5, -1), Y (-1,
2) and Z (0, 5) and perpendicular to the side XZ. Ans: 5x + 6y = 2
Find the equation of line passing through the centroid of triangle LMN with vertices L (-2, -3), M (-4,
5) and N (3, -8) and perpendicular to the side LN. Ans:x – y + 1= 0
Find the equation of a straight line perpendicular to the line 2x + y =1 and passing through the mid-
point of the line joining the points (0, 7) and (6, 1). Ans: x-2y+5 = 0
Find the equation of a straight line perpendicular to the line x + 3y = 4 and passing through the mid-
point of the line joining the points (3, 1) and (1, 3). Ans: 3x – y = 4
Find the equation of a straight line perpendicular to the line 9x +7y = 2 and passing through the mid-
point of the line joining the points (-1, 4) and (1, -4). Ans:7x- 9y = 0
Find the equation of the altitude of triangle PQR with vertices P (2, 3), Q (-4, 1) and R (2, 0) drawn
from the vertex P. Ans: 6x – y = 9
The points M (3, 4), N (-1, 1) and P (5, -1) are the vertices of a MNP. Find the equation of altitude
of the MNP drawn from the point N (-1, 1) Ans: 5y -2x +7 = 0
Find the equation of altitude of ABC with vertices A (2, 3), B (-4, 1) and C (2, 0) drawn from the
vertex A (2, 3) Ans: 6x – y = 9
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 29
10. (a) M (4, 7) and N (5, -2) are two given points. Find the equation of the line PQ which is perpendicular to
MN and passes through the point (2, 3) Ans:x – 9y + 25 = 0
(b) A (-3, 2) and B (5, -3) are two given points. Find the equation of the line XY which is perpendicular to
AB and passes through the point (-6, 2) Ans: 8x -5y + 58 = 0
(c) Find the equation of a straight line passing through origin asnd perpendicular to the line joining the the
points (-5, -1) and (-7, -4). Ans: 2x + 3y = 0
11. (a) Find the equation of the perpendicular bisector of line segment joining the points
(i) (3, 5) and (9, 3) Ans: 3x – y = 14 (ii) (2, 4) and (8, 10) Ans: x + y = 12
(iii) (3, 5) and (-7, 3) Ans: 5x + y + 6 = 0 (iv) (2, 5) and (8, -1) Ans: x – y = 3
12. (a) A (3, 5) and C (7, 9) are the opposite vertices of a rhombus ABCD, find the equation of the diagonal
BD. Ans: x + y = 12
(b) P (3, -7) and R (-5, 3) are the opposite vertices of a rhombus PQRS, find the equation of the diagonal
QS. Ans: 4x – 5y = 6
(c) If P (5, 7) and R (7, -2) are the opposite vertices of a square PQRS, find the equation of the diagonal
QS. Ans:4x – 18y + 21 = 0
(d) If S (4, -3) and T (8, 5) are the opposite vertices of a square SITA, find the equation of the diagonal
IA. Ans:x + 2y = 8
HIGH ABILITY (HA) BASED PROBLEMS
1. (a) Find the equation of the straight line passing through the point of intersection of the lines 3x + y = 7
(b)
(c) and 3y = 4x – 5 and parallel to the line 2x – y = 3. Ans: 2x – y = 3
2. (a) Find the equation of the straight line passing through the point of intersection of the lines 2x + y = 3
(b)
and 7x = 5y + 2 and parallel to the line 4x + 5y = 9 Ans: 4x + 5y = -9
3. (a)
(b) Find the equation of the straight line passing through the point of intersection of the lines x – 2y = 1
(c)
(d) and 2x – 3y = 3 and parallel to the line 3x -7y = 1 Ans: 3x -7y = 2
4. (a) Find the equation of the line passing through the point of intersection of the lines 3x + 4y = 7 and 5x –
(b)
(c) 2y = 3 and perpendicular to the line 2x + 3y = 5. Ans: 3x – 2y = -1
5. (a) Find the equation of the straight line passing through the point of intersection of the lines x + y = 3 and
(b)
x – y = 1 and perpendicular to the line 2x – y = 1. Ans: x + 2y = 3
6. (a)
Find the equation the lines passing through the point (2, 3) and making an angle of 450 with the line x
– 3y = 5. Ans: 2x – y = 1, x + 2y = 8
Find the equation the lines passing through the point (1, -4) and making an angle of 450 with the line
2x + 3y = 4. Ans: 5x + y = 1, x – 5y = 21
Find the equation the lines passing through the point (1, 0) and inclined at an angle of 300 with the line
x – 3 y = 4. Ans: y = 0, 3 x – y = 3
Find the equation the lines passing through the point (2, -3) and inclined at an angle of 600 with the
line 3 x + y = 4. Ans: y + 2 = 0, 3 x – y = 3 3 + 2
If B (1, 5) and D (7, 0) are the ends of diagonal BD of a square ABCD, find the equations of sides BC
and CD. Ans: x – 11y + 54 = 0, 11x + y = 16
ABCD is a square having a vertex A (1, 2). If the equation of the diagonal AC is x – 2y = 3, find the
equations of sides AB and AD. Ans: 3x + y = 5, x – 3y + 5 = 0
A square PQRS has a vertex P (1, -4) and the equation of diagonal PR is 2x + 3y = 5, find the
equations of the sides through P. Ans: 5x = y =1, x – 5y = 21
Find the equation of the sides of an equilatereal triangle whose vertex is (-1, 2) and base is y = 0.
Ans: 3 x – y + ( 3 + 2) = 0, 3 x + y + ( 3 - 2) = 0
Find the equation of the remaing sides of a right angled isosceles triangle whose vertex is (-2, -3) and
base is x = 0. Ans: x – y = 1, x + y + 5 = 0
If the line x + y 1 passes through the point of intersection of the lines x + y = 3 and 2x – 3y = 1 and
ab
is parallel to the line y = x – 6, find the values of aand b. Ans: a = 1, b = -1
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 30
(b) If the line x + y 1 passes through the point of intersection of the lines x - y = 0 and 4x – 3y = 1 and
ab
(c)
7. (a) is parallel to the line 2x – 2y = 5, find the values of aand b. Ans: a = 2, b = 2
(b) The line y = mx + c passes through the point of intersection of the lines x+ y = 6 and 6x – y = 1 and
(c)
8. (a) is parallel to the line 2x – y + 1 = 0, then find the values of m and c. Ans: m = 2, c = 3
(b)
(c) Find the equation of line which is parallel to the line 4x + 5y = 6 and make s the interepts on the axes
9. (a)
(b) whose sum is 9. Ans: 4x+5y = 20.
Find the equation of line which is parallel to the line 2x – 3y = 7 and makes the interepts on the axes
whose sum is 5. Ans: 2x – 3y = 30
Find the equation of line which is parallel to the line x – 8y = 1 and makes the interepts on the axes
whose sum is 14. Ans: x – 8y = 16
Find the equation of line which is perpendicular to the line 3x - 5y = 1 and makes the interepts on the
axes whose sum is 8. Ans: 5x + 3y =15
Find the equation of line which is perpendicular to the line 2x – 7y = 4 and makes the interepts on the
axes whose sum is 18. Ans: 7x + 2y = 28
Find the equation of line which is perpendicular to the line x – 4y = 3 and makes the interepts on the
axes whose sum is 15. Ans: 4x + y = 12
Find the angle between the straight lines y = m1x + c1 and y = m2x + c2. Also, derive the conditions of
the lines of perpendicularity and parallelism.
Show that the equation of line passing through the point (acos3, asin3) and perpendicular to the line
xsec + ycosec = a is xcos - ysin = acos2.
B. PAIR OF STRAIGHT LINES
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Write the formula to find the angle between the pair of lines represented by the equation ax2 + 2hxy +
by2 = 0, a 0, b 0.
(b) If be the angles the pair of lines represented by ax2 + 2hxy + by2 = 0; a, b 0, what is the value of
tan?
2. (a) If the pair of lines represented by ax2 + 2hxy + by2 = 0, a 0, b 0are coincident (parallel) to each
other, write the relation between a, b and h..
(b) Write the condition under which the pair of lines represented by ax2 + 2hxy + by2 = 0, are
perpendicular to each other.
3. (a) In the homogenous equation ax2 + 2hxy + by2 = 0, a 0, b 0 representing a pair of lines, if a + b = 0,
what type of straight lines are they?
(b) In the equation ax2 + 2hxy + by2 = 0, if h2 – ab = 0, mention the relation of lines.
4. (a) Write the single equation to represent x-axis and y-axis.
Write the single equation to represent the lines y = 0 and x - y = 0.
(b) Find the separate equations of the lines represented by (x – y) (x + y) = 0.
5. (a) Find the separate equations of the lines represented by the equation x (x + 2) = 0.
What is the angle between the lines x = 0 and y = 0?
(b) What is the angle between the lines x = 0 and x = y?
6. (a)
(b)
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Find the single equation for the pair of lines represented by 3x + 2y = 0 and 2x – 3y = 0.
(b) [SEE MODEL-2076] Ans: 6x2 – 5xy - 6y2 = 0
(c) Find the single equation for the pair of straight lines represented by x + y = 0 and x – 2y = 0.
(d)
2. (a) Ans: x2 – xy – 2y2 = 0
(b)
(c) Find the single equation which represents the pair of lines x = 2y and x-axis. Ans: xy – 2y2 = 0
Find the single equation which represents the pair of lines x + y = 0 and y-axis. Ans: x2 + xy = 0
Find the separate equation of a pair of lines represented by x2 + 7xy + 6y2 = 0 Ans: x + 6y = 0, x + y = 0
Find the separate equation of lines represented by 2x2 + xy – y2 = 0 Ans: x + 2y = 0, x – y = 0
Find the separate equation of lines represented by x2 + 2x + 2y – y2 = 0 Ans: x + y = 0, x – y + 2 = 0
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 31
(d) Find the separate equation of lines represented by x2 – y2 – 4x + 4 = 0 Ans: x + y - 2= 0, x – y – 2 = 0
3. (a) Find the separate equation of lines represented by x2 – y2 – 6y – 9 = 0 Ans: x + y + 3= 0, x – y – 3 = 0
Find the separate equation of lines represented by x (x – 1) – y (y – 1) = 0 Ans: x - y = 0, x + y – 1 = 0
(b) Find the separate equation of lines represented by ab (x2 – y2) – (a2 – b2) xy=0 Ans: px + qy = 0, qx – py = 0
(c)
4. (a) Find the separate equation of lines represented by x2 + 2xysec + y2 = 0 Ans: x + (sec tan) y = 0
(b) Find the separate equation of lines represented by x2 + 2xycosec + y2 = 0 Ans: x + (cosec cot) y = 0
(c) Find the separate equation of lines represented by x2 + 2xytan - y2 = 0 Ans: x (sec- tan) y = 0
(d)
5. (a) Find the separate equation of lines represented by x2 + 2xycot -y2 = 0 Ans: x (cosec- cot) y = 0
(b)
6. (a) Find the acute angle between the pair of lines represented by 3x2 + 7xy + 2y2 = 0 Ans: 450
(b)
7. (a) Find the acute angle between the pair of lines represented by x2 + 4xy + y2 = 0 Ans: 600
(b) Find the obtuse angle between the pair of lines represented by 12x2 – 23xy + 5y2 = 0 Ans:1350
8. (a) Find the obtuse angle between the pair of lines represented by 3 x2 + 4xy + 3 y2 = 0. Ans:1500
(b)
What is the angle between the lines represented by ax2 + 2hxy + by2 = 0? Also, write the conditions of
9. (a)
the lines of perpendicularity and coincident.
(b)
Find the angle between the lines represented by px2 + 2qxy + ry2 = 0. Also, write the conditions of the
(c)
lines of perpendicularity and coincident.
(d) Show that the pair of straight lines represented by 3x2 + 4xy – 3y2 = 0 are perpendicular to each other.
Prove that the pair of straight lines represented by x (x + 2) + y (3 – y) = 0 are perpendicular to each
10. (a)
(b) other.
(c) If the pair of straight lines represented by (k + 1) x2 – 3xy – 5y2 = 0 are perpendicular to each other,
11. (a) find the value of k? Ans: 4
(b)
(c) If the lines represented by the equation (4k + 5) x2 + 6xy – k2y2 = 0 are mutually per pendicular to
(d)
each other, find the value of k. Ans: 5, -1
12. (a)
For what value of m, the two lines represented by (3 – 2m) x2 – 5xy + my2 = 0 are perpendicular to
(b)
each other? Find it. Ans: 3
(c)
For what value of p, the two lines represented by (p + 1) x2 – 7xy – 2py2 = 0 are perpendicular to each
other? Find it. Ans: 1
Show that the pair of straight lines represented by 4x2 + 12xy + 9y2 = 0 are coincident to each other.
Show that the pair of straight lines represented by x2 – 14xy + 49y2 = 0 are coincident to each other.
Show that the pair of straight lines represented by 4x2 – 4xy + y2 = 0 are coincident to each other.
A pair of straight lines represented by px2 -12xy + 9y2 = 0 are coincident, find the value of p. Ans:4
A pair of straight lines represented by x2 - 4xy + (k-1) y2 = 0 are coincident, find the value of k.Ans:5
A pair of straight lines represented by 4x2- 3kxy + 9 y2 = 0 are coincident, find the value of k.Ans:4
A pair of lines represented by (m-6) x2 + 8xy + my2 = 0 are coincident; find the values of m. Ans: 8, -2
If the angle between the pair of lines represented by kx2 + 7xy + 2y2 = 0 is 450; find the positive value
of k. Ans: 3
If the angle between the pair of straight lines represented by x2 – 4xy + py2 = 0 is 600; find the positive
value of p. Ans: 1
If the angle between the pair of straight lines represented by 3 x2 + mxy + 3 y2 = 0 is 600; find the
positive value of m. Ans: 4
APPLICATION (A) BASED PROBLEMS
1. (a) Find the separate equations of lines represented by 6x2 –xy – y2 = 0. Also, find the angle between
(b)
(c) them. Ans:3x – y = 0, 2x + y = 0, 450, 1350
(d)
Find the separate equations of lines represented by 2x2 + 7xy + 3y2 = 0. Also, find the angle between
2. (a)
them. Ans:2x + y = 0, x +3y = 0, 450, 1350
(b)
Find the separate equations of lines represented by ab (x2 – y2) + (a2 – b2) xy = 0. Also, find the angle
between them. Ans:ax – by = 0, bx + ay = 0, 900
Find the angles between the lines represented by (3b2 – a2) x2 – 8abxy + (3a2 – b2) y2 = 0. Ans: 600, 1200
Find the separate equations of lines represented by x2 + 2xysec + y2 = 0. Also, find the angle between
them. Ans:x + (sec tan) y = 0,
Find the separate equations of lines represented by x2 – 2xycosec + y2 = 0. Also, find the angle
between them. Ans:x – (cosec cot) y = 0, (900 )
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 32
(c) Find the separate equations of lines represented by x2 – 2xycot2 –y2 = 0. Also, find the angle between
them. Ans:x – (cot2 cosec2) y = 0, 900
3. (a) If be the acute angle between the pair of lines represented by the equation x2 + 2xysec + y2 = 0,
prove that: =
(b) If the acute angle between the pair of lines x2 - 2xycosec + y2 = 0 is , prove that: = 900 - .
(c) If the acute angle between the line pair x2 + 2xysec + y2 = 0 is , prove that: = 900 + .
4. (a) Find the single equation of the pair of straight lines passing through the origin and perpendicular to the
lines represented by 2x2 - 5xy + 2y2 = 0. [SEE MODEL-2076] Ans: 2x2+ 5xy + 2y2 = 0
(b) Find the single equation of the pair of straight lines passing through the origin and perpendicular to the
lines represented by x2 + xy – 2y2 = 0. Ans: 2x2 + xy – y2 = 0
(c) Find the single equation of the pair of straight lines passing through the origin and perpendicular to the
lines represented by x2 + 2xy –143y2 = 0. Ans: 143x2 + 2xy – y2 = 0
(e) Find the single equation of the pair of straight lines passing through the origin and perpendicular to
lines represented by ax2 + 2hxy + by2 = 0. Ans: bx2 – 2hxy + ay2 = 0
5. (a) Find the single equation of the pair of straight lines passing through the point (1, 2) and parallel to the
lines represented by x2 + 3xy + 2y2 = 0. Ans: x2 + 3xy + 2y2 – 8x – 11y + 15 = 0
(b) Find the equations of the pair of straight lines passing through the point (2, 1) and parallel to the lines
represented by 20x2 –11xy –3y2 = 0. Ans: 4x – 3y – 5 = 0, 5x + y – 11 = 0
(c) Find the equations of the pair of straight lines passing through the point (4, 3) and parallel to the lines
represented by x2 + 2xy –15y2 = 0. Ans: x + 5y – 19 = 0, x – 3y +5 = 0
6. (a) Find the single equation of pair of straight lines passing through the point (2, 1) and perpendicular to
the pair of straight lines represented by x2 + 3xy – 4y2 = 0 Ans: 4x2 + 3xy – y2 – 19x – 4y + 21 = 0
(b) Find the single equation of pair of straight lines passing through the point (1, 2) and perpendicular to
the pair of straight lines represented by x2 – xy – 2y2 = 0 Ans: 2x2 – xy – y2 – 2x + 5y – 4 = 0
(c) Find the equations of the pair of straight lines passing through the point (4, 3) and perpendicular to the
lines represented by x2 + 2xy –15y2 = 0. Ans:5x – y = 17, 3x + y – 15 = 0
7. (a) If two straight lines represented by an equation 3x2 + 8xy + my2 = 0 are perpendicular to each other,
find the separate equation of two lines. Ans: x + 3 y = 0, 3x – y = 0
(b) If an angle between the lines represented by 2x2 + kxy + 3y2 = 0 is 450, find the positive value of k and
then separate equation of lines. Ans:7, 2x + y = 0, x + 3y = 0
HIGH ABILITY (HA) BASED PROBLEMS
1. (a) Find the separate equations of pair of lines represented by x2 + 2xy + y2 – 2x – 2y – 15 = 0
Ans:x + y + 3 = 0, x + y = 5
(b) Find the equation of a pair of lines represented by 2x2 – 5xy – 3y2 + 3x + 19y – 20 = 0.
Ans: x – 3y + 4 = 0, 2x + y = 5
2. (a) Find the angle between the straight lines represented by ax2 + 2hxy + by2 = 0; b 0. Also, deduce the
conditions of lines being perpendicular and coincident.
(b) Prove that the pair of lines represented by the equation ax2 + 2hxy + by2 = 0; b 0 pass through the
3. (a) origin.
Prove that the angle between the pair of straight lines represented by the homogeneous equation (p2 –
3q2) x2 + 8pqxy + (q2 – 3p2) y2 = 0 are 600 and 1200.
(b) Prove that the angle between the pair of straight lines represented by the homogenous equation (x2 +
y2) sin2 = (x cos - y sin) 2 are 2 and (1800 - 2).
C. CONIC SECTIONS
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define conic section. (b) Define circle. (c) Define ellipse.
(d) Define parabola. (e) Define hyperbola
2. (a) Which geometrical figure will form if a plane intersects a cone parallel to its base? [SEE MODEL-2076]
(b) Which geometrical figure will form if a plane intersects a cone perpendicular to its axis?
(c) Which geometrical figure will form if a plane intersects a cone parallel to its generator?
(d) If the plane cuts the cone being parallel to its axis, what conic section will form?
(e) Which geometrical figure will form if a plane intersects a cone parallel to its generator?
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 33
(f) Name the conic section so formed when an intersecting plane is neither parallel nor perpendicular to
the base.
(g) Name the conic section so formed when an intersecting plane is neither parallel nor perpendicular to
its axis.
3. (a) What type of conic section is formed if the angle between intersecting plane and axis of cone is a right
angle (900)?
(b) What type of conic section is formed if the angle between intersecting plane and axis of cone is 00?
(c) What type of conic section is formed if the angle between intersecting plane and one of the generators
of cone is 00?
(d) What type of conic section is formed if the angle between intersecting plane and axis is more than
semi-vertical angle?
4. (a) In which condition, the conic section forms the circle?
(b) In what condition, the ellipse is formed as conic?
(c) In what condition, the ellipse becomes a circle?
(d) Write the condition in which conic section forms the parabola?
(e) Write the condition in which conic section forms the hyperbola?
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Define circle with figure. (b) Define parabola with figure.
(c) Define ellipse with figure. (d) Define hyperbola with figure.
2. (a) Differentiate between axis and generator of cone.
(b) Write any two differences between cone and ellipse.
(c) Write any two differences between parabola and hyperbola.
(d) Write any two differences between circle and ellipse.
(e) Write any two differences between cicle and hyperbola.
D. CIRCLE
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Name the locus represented by the equation x2 + y2 = r2
(b) What is the name the conic section represented by (x – h) 2 + (y – k) 2 = r2?
(c) Write the name of the locus represented by(x – x1) (x – x2) + (y – y1) (y – y2) = 0.
Define concentric circle.
2. (a) What do you mean by equal circles?
(b) Name the point of intersection of the diameters of a circle.
(c) What is the equation of the circle with centre (0, 0) and radius r units?
What is the equation of the circle with centre at origin and radius p units?
3. (a) What is the equation of the circle having centre (a, b) and radius c units?
(b) Write the equation of the circle having ends of a diameter are (x1, x2) and (y1, y2).
(c) In which condition, the equation of circle is given by (x – h) 2 + (y – k) 2 = r2?
(d) In what case, the equation of circle is (x – x1) (x – x2) + (y – y1) (y – y2) = 0?
What is the centre and radius of a circle with equation (x – p) 2 + (y – q) 2 = k 2?
4. (a) What is the centre and radius of a circle x 2 + y 2 + 2gx + 2fy + c = 0?
(b) What is centre of circle concentric with the circle x 2 + y 2 + 2ax + 2by + c = 0?
What is the centre of the circle whose ends of a diameter are (x1, x2) and (y1, y2)?
5. (a) What will be the length of radius of a circle having centre (h, k) and touches the x-axis?
(b) What will be the length of radius of a circle having centre (h, k) and y-axis is tangent to the circle?
(c) What will be the length of radius of a circle having centre (h, k) and touches both the axes?
(d) Under what condition will radius r = hin a circle centered at (h, k)?
Write the condition under which radius r = kin a circle with centre (h, k)?
6. (a) State the condition under which radius r = h= kin a circle with centre (h, k)?
(b) Two circles with cenrtres A and B have radii R1 and R2. If they intersect externally at P, write the
(c) relation among AB, R1 and R2.
Two circles with cenrtres A and B have radii R1 and R2. If they intersect internally at P, write the
7. (a) relation among AB, R1 and R2.
(b)
(c)
8. (a)
(b)
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 34
(c) Two circles with cenrtres A and B have radii R1 and R2. If they intersect to each other at P and Q, write
the relation among AB, R1 and R2.
(d) Two circles with cenrtres A and B have radii R1 and R2. If they are disjoint, write the relation among
AB, R1 and R2.
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Find the coordinates of centre and length of radius of circle (x – 2)2 + (y – 3)2 = 16.Ans: (2, 3); 4 units
(b) Find the coordinates of centre and length of radius of circle (x +1)2 + (y – 2)2 = 9. Ans: (-1, 2); 3 units
(c) Find the coordinates of centre and length of radius of circle x2 + (y + 1)2 = 18. Ans: (0, -1); 3 2 units
(d)
2. (a) If the equation of circle is (4x + 1)2 + (4y – 3)2 = 16, find its centre and radius. Ans: (-1/4, 3/4), 1 unit
(b)
(c) Find the centre and length of radius of circle x2 + y2 + 4x – 6y + 4 = 0. Ans: (- 2, 3); 3 units
3. (a)
Find the centre and length of radius of circle x2 + y2 – 2x – 4y + 1 = 0. Ans: (1, 2); 2 units
Find the centre and length of radius of circle x2 + y2 – 8x + 6y = 0. Ans: (4, -3); 5
Find the circumference of the circle x2 + y2 – 2y – 48 = 0. Ans: 44 units
(b) If the equation of circle is x2 + y2 + 10x - 171= 0, find its circumference. Ans: 88 units
4. (a)
If the centre of a circle x2 + y2 – ax + by + 1 = 0 is (2, 3), find the values of ‘a’ and ‘b’. Ans: a = 4, b = - 6
(b)
(c) If the centre of a circle is x2 + y2 + mx-ny = 7 is (- 4, 1), find the values of ‘m’ and ‘n’. Ans: m = 8, n = 2
(d) If the centre of a circle is x2 + y2 + mx – 4y = 7 is (- 4, n), find the values of ‘m’ and ‘n’.Ans: m = 8, n = 2
5. (a)
If the centre of a circle is x2 + y2 – 6x – py = 7 is (q, 2), find the values of ‘p’ and ‘q’. Ans: p = 4, q = 3
(b)
(c) If the radius of a circle with equation x2 + y2 – ay = 24 is 5 units, find the value of ‘a’. Ans: a = 2
6. (a)
If the radius of a circle x2 + y2 –4x – 6y – k = 0 is 4 units, find the value of ‘k’. Ans: k = 3
(b)
(c) The diameter of a circle x2 + y2 + kx – 2y + 1= 0 is 8 units; find the value of ‘k’. Ans: k = 8
(d) Find the equation of circle with centre (1, 0) and radius 3 units. Ans: x2 + y2 – 2x = 8
Find the equation of circle with centre (0, 3) and radius 4 units. Ans: x2 + y2 – 4y = 7
Find the equation of circle with centre (2, 3) and diameter 10 units. Ans: x2 + y2 – 4x – 6y = 12
Find the equation of circle with centre (1, 4) and diameter 6units. Ans: x2 + y2 – 2x – 8y + 8 = 0
7. (a) Find the equation of circle with centre (4, -3) and circumference 44 units. Ans: x2 + y2 -8x + 6y = 24
(b) Find the equation of circle with centre (9, 10) and circumference 88 units. Ans: x2 + y2-18x-20 y = 15
(c)
(d) Find the equation of circle with centre (4, 5) and area 154 sq. units. Ans: x2 + y2 – 8x– 10 y = 8
8. (a)
Find the equation of circle with centre (8, 9) and area 616 sq. units. Ans: x2 + y2 – 16x– 18y = 24
(b)
(c) Find the equation of a circle with centre (2, 3) and x-axis is tangent to it. Ans: x2 + y2 – 4x – 6y + 4 = 0
Find the equation of a circle with centre (–6, 5) and tangent to x-axis. Ans: x2 + y2 + 12x-10y + 36 = 0
Find the equation of a circle with centre (2, 1) and touching y-axis. Ans: x2 + y2 – 4x – 2y + 1 = 0
9. (a) Find the equation of a circle with radius 3 units, touching both the positive axes. Ans: x2 + y2- 6x-6y + 9 = 0
(b) Find the equation of a circle with radius 5 units, touching both the positive axes.Ans: x2 + y2-10x-10y + 25 = 0
(c) Find the equation of a circle with radius 4 units, touching both the negative axes. Ans: x2 + y2 + 8x + 8y + 16 = 0
(d) Find the equation of a circle with radius 5 units, touching both the negative axes.Ans: x2 + y2 + 10x + 10y + 25 = 0
10. (a) Find the equation of a circle whose ends s of a diameter are (0, 6) and (8, 0). Ans: x2 + y2-8x-6y = 0
(b) Find the equation of a circle whose ends s of a diameter are (0, 3) and (4, 0). Ans: x2 + y2-4x-3y = 0
(c) Find the equation of a circle whose ends s of a diameter are (2, 3) and (–1, 4).Ans: x2 + y2-x-7y + 10 = 0
(d) Find the equation of a circle whose end points of a diameter are (3, 4) and (0, -2). Ans: x2 + y2-3x-2y-8 = 0
11. (a) Find the equation of a circle having centre (0, 0) and passes through the point (2, -3). Ans: x2 + y2 = 13
(b) Find the equation of a circle with centre (2, 3) and passes through the point (-2, 0). Ans: x2 + y2-4x -6y = 12
(c) Find the equation of a circle with centre (4, 0) and passes through the point (4, 5). Ans: x2 + y2 – 8x = 9
(d) Find the equation of a circle with centre (2, 3) and passes through the point (4, 5). Ans: x2 + y2-4x-6y = 5
12. (a) Find the centre of a circle having equations of two diameters x + y = 5 and 2x – y = 1 Ans: (2, 3)
Find the centre of a circle having equations of two diameters x – y = 3 and x + 2y = 15 Ans: (7, 4)
(b) Find the centre of the circle which is concentric with the circle x2 + y2 – 8x– 4 y = 7. Ans: (4, 2)
13. (a) Find the centre of the circle which is concentric with the circle x2 + y2 – 2x + 6 y = 1. Ans: (1, -3)
(b)
14. (a) Find the equation of circle with centre (3, 1) and that touches the line having equation 5x + 12y - 22= 0
Ans: x2 + y2 – 6x + 2y + 9= 0
(b) Find the equation of circle with centre (1, 2) and that touches the line having equation 3x – 4y + 10 = 0.
Ans: x2 + y2 – 2x – 4y + 4 = 0
APPLICATION (A) BASED PROBLEMS
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 35
1. (a) Find the centre and the radius of the circle 9x2 + 9y2 – 36x + 6y = 107. Ans: (2, -1/3), 4units
(b) Find the centre and the radius of the circle 4x2 + 4y2 – 24x – 20y = 3. Ans: (3, 5/2), 8units
(c)
Find the centre and the radius of the circle 2x2 + 2y2 – 8x – 12y + 1= 0. Ans: (2, 3), 5/ 2 units
2. (a) If (3, 4) is one end of a diameter of a circle x2 + y2 – 4x – 6y + 11= 0, find the other end. Ans: (1, 2)
(b) If (2, 2) is one end of a diameter of a circle x2 + y2 + 2x – 8y + 5 = 0, find the other end. Ans: (-4, 6)
(c) If the coordinates of one end of a diameter of the circle x2 + y2 + 2x + 4x – 6y + 8 = 0 is (0, 2) then find
(d)
the coordinates of other end of the diameter. Ans: (-4, 4)
3. (a)
(b) If the coordinates of one end of a diameter of the circle x2 + y2 + 2x + 4x – 6y + 8 = 0 is (-4, 4) then find
(c)
(d) the coordinates of other end of the diameter. Ans: (0, 2)
(e)
Find the equation of the circle which passes through the point (1, 4) and equations of two diameters are
4. (a)
(b) 2x + y = 5 and x – y = 1. Ans: x2 + y2 – 4x – 2y = 5
(c)
(d) Find the equation of the circle which passes through the point (0, 2) and equations of two diameters are
5. (a) 4x – 2y = 5 and 8x + 6y = -5. Ans: x2 + y2 – x + 3y = 10
(b)
(c) Find the equation of the circle which passes through the point (2, 4) and equations of two diameters are
(d)
x – y = 4 and 2x + 3y + 7 = 0 Ans: x2 + y2 –2y – 6y = 40
(e)
6. (a) Find the equation of the circle which passes through the point (4, 6) and equations of two diameters are
(b) 2x + y – 4 = 0 and 2y – x – 3 = 0. Ans: x2 + y2 –2x – 4y = 20
(c)
(d) Find equation of the circle which has radius 5 units and the centre as the point of intersection of the
7. (a)
straight lines 2x – y = 5 and x – 3y = -5. Ans:x2 + y2 – 8x – 6y = 0
(b)
Find the equation of the circle with centre (3, 4) and passing through the point of intersection of the
lines given by 3x – y = 14 and 2x – y = 7 Ans: x2 + y2 – 6x – 8y = 0
Find the equation of the circle with centre (1, 2) and passing through the point of intersection of the
lines given by x + 2y = 3 and 3x + y = 4 Ans: x2 + y2 – 2x – 4y = 13
Find the equation of the circle with centre (2, 0) and passing through the point of intersection of the
lines given by 2x + y = 5 and x + 2y = 4 Ans: x2 + y2 – 4x = 1
Find the equation of the circle with centre (0, 3) and passing through the point of intersection of the
lines given by 3x – y = 4 and 2x + y = 1 Ans: x2 + y2 – 6y = 17
Find the equation of the circle with centre (3, 2) and passing through the centre of circle x2 + y2– 2x +
4y + 5 = 0. Ans:x2 + y2– 6x – 4y + 5 = 0
Find the equation of a circle with centre (4, 3) and passes through the centre of the circle x2 + y2 – 6x +
2y – 7 = 0. Ans: x2 + y2 – 8x – 6y + 8 = 0
Find the equation of a circle with centre (-1, 2) and passes through the centre of the circle x2 + y2 – 6x –
10y – 2 = 0. Ans: x2 + y2 + 2x – 4y – 20 = 0
A circle A with centre X passes through the centre Y of the circle B. If the equation of the circle B is x2
+ y2 – 4x + 6y – 12 = 0 and the coordinates of X are (-4, 5), find the equation of circle A.
Ans: x2 + y2 + 8x – 10y = 59
Find the equation of the circle having centre (4, 6) and passing through the mid-point of the line joining
the points (-1, 3) and (3, 1). Ans:x2 + y2 – 8x – 12y + 27 = 0
Find the equation of circle concentric with the circle x2 + y2 – 6x + y = 1 and passing through the point
(4, -2). Ans:x2 + y2 – 6x + y + 6 = 0
Find the equation of circle passing through the point (4, 3) concentric with the circle x2 + y2 + 6x – 8y -
11 = 0. Ans: x2 + y2 + 6x – 8y – 25 = 0
A circle P passes through (2, 1) and concentric with the circle Q having equation x2 + y2 –x-2y + 1= 0,
find the equation of circle P. Ans: x2 + y2 – x – 2y – 1 = 0
Find the equation of circle concentric with the circle 2x2 + 2y2 + 4x – 2y + 1 = 0 and passing through the
point (4, -2) Ans: x2 + y2 + 2x – y = 30
The centres of two equal circles A and B are X and Y respectively. If the coordinates of X are (2, 3) and
the equation of circle B is x2 + y2 –2x + 6y + 1 = 0 then find the equation of circle A.
Ans: x2 + y2 – 4x – 6y + 4 = 0
Find the equation of circle having centre (8, 3) and congruent to the circle having equation x2 + y2 + 4x
– 6y – 3 = 0. Ans: x2 + y2 – 16x – 6y + 55 = 0
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 36
(c) Find the equation of circle having centre (– 4, 1) and has the same radius as the circle x2 + y2 – 2x + 2y
(d) = 7. Ans: x2 + y2 + 8x – 2y = 8
8. (a) A circle is concentric with the circle 25x2 + 25y2 – 200x – 250y + 989 = 0 and has radius half of the
(b) radius of given circle. Find the equation of the circle. Ans: 25x2 + 25y2 – 200x – 250y + 1016 = 0
(c) Find the equation of the image of the circle x2 + y2– 2x + 4y - 5 = 0 under reflection on x-axis.
9. (a)
(b) Ans:x2 + y2–2x – 4y – 2 = 0
10. (a) Find the equation of the image of the circle x2 + y2– 4x – 6y + 1 = 0 under reflection on y-axis.
(b)
(c) Ans:x2 + y2+ 4x – 6y + 2 = 0
Find the equation of the image of the circle x2 + y2– 4x – 2y + 1 = 0 under roation through 900 about
origin. Ans:x2 + y2 + 2x – 4y + 1 = 0
Find the equation of the circle having centre at (b, 4) and passing through the points (-2, 0) and (4, 0) of
x-axis. Ans:x2 + y2 – 2x – 8y = 8
Find the equation of the circle which touches the x-axis at (3, 0) and passing through the point (1, 2).
Ans:x2 + y2 – 6x – 4y + 9 = 0
Find the equation of the circle which passes through the points (2, 3) and (-1, 2) and its centre lies on
the straight line 2x – 3y + 1 = 0. Ans: x2 + y2 – 2x – 2y – 3 = 0
Find the equation of the circle which passes through the points (2, 3) and (5, 4) and its centre lies on the
straight line 2x + 3y – 7 = 0. Ans: x2 + y2 – 10x + 2y + 1 = 0
Find the equation of the circle which passes through the origin and the point (4, 2) and its centre lies on
the line x + y = 1. Ans: x2 + y2 – 8x + 6y = 0
HIGH ABILITY (HA) BASED PROBLEMS
11. (a) On a wheel there are three points (5, 7), (-1, 7) and (5, -1) located such that the distance from a fixed
point to these points is always equal. Find the coordinates of the fixed point and then derive the
equation of representing the locus that contains all three points. [SEE MODEL-2076]
Ans: (2, 3), x2 + y2 – 4x – 6y – 12 = 0
(b) A cow is tight to a stake by a rope. If the cow moves through the points (2, -2), (5, 7) and (6, 6) in a
circular path always keeping the rope tight then find the coordinates of the point at which the stake is
fixed and the equation repseenting the locus of cow. Ans: (2, 3), x2 + y2 – 4x – 6y – 12 = 0
12. (a) Find the equation of circle which passes through the points (2, 0), (0, 2) and (-2, 0). Ans:x2 + y2 = 4
(b) Find the equation of circle passing through the points (2, -2), (5, 7) and (6, 6). Ans: x2 + y2- 4x-6y-12 = 0
13. (a) Find the equation of the circle which passes through the origin and making intercepts of lengths 6 and
8 units on the positive x- axis and y- axis. Ans:x2 + y2 – 6x – 8y = 0
(b) A circle passing through the origin and makes intercepts of lengths 4 and 2 units on the positive x-
14. (a) axis and y- axis. Find the centre and the equation of the circle. Ans: (2, 1) units, x2 + y2 – 4x – 2y = 0
If the line x + y = 1 cuts the circle x2 + y2 = 1 at two points, find the distance between the points.
Ans: 2 units
(b) If the line x + y = 5 cuts the circle x2 + y2 = 25 at two points A and B, find the length of AB.
Ans: 5 2 units
15. (a) Find the equation of circle which touches x- axis at the point (4, 0) and cuts off an intercept of 6 units
from the positive y-axis. Ans: x2 + y2 – 8x – 10y + 16 = 0
(b) Find the equation of circle which lies in the fist quadrant, touches x- axis at the point (3, 0) and cuts
off an intercept of 8 units from y-axis. Ans: x2 + y2 – 6x –10y + 9 = 0
16. (a) Find the equation of circle whose diameter is the intercept between the coordiantre axes of the straight
line 3x +2y = 6. Ans: x2 + y2 – 2x – 3y = 0
(b) Find the equation of circle whose diameter is the intercept between the coordiantre axes of the straight
line 4x – 6 y = 12. Ans: x2 + y2 – 3x + 2y = 0
UNIT-5 TRIGONOMETRY
SEE SPECIFICATION GRID KUA HA Total Marks
Contents
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 37
1. Multiple & Sub-multiple of Angles 21=2 32=6 34=12 2+6+12 = 20
2. Transformation of trigonometric Formulae
3. Conditional Identities
4. TrigonometricEquation
5. Height & Distance
A. MULTIPLE ANGLES
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Express sin2A in terms of sinA and cos A. (b) Express sin2A in terms of tanA. [SEE MODEL-2076]
(c) Express sin2A in terms of cot A. (d) Write sin3A in terms of sinA.
2. (a) Express cos2A in terms of cosA and sin A. (b) Express cos2A in terms of cosA.
(c) Write cos2A in terms of sinA. (d) Write cos2A in terms of tanA.
(e) Write cos2A in terms of cotA. (f) Express cos3A in terms of cosA.
3. (a) Express tan2A in terms of tan A . (b) Express tan3A in terms of tanA.
UNDERSTANDING (U) BASED PROBLEMS
1. (a) If sinA = 3/4, find the value of cos2A. Ans: 7/25 (b) If 5sin = 4, find the value of cos2. Ans: -7/25
(c) If cos = 3/4 find the value of cos2. Ans: 1/8 (d) If tan = 3/4, find the value of cos2. Ans:7/25
2. (a) If sinA = 4/5, find the value of sin2A. Ans: 24/25 (b) If 5sin = 3 find the value of sin2. Ans:-8/25
(c) If cosA = 12/13 find the value of sin2A. Ans: 120/169 (d) If tan = 1/3, find the value of sin2.Ans: 3/5
3. (a) If tan = 4/3, find the value of tan2. Ans: -24/7 (b) If 3tan = 2, find the value of tan2. Ans: 12/5
4. (a) If cos = 3/5, find the value of cos3. Ans: -117/125 (b) If 2cos = 3 , find the value of cos3. Ans: 0
5. (a) If 2sin = 1, find the value of sin3. Ans:1 (b) If 5sinA = 4, find the value of sin3A. Ans: 44/125
6. (a) If cosA = 1 a + 1 then show that cos2A = 1 a2 + 1 .
2 a 2 a2
(b) If cosA = 1 x + 1 then show that cos3A = 1 x3 + 1 .
2 x 2 x3
7. (a) If tan2 = 1 + 2tan2, prove that cos2 = 1 + 2cos2 (b) If tan = b/a, prove that:acos2 + bsin2 = a
8. (a) Prove that: 2cos2(450 – A) = 1 + sin2A (b) Prove that: 2cos2 (450 + ) = 1 – sin2
(c) Prove that: 1 – 2sin2 (450 – ) = sin2 (d) Prove that: 1 – sin2A = 2sin2 (450 +A)
(e) Prove that: 1 + cos2A = 2sin2 (900 –A) (f) Prove that: 1 + cos2A = 2sin2 (900 + A)
9. (a) Prove that: cos4A = 8cos4A – 8cos2A + 1 (b) Prove that: cos4 = 8sin4 – 8sin2 + 1
10. (a) Prove that: 1 tan2 450 A = cosec2A (b) Prove that: 1 tan2 (450 α) sin2α
1 tan2 (450 α)
1 tan2 450 A
(c) Prove that: cos2 π A sin 2 π A = sin2A
4 4
11. (a) Prove that: sin 2θ 2 tan θ (b) Prove that: cos 2θ 1 tan2 θ
1 tan2θ 1 tan 2θ
(c) Prove that: sin 3θ 3sin θ 4sin3 θ (d) Prove that: cos 3θ 4 cos3 θ 3cos θ
cos3x cos3x (b) Express sin3 A + sin3A in terms of cotangent.
12. (a) Express sin3x + sin3x in terms of tangent. cos3A cos3A
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 38
(c) Express sin2A cos2A in terms of secant. (d) Express 1 sinα + sin2α in terms of tangent.
sinA cosA + cosα + cos2α
(e) Express sin2A sinA in terms of tangent. (f) Express 1+ sin2A cos2A in terms of tangent.
cos2A cos A+1 1+ sin2A + cos2A
(g) Express sin α 1 sin2α in terms of cotangent. (h) Express cosα 1 sin2α in terms of tangent
cosα 1 sin2α sinα 1 sin2α
(i) Express cosec2A – cot2A in terms of tangent. (j) Express cosec2A + cot2A in terms of cotangent.
13. (a) Prove that: cos3α + sin3α = 1 1 sin2α (b) Prove that: cos3A sin3A = 1 1 sin2A
cosα + sinα 2 cosA sinA 2
14. (a) Prove that: sin A.cos 2A 1 sin 4A.sec A (b) Prove 1 2sin A.cos A sin2 πc A
4 2 4
that:
(c) Prove that: 1 2sin θ.cos θ sin 2 πc θ
2 4
APPLICATION (A) BASED PROBLEMS
1. (a) Prove that: 3 cosec200 – sec200 = 4 (b) Find the value of cosec100 – 3 sec100
(c) Find the value of sec400 + 3 cosec400 (d) Find the value ofsec800 – 3 cosec800
(e) Find the value of cosec500 + 3 sec500
2. (a) Prove that: cos2α + sin2α.cos2β = cos2β + sin2β.cos2α
(b) Prove that: cos2A + sin2A.cos2B = cos2B + sin2B.cos2A
(c) Prove that: sin2A cos2A.cos2B = sin2B cos2B.cos2A
3. (a) Prove that: cos6θ sin6θ = cos2θ 1 1 sin 2 2θ
4
(b) Prove that: cos6α sin6α = 1 (3cos2α + cos3 2α)
4
(c) Prove that: cos6θ + sin6θ =1 3 sin2 2θ
4
(d) Prove that: cos6θ + sin6θ = 1 (5 + 3cos4θ)
8
4. (a) Prove that: cos8θ sin8θ = cos2θ 1 1 sin 2 2θ
2
(b) Prove that: cos8θ sin8θ = 1 cos2θ 3 + cos4θ
4
(c) Prove that: cos8A – sin8A = 1 (3cos2A + cos2A.cos4A)
4
(d) Prove that: cos8 + sin8 = 1 – sin22 + 1 sin42
8
5. (a) Prove that: sin4x = 1 (3 – 4cos2x + cos4x)
8
(b) Prove that: cos4 = 1 (3 + 4cos2 + cos4)
8
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 39
(c) Prove that: 3 + 4 cos2 + cos4 = 8cos4
(d) Prove that: 3 – 4 cos2 + cos4 = 8sin4
6. (a) Prove that:coec2A + cot4A = cotA – cosec4A
(c) Prove that: coec4A + cot8A = cot2A – cosec8A
(b) Prove that: cosec2A + cosec4A = cotA – cot 4A
(c) Prove that: cosec4A + cosec8A = cot2A – cot8A
7. (a) Prove that: sec4A 1 = tan4A.cotA
sec2A 1
(b) Prove that: sec8θ 1 = tan8θ
sec4θ 1 tan 2θ
8. (a) If 2tanα = 3tanβ , prove that: tan (α + β) = 5sin2β
5cos2β 1
(b) If 2tanα = 3tanβ , prove that: tan (α β) = 5 sin2β
cos2β
9. (a) Prove that: tanθ + 2tan2θ = cotθ (b) Prove that: tanθ + 2tan2θ + 4cot4θ = cotθ
(c) Prove that: tanθ + 2tan2θ + 4tan4θ + 8cot8θ = cotθ
10. (a) Prove that: (1+ sec2θ)(1+ sec4θ)(1+ sec8θ) = tan8θ.cotθ
(b) Prove that: 2cosθ +12cosθ 12cos2θ 1 = 2cos4θ +1
(c) Prove that: 2cosθ +12cosθ 12cos2θ 12cos4θ 1 = 2cos8θ +1
11. (a) Prove that: 2 + 2 +2cos4θ = 2cosθ (b) Prove that: 2 + 2 + 2 + 2cos8θ = 2cosθ
12. (a) Prove that: cot(θ 450 ) tan(θ 450 ) = 2 cos 2θ
1 sin 2θ
(b) Prove that: cot A + π tan A π = 2cosA
2 4 2 4 1 + sinA
13. (a) Prove that: 4 cos3150 + sin3150 =3 cos150 + sin150
(b) Prove that: 4 cos3 200 + sin3100 =3 cos200 + sin100
(c) Prove that: 4 cos3 200 + sin3 500 =3 cos200 + sin500
(d) Prove that: 4 sin3100 + sin3 700 =3 sin100 + sin700
14. (a) Prove that: cos3A.cos3A + sin3A.sin3A = cos3 2A
(b) Prove that: cos3A.sin3A + sin3A.cos3A = 3 sin4A
4
15. (a) Prove that: sin5θ =16sin5θ 20sin3θ + 5sinθ
(b) Prove that: cos5θ =16cos5θ 20cos3θ + 5cosθ
16. (a) Prove that: sin πc sin 3πc 1 sin 5πc sin 7πc = 1
1+ 8 1+ 8 8 1 8 8
(b) Prove that: πc 3πc 5πc 7πc 1
1+cos 8 1+cos 1+cos 1+cos =
8 8 8 8
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 40
17. (a) Prove that: sin4 πc + sin4 3πc + sin4 5πc + sin4 7πc 3
=
8 8 8 82
(b) Prove that: cos4 πc + cos4 3πc + cos4 5πc + cos4 7πc 3
=
8 8 8 82
cos2 cos2 πc 2 πc 3
(c) Prove that: A+ A + 3 +cos A 3 =
2
(d) Prove that: cos3A+ cos3 1200 +A + cos3 2400 +A = 3 cos3A
4
πc 2πc 3πc 1
(e) Prove that: cos .cos .cos
77 78
(f) Prove that: cos πc 2πc 4πc 1
.cos .cos
77 78
18. (a) Prove that: 2sinx + 2sin3x + 2sin9x = tan27x tanx
cos3x cos9x cos27x
(b) Prove that: 2cosx + 2cos3x + 2cos9x = cotx cot27x
sin3x sin9x sin27x
B. SUB-MULTIPLE ANGLES
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Express sinA in terms of sinA/2 and cos A/2.
(b) Express sinA in terms of tanA/2
(c) Write cos in terms of cos/2
(d) Express cos in terms of sin/2
(e) Express cos in terms of tan/2
(f) Write tan in terms of tan/2
(g) Express sin in terms of sin/3
(h) Express cosA in terms of cosA/3
UNDERSTANDING (U) BASED PROBLEMS
2. (a) α4 Ans:7/25
(b) If cos = , find the value of cos. Ans: 0
(c) Ans:-1/8
(d) 25 Ans:1/2
(e) Ans: 7/25
If cos θ = 1 , find the value of cos.
22 3
If sin A = 3 , find the value of cosA. Ans:
24
2
θ
If 2 sin =1 , find the value of cos.
2
If tan A = 3 , find the value of cosA.
24
(f) If tan C = 3 , find the value of cosC.
24
3. (a) θ = 1 , find the value of sin. Ans: 1
(b) If sin Ans: 44/125
(c) 32 Ans: 117/125
If sin A = 4 , find the value of sin.
35
If sin A = 3 , find the value of sin.
35
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 41
4. (a) If cos θ = 3 , find the value of cos. Ans: -117/125
(b) 35 Ans: 44/125
5. (a) If cosA/3 = 4/5, find the value of cosA.
If cosA = 1 a + 1 then show that cos2A = 1 a 2 + 1 .
2 a 2 a2
(b) If sinθ = 1 p2 + 1 then show that cos2θ = 1 p4 + 1 .
2 p 2 p2
(c) α = 1 x + 1 prove that cosα = 1 x2 + 1
If cos 2 x 2 x2
2
(d) If sin A = 1 x + 1 prove that cos A + 1 x2 + 1 0
2 2 x 2 x2
(e) If cos = 1 k + 1 , prove that cos = 1 k3 + 1
3 2 k 2 k3
(f) If sin θ 1 m 1 prove that sin + 1 m 3 1 = 0
3 2 m 2 m3
(g) If cos θ 1 t 1 prove that cos θ 1 t 2 1 = 0
4 2 t 2 2 t2
6. (a) If cos3300 = 3 , prove that: (i) sin1650 = 1 2 3 (ii) sin1650 = 3 1
2 2 22
(b) If cos300 = 3 , prove that: sin150 = 3 1
2 22
(c) If cos300 = 3 , find the value of cos150 Ans: 3 1
2 22
(d) If cos300 = 3 ,find the value of tan150 Ans: 2 3
(e)
(f) 2
(g)
7. (a) If cos450 = 1 , show that: cos 22 1 0 =1 2 2
(b) 2 2 2
(c)
If cos450 = 1 , show that: sin 22 1 0 =1 2 2
2 2 2
If cos450 = 1 , show that: tan 22 1 0 = 32 2
2 2
Express sinA in terms of sub-multiple angle of tangent. [SEE MODEL-2076]
1+cosA
Express 1-cosA in terms of sub-multiple angle of tangent.
sinA
Express 2sinβ sin2β in terms of sub-multiple angle of tangent.
2sinβ + sin2β
sin α sin α α sin ψ sin ψ ψ
that: 2 2 2 2
8. (a) Prove α tan (b) Prove that: cos ψ tan
(c) 1 cos α cos (d)
2 1 cos ψ 2
1+ cosA + cos A 1+ θ
that: 2 1 tan
Prove c ot A Prove that: 2 = cosθ
sinA + sin A 2 θ 1 sinθ
2 tan
2
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 42
θ 1+ sinθ cosθ (f) Prove that: cos A 1 sinA tan A
(e) Prove that: tan 2 = 1+ sinθ + cosθ 2
9. (a) Prove that: 2cot2A =cot A tan A 1 sinA 2
sin A
2
(b) Prove that: cotα = 1 cot α tan α
2 2 2
10. (a) Find the value of cot22 1 0 tan22 1 0 (b) Find the value of tan67 1 0 cot67 1 0
22 22
C. TRANSFORMATION OF TRIGONOMETRIC FORMULAE
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Express 2sinA.cosB in terms of sine. (b) Express 2cosx.siny in terms of sine.
(c)
Convert 2cosA.cosB in terms of cosine. (d) Express 2sinP.sinQ in terms of cosine.
2. (a)
(b) Convert sin C + sin D in terms of product of sine or cosine.
(c)
(d) Convert sin C - sin D in terms of product of sine or cosine.
Convert cos + cos in terms of product of cosine.
Convert cos X - cos Y in terms of product of sine.
UNDERSTANDING (U) BASED PROBLEMS
1. (a) Convert sin6A.cos4A into sum or difference of sine or cosine. [SEE MODEL-2076] Ans: 1 (sin10A+sin2A)
(b) 2
(c) Convert cos4A.sin2A into sum of difference of sine or cosine. Ans: 1 (sin6A-sin2A)
(d) 2
2. (a) Convert cos9.cos5 into sum of difference of sine or cosine. Ans: 1 (cos14θ cos 4θ)
(b) 2
(c)
(d) Convert sin5.sin20 into sum of difference of sine or cosine. Ans: 1 (cos15α cos 25α)
(a)
2
3.
(c) Express sin8A + sin2A into product of sine or cosine. Ans: 2sin5A.cos3A
4. (a) Express sin25A – sin15A into product of sine or cosine. Ans: 2cos20A.sin5A
(b)
Express cos5A + cos3A into product of cosine. Ans: 2cos4A.cosA
5. (a) Express cos40A – cos60A into product of sine. Ans: 2sin50A.sin10A
(c)
(e) Find the value of sin750 + sin150 Ans: 3 (b) Find the value of sin750 – sin1050 Ans: 1
6. (a) 2 2
(c)
(e) Find the value of cos150 + cos750 Ans: 3 (d) Find the value of cos150 – cos750 Ans: 1
7. (a) 2 2
(c) Find the value of sin750. sin150 Ans: 1 / 4 (b) Find the value of 4cos1050 .cos150 Ans: -1
(e) Find the value of 4sin1050 .cos750 Ans: 1 Prove that: 2sin500 .sin400 = cos100
Prove that: 2cos800 .sin100 = 1 – sin700
Prove that: 2cos700 .cos200 = cos500 (b) Prove that:2cos(450 + ).sin(450 + ) = cos2
Prove that:cos400 + cos800 + cos1600 = 0
Prove that: 2sin550 .cos350 = 1 + sin200 (d)
Prove that: 2cos (450 + A). cos (450 – A) = cos2A (f)
Prove that:cos100 + cos1100 + cos1300 = 0 (b)
Prove that:cos200 – cos1000 – 3 sin400 = 0 (d) Prove that:sin500 + sin700 = 3 cos100
Prove that: cos400 + sin400 = 2 cos50 (f) Prove that:cos180 – sin180 = 2 sin270
Prove that: sinA + sin5A = tan3A (b) Prove that: cosA cos5A = tan3A
cosA + cos5A sin5A sinA
Prove that: cos100 sin100 = cot550 (d) Prove that: cos140 sin140 = tan310
c os100 sin100 c os140 sin140
Prove that: sin 800 sin100 = cot350 (f) Prove that: sin (p + 2)θ sinpθ cot(p 1) θ
c os100 cos 800 cospθ cos (p + 2)θ
APPLICATION (A) BASED PROBLEMS
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 43
1. (a) Prove that: (cosA + cosB)2 + (sinA + sinB)2 = 4cos2 AB
2
(b) Prove that: (cosA cosB)2 + (sinA sinB)2 = 4sin 2 AB
2
2. (a) Prove that: sin θ.sin (600 θ).sin (600 +θ) = 1 sin3θ
4
(b) Prove that: cos (600 θ).cos(600 θ).cosθ = 1 cos3θ
4
(c) Prove that: 4cos (600 A).sin(300 +A).cosA = cos3A
3. (a) Prove that: sin 2 α sin2β tan(α β)
sinα.cosα sinβ.cosβ
(b) Prove that: cos2 α sin2β cot(α β)
sinα.cosα sinβ.cosβ
4. (a) Prove that: 16sin200.sin400.sin600.sin800 = 3 (b) Prove that: sin100.sin300.sin500.sin700 = 1
(c) Prove that: sin100.sin500.sin700 = 1 (d) 16
8
Prove that: cos100.cos300.cos500.cos700 = 3
16
(e) Prove that: 8cos100.cos500.cos700 = 3 (f) Prove that: cos400.cos1000.cos1600 = 1
(g) Prove that: 8cos400.cos1000.cos1600 = 1 8
(h) Prove that: 16 sin60.sin420.sin660. sin780 = 1
(i) Prove that: tan200.tan400.tan800 = 3
5. (a) Find the value of sin200.sin300.sin400.sin800 [SEE MODEL-2076] Ans: 3
16
(b) Find the value of sin100.sin300.sin500.sin700 Ans: 1
16
(c) Find the value of cos100.cos300.cos500.cos700 Ans: 3
16
6. (a) Prove that: sin2A + sin5A sinA = tan2A (b) Prove that: cosA - cos2A cos3A cot2A
cos2A + cos5A + cosA sinA - sin2A sin3A
7. (a) Prove that: sinA sin2A sin4A sin5A tan3A (b) Prove that: sin2A sin5A sin7A sin9A tan6A
cosA cos2A cos4A cos5A cos3A cos5A cos7A cos9A
8. (a) Prove that: (sin4A sin2A)(cos 4A - cos8A) 1 (b) Prove that: (cosA - cos3A)(sin 8A sin2A) 1
(sin7A sin5A)(cos A - cos5A) (cos4A - cos8A)(sin 5A - sinA)
9. (a) Prove that: sin8A.cosA sin6A.cos3 A tan2A (b) Prove that: cos4A.cos3A - cos5A.cos2A tan2A
cos2A.cosA sin2A.sin5 A
cos4A.sin3 A - cos5A.sin2 A
10. (a) Prove that: sec π + θ .sec π θ = 2secθ (b) Prove that: cosec π + θ .cosec π θ = 2secθ
4 2 4 2 4 2 4 2
(c) Prove that: 2cos πc 9πc cos 3πc cos 5πc 0
.cos
13 13 13 13
11. Prove that: (a) cos3x.sin2x = 1 (2cosx cos3x cos 5x) (b) cos2x.sin3x = 1 (2sinx + sin3x sin 5x)
16 16
12. (a) If 1 + 1 = 1 + 1 , prove that: cot A + B = tanA.tanB
sinA cosA sinB cosB 2
(b) If sinA + sinB = a and cosA + cosB = b, prove that: tan A + B a
2 b
D. CONDITIONAL TRIGONOMETRIC IDENTITIES
UNDERSTANDING (U) BASED PROBLEMS
1. (a) If A + B + C = 1800, convert sin (A + B) in terms of C.
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 44
(b) If A + B + C = 200g, express cos (A + B) in terms of C.
(c) If A + B + C = c, find tan (A + B) in terms of angle C.
(a) If A + B + C = 900, find the value of sin(A2 + B2) (b) If + + = 900, find the value of cos(A2 + B2)
2.
APPLICATION (A) BASED PROBLEMS
1. If A, B and C are the intrereior angles of a triangle ABC, prove that:
a) cot A .cot B .cot C = cot A + cot B + cot C b) tan A tan B + tan B .tan C + tan C .tan A =1
222 2 22 22 22 22
2. If A + B + C = c, then prove that:
a) sin2A + sin2B + sin2C = 4sinA.sinB.sinC b) sin2A – sin2B + sin2C = 4cosA.sinB.cosC
c) sin2A + sin2B + sin2C = 32sin A .sin B .sin C d) sin2A + sin2B + sin2C = 8cos A .cos B .cos C
cos A .cos B .cos C 222 4sin A .sin B .sin C 222
222 222
e) sin (B + C – A) + sin (C + A – B) + sin (A + B – C ) = 4 sinA.sinB.sinC
f) cosA + cosB + cosC = 2
sinB.sinC sinC.sinA sinA.sinB
3. If A + B + C = c, then prove that:
a) cos2A + cos2B + cos2C = - (1 + 4cosA.cosB.cosC) b) cos2A – co2B – cos2C = 4cosA.sinB.sinC – 1
c) cos (B + C – A) + cos (C + A – B) + cos (A + B – C ) = 4 cosA.cosB.cosC + 1
4. If A, B and C are the intrereior angles of a triangle ABC, prove that:
a) sinA.cosB.cosC + sinB.cosC.cosA + sinC.cosA.cosB = sinA.sinB.sinC
b) cosA.sinB.sinC + cosB.sinC.sinA + cosC.sinA.sinB = 1 + cosA.cosB.cosC
c) If P + Q + R = c, then prove that: sinP.cosQ.cosR + sinQ.cosR.cosP + sinR.cosP.cosQ = 1
sinP.sinQ.sinR
5. If A + B + C = c, then prove that:
a) sin2A + sin2B + sin2C = 2 + 2cosA.cosB.cosC b) sin2A + sin2B – sin2C = 2sinA.sinB.cosC
c) sin2A – sin2B + sin2C = 2sinA.cosB.sinC [SEE MODEL-2076]
d) sin2A – sin2B – sin2C = -2cosA.sinB.sinC
6. If A, B and C are the intrereior angles of a triangle ABC, prove that:
a) cos2A + cos2B + cos2C = 1 2cosA.cosB.cosC b) cos2A cos2B + cos2C + 2sinA.cosB.sinC = 1
c) cos2A + cos2B cos2C = 1 2sinA.sinB.cosC d) cos2A cos2B cos2C + 1= 2cosA.sinB.sinC
7. If A + B + C = 1800, then prove that:
a) sinA + sinB + sinC = 4cos A .cos B .cos C b) sinA + sinB sinC = 4sin A .sin B .cos C
2 22 2 22
c) sinA + sinB +sinC = 4cos A .sin B .sin C
2 22
8. If A, B and C are the intrereior angles of a triangle ABC, prove that:
a) cosA + cosB + cosC = 1+ 4sin A .sin B .sin C
2 22
b) cosA + cosB + cosC = 1+ 4sin A .sin B .sin C
2 22
c) cosA + cosB + cosC + 1 = 4sin A .cos B .cos C
2 22
9. If A + B + C = 1800, then prove that:
a) sin2 A + sin2 B +sin2 C = 1 2sin A .sin B .sin C
2 22 2 22
b) cos2 A + cos2 B +cos2 C = 2 1 sin A .sin B .sin C
2 2 2 2 2 2
10. a) If A + B + C = 1800, then prove that: A BC πc -A πc -B πc -C
sin + sin +sin = 1+ 4sin .sin .sin
2 2 2 4 4 4
b) If A + B + C = 1800, prove that: sin A + sin B + sin C 1 = 4sin A + B .sin B + C .sin C + A
2 2 4 4 4
2
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 45
c) If A + B + C = 1800, prove that: sin (A 2B)sin (B 2C) sin(C 2A) 2sin A B .sin B C .sin C A
2 2 2
11. If A + B + C = 1800, prove that:
a) cos A + cos B +cos C πc -A πc -B πc -C
2 2 2 = 4cos .cos .cos
4 4 4
b) cos A + cos B cos C πc + A πc + B πc - C
2 22 = 4cos .cos .cos
4 4 4
c) cos A cos B cos C πc + A πc - B πc + C
2 22 = 4cos .cos .cos
4 4 4
E. TRIGONOMETRIC EQUATIONS
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define trigonometric equation.
(b) What is trigonometric equation differ from trigonometric identities?
2. (a) If x sin y, what are the values of x and y?
(b) If x cosA y, what are the values of x and y?
(c) If x tansA y, what are the values of x and y?
3. (a) If the value of sin is positive then in which quadrants does lie?
(b) If the value of sinA is negative then in which quadrants does A lie?
(c) If the value of cos is positive then in which quadrants does lie?
(d) If the value of cos is negative then in which quadrants does lie?
(e) If the value of tanA is positive then in which quadrants does A lie?
(d) If the value of tan is negative then in which quadrants does lie?
4. (a) If sin (1800 – A) = x, what is the value of x? (b) If sin (1800 + A) = p, what is the value of p?
(c) If sin (3600 - ) = m, what is the value of m? (d) If sin (3600+) = a, what is the value of a?
5. (a) If cos (1800–) = x, what is the value of x? (b) If cos (1800 +) = y, what is the value of y?
(c) Ifcos (360 0- ) = h, what is the value of h? (d) Ifcos (3600+) = k, what is the value of k?
6. (a) If tan (1800– A) = b, what is the value of b? (b) If tan (1800 + A) = d, what is the value of d?
If tan (3600+A) = q, what is the value of q?
(c) If tan (360 0- A) = p, what is the value of p? (d)
7. (a) If sinA = 1 / 2, what is the acute value of A?
(b) If tan = 1, what is the value of between 00 and 900?
(c) If cos = -1, what is the value of ? [00 < < 3600]
(d) If sin = -1, what is the value of .?[1800 < < 3600]
UNDERSTANDING (U) BASED PROBLEMS
Range: (00 1800)
1. (a) If 2sin2 = 3 , find the value of . [SEE MODEL-2076] Ans: 300, 600
(b) If 2cos2 = 3 , find the value of . Ans: 300 , 1650
(c) If 3 tan3A – 3 = 0, find the value of . Ans: 200 , 800
2. (a) If sin = cos , find the value of . (00 1800) Ans: 450
(b) If cosec = sec , find the value of (00 1800) Ans: 450
(c) If tan = cot , find the value of .(00 1800) Ans: 450
3. (a) If tan2 = 3 , find the value of .(00 1800) Ans: 600,1200 Page 46
(b) If cot2 = 3 , find the value of .(00 1800) Ans: 300,1500
(c) If 4sin2 = 1 ,find the value of .(00 1800) Ans: 300,1500
(d) If 4sin - 3cosec = 0, find the value of . Ans: 600,1200
(e) If 4cos - 3sec = 0, find the value of . Ans: 300,1500
Ans: 450
4. (a) If tanx + cotx = 2, find the value of x. Ans: 450
(b) If sec2 + cosec2 = 4, find the value of . Ans: 450
(c) If sin2 = sin2 + cos2 , find the value of . Ans: 330
5. (a) If sin (x + 240) = cosx, find the value of x. BY: TARA BDR MAGAR
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077
(b) If sec (x - 60) = cosec3x, find the value of x. Ans: 240
(c) If sin (2x – 150) =cos (2x – 150) find x. Ans: 600
6. (a) If sin2 = sin2 + cos2 , find the value of . Ans: 450
(b) If cos22A - 4cos2A + 1 = 0 , find A. Ans: 300,1500
(c) If sin22A - 4sin2A + 1 = 0 , find A. Ans: 150,750
(d) If cos2 θ cos θ 1 0 , find the value of . Ans: 1200
2 24 Ans: 600
(e) If sin2 θ sin θ 1 0 , find the value of .
2 24
APPLICATION (A) BASED PROBLEMS
1. Solve:
a) cos2θ sinθ 1 = 0 (00 θ 3600 ) Ans: 300, 1500
4
b) 3sin2θ 4 cos θ = 4 (00 θ 3600 ) Ans: 00, 3600, cos1 1
3
c) 6sin2A cos A = 5 (00 A 3600) Ans: 600, 3000, cos1 1
3
d) cos2x 3sin2x sinx = 2 (00 x 3600 ) Ans: 300, 1500, 2700
e) 4cos2θ + 4sinθ 5 = 0 (00 θ 3600 ) Ans: 300, 1500
f) cot2θ cosecθ = 3 (00 θ 3600 ) Ans: 300, 1500, sin1 1
3
g) 3tan2θ - 4secθ - 1 = 0 (00 θ 3600 ) Ans: 600, 3000
h) 3tan2θ - 4secθ - 1 = 0 (00 θ 3600 ) Ans: 600, 3000
i) 4sec2θ 7tan2θ = 3 (00 θ 3600 ) Ans: 300, 1500, 2100, 3300
j) 7sin2θ +3cos2θ = 4 (00 θ 3600 ) Ans: 300, 1500, 2100, 3300
2. Solve: (00 θ 3600 ) Ans: 300, 3300
a) 2 3sin2θ = cosθ
b) 2 3 cos2 θ = sinθ (00 θ 3600 ) Ans: 600, 1200
c) 2 3sin2α + cosα = 0 (00 α 3600) Ans: 1500, 2100
d) 2 3 cos2 A + sinA = 0 (00 A 3600 ) Ans: 2400, 3000
3. Solve (00 θ 3600 ) :
a) secθ.tanθ = 2 Ans: 450, 1350 b) cosecθ.cotθ = 2 Ans: 450, 3150
4. Solve (00 θ 3600 ) :
a) cot 2 θ + 3 1 cotθ 1 0 Ans: 1200, 1500, 3000, 3300
3
b) cot 2 θ + 3 1 cotθ = 1 Ans: 600, 1500, 2400, 3300
3
c) tan2θ 3 1 tanθ 1 0 Ans: 300, 600, 2100, 2400
3 Ans: 450, 600, 2250, 2400
5. a) tan2θ 1 3 tanθ 3 0
b) tan2θ 1 3 tanθ 3 0 Ans: 600, 1350, 2400, 3150
c) 1 3 tanθ + 1 + 3 3 sec2 θ Ans: 300, 1350, 2100, 3150
d) 3 1 tanθ + 3 1 3 sec2 θ Ans: 450, 1500, 2250, 3300
6. Solve for the angles upto 3600:
a) 3sinA + cosA = 2 Ans: 150, 1050 b) sinA = 3(1 cos A) Ans: 00, 600, 3600
c) 3sinx cosx = 2 Ans: 750, 1650 d) sinA + cosA = 1 Ans: 00, 900, 3600
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 47
e) 3 cos x sin x 3 Ans: 00, 600, 3600 f) 3tanθ = secθ 1 Ans: 00, 1200, 3600
h) 3(cotθ cosecθ) + 1 = 0 Ans: 00, 600, 3600 i) Solve: sinx 3cosx = 2 Ans: 750, 1650
6. Solve for the angles upto 3600: Ans: 450, 1200, 1350
a) cosθ + cos2θ + cos3θ = 0
b) cos3θ cos2θ + cosθ = 0 Ans: 450, 600, 1350, 3000
c) sinA + sin3A = sin2A Ans: 00, 600, 900, 1800, 2700, 3000, 3600
7. Solve:
a) tanθ + tan2θ + 3tanθ.tan2θ = 3 Ans: 200, 800, 1400
b) cotθ + cot2θ 3cotθ.cot2θ + 3 0 Ans: 200, 800, 1400
8. a) If tanx + tany = 2 and 2cosx.cosy = 1, find the values of x and y. Ans: x = y = 450
b) If cotA + cotB = 2 and 2sinA.sinB = 1, find the values of A and B.Ans x = y = 450
E. HEIGHT AND DISTANCE
KNOWLEDGE (K) BASED PROBLEMS
1. (a) Define angle of elevation. [SEE MODEL-2076] (b) Define angle of depression.
2. (a) What angle is subtended by the tree to the top of its shadow when sun’s altitude is ‘’?
(b) What is the angle of elevation of the bird sitting on the top of a tower as observed by a boy from a
certain distance from the foot of the tower when the angle of depression of the boy as observed by the
bird at the same time is found to be ?
(c) If he angle of elevation of the girl sitting on the top of a building as observed by a boy from a certain
distance away from the foot of the building was and the angle of depression of the boy as observed
by the girl at the same time is found to be , what is the relation between and ?
3. (a) What are the conditions under which the angle of elevation becomes smaller?
(b) What are the conditions under which the angle of depression becomes larger?
APPLICATION (A) BASED PROBLEMS
1. (a) The angle of elevation of the top of a tower was observed to be 600 from a point. On walking 200m away
from the point it was found to be 300. Find the height of the tower. Ans:173.2m
(b) The angle of elevation of the top of a tower from two places due east of its foot are 450 and 300. If two
places are 180 m apart, find the height of the tower. Ans: 245.9m
(c) A pole subtends an angle of 300 and 600 at two points on a road. If the distance between the points is
40m, find the height of the pole. Ans: 20 3
(d) The shadow of a tower on the ground is found to be 45m longer when sun’s altitude is 450 than when it is
600, find the height of the tower. Ans: 106.46m
2. (a) The angle of depression and elevation of the top of a building 40m high from the top and bottom of a
tower are found to be 600 and 300 respectively, find the height of the tower. Ans:160m
b) The angle of depression and elevation of the top of a building 25m high from the top and bottom of a
cliff are found to be 600 and 300 respectively, find the height of the cliff. Ans:100m
c) From the top of a tree 30m high, the angles of depression of the top and bottom of a pole are observed to
be 450 and 600 respectively. Find the height of the pole the distance between the tree and the pole.
Ans:12.68m, 17.32m
d) From the top of a cliff 100m high, the angles of depression of the top and bottom of a building are
observed to be 300 and 450 respectively. Find the height of the building. Also, find the distance between
the tower and the building. Ans:42.27m
e) From the top of a tower 60m high, the angles of depression of the top and bottom of a building are
observed to be 300 and 600 respectively. Find the height of the building. Also, find the distance between
the tower and the building. Ans:40m, 20 3 m
3. a) A poster hanging on the wall has a vertical height 3.66m. From the point 5m away from the wall on the
same plane, the angle of elevation of the bottom edge of the poster was found to be 450. What was the
angle of elevation of the top edge of the poster, as observed from the same place? Ans:600
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 48
b) There are two posts of height 90m and 30m respectively. From the foot of the second post, the angle of
elevstion of the first post is found to be 600. Fi nd the angle of elevation of the top f the second post from
the foot of the first post. Ans: 300
c) A pole AB is divided by a point C such that AC: CB = 2: 1where B is the foot of the pole. The angle of
elevation of C from a point on the ground is 300. Find the angle of elevation of A from the same point?
Ans: 600
4. (a) From a place at the ground level in front of a tower the angle of elevations of the top and bottom of
flagstaff 6m high situated at the top of a tower are observed 60° and 45° respectively. Find the height of
the tower and the distance between the base of the tower and point of observation. [SEE MODEL-2076]
Ans: 8.19 m, 8.19 m
(b) A flagstaff of height 7m stands on the top of a tower. The angles subtended by the tower and the flagstaff
at a point on the ground are 450 and 150 respectively, find the height of the tower.Ans:9.56m
(c) A flagstaff stands on the top of a 10m high pillar. The angles elevation of the top of the flagstaff from a
point on the ground is 600 and from the same point the angle of elevation of the top of the pillar is 450;
find the height of the flagstaff. Ans: 7.32m
(d) The angle of elevation of the top of an incomplete tower from a distance of 100m is 450. What height
should it be raised so that the angle of elevation of the may change to 600? Ans: 73.20m
5. (a) The angles of elevation of the top of the tower as observed from the distances of 20m and 45m from its
foot are found to be complementary. Find the height of the tower. Ans:30m
(b) The angles of elevation of the top of the tower as observed from the distances of 18m and 32m are found
to be complementary. Find the height of the tower. Ans: 24m
(c) The angles of elevation of the top of the cliff as observed from the distances of 144 m and 441 m from its
foot of the same side and are found to be complementary. Show that the height of the cliff is 252m.
(d) The angles of elevation of the top of the tower as observed from the two places from its foot are found to
be complementary. If the distance of a place from the tower is 36m and the other place is 13m away from
the first place, find the height of the tower. Ans: 42m
6. (a) Two posts are 180m apart and the height of one is double that of the other. From the mid point of the
line joining their feet, an observer finds the angles of the elevation of their tops to be complementary,
find the height of the longer post. Ans: 127.28m
(b) Two poles are 40m apart and the height of one is double that of the other. The angles of elevation of the
top of poles from the mid way between them are complementary; find the height of the longer post.
Ans: 10 2 m, 20 2 m
7. (a) A vertical tower is divided by any point in the ratio 9: 1 from the top. If both the segments subtend equal
angles to each other at a distance of 20m away from the foot of the pole, find the height of the tower.
Ans:178.9m
(b) A vertical tower is divided by any point in the ratio 3: 1 from the top. If both the segments subtend equal
angles to each other at a distance of 10m away from the foot of the pole, find the height of the tower.
Ans: 20 2 m
8. (a) A ladder 10m long reaches a point 10m below the top of a vertical flagstaff. From the foot of the ladder
the angle of elevation of the flagstaff is 600, find the height of the flagstaff. Ans:15m
(b) A ladder 6m long reaches a point 6m below the top of a vertical flagstaff. From the foot of the ladder the
angle of elevation of the flagstaff is 600, find the height of the flagstaff. Ans: 9m
9. (a) An aero-plane flying horizontally 750m above the ground is observed at an elevation of 600. After 5
seconds, the elevation of the plane changes to 300, find the speed of the aero-plane in km/hr.
Ans:623.54km/hr
(b) From the top of lighthouse on a sea-shore, a boy observes a boat coming in a uniform speed to the shore
at an angle of depression of 300. After 10 minutes, he finds the angle of depression 600. What time will
the boat take to reach the bottom of the lighthouse on the shore from the starting point? Ans: 15 minutes
(c) At the foot of the mountain, the angle of elevation of its summit is 450, after ascending 1km forwards at
10. (a)
an inclination of 300, the elevation changes to 600, find the height of the mountain. Ans: 1.366km
The angle of elevation of a bird from a point 100 feet above a lake is 300 and the angle of depression on
its image in the lake is 600; find the height of the bird from the water surface. Ans: 200 ft.
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 49
(b) The angle of elevation of a bird from a point 400 feet above a lake is 300 and the angle of depression on
its image in the lake is 600; find the height of the bird from the water surface. Ans: 800 ft.
UNIT-6 VECTOR
SEE SPECIFICATION GRID U A HA Total Marks
Contents K
22=4 51=5 1+4+5 = 10
1. Scalar Product
2. Vector geometry 11=1
KNOWLErDGE r(K) BASED PROBLEMS [SEE MODEL-2076]
1. (a) What is the scalar product of two vectors a and b if the angle between them is ?
(b) Define scalar product of two vectors.
2. (a) Write the condition of parallelism of vectors in term of their scalar product.
(b) Under what the condition two vectors are perpendicular (orthogonal) to each other? Write in term of
their scalar product.
rr r r
3. (a) If a.b 0 then what is the relation between a and b ?
rr
rr r r
(b) If a.b a b what is the relation between a and b ?
rr r r
(c) If a.b 0 then what is the angle between a and b ?
rr r r
(d) If a b then what is the angle between a and b ?
r rr
4. (a) If i the standard unit vector along x-axis, what is the value of i.i ?
rr
(b) If j the standard unit vector along y-axis, what is the value of i 2 ?
rr rr
(c) If i and j are the standard unit vectors along x-axis and y-axis respectively, what is the value of i. j
uuur uuur
5. (a) In a triangle ABC, if AM BC, what is the value of AM.BC ?
uuur uuur
(b) In a rectangle PQRS, what is the value of PQ.QR?
r r2 ur ur 2
6. (a) If a 5 , what is the value of a ? (b) If p 7 , what is the value of p ?
rr r2 rr r2
(c) If a 2i , what is the value of a ? (d) If a 6 j , what is the value of a ?
7. (a) If M is the mid-point of line segment joining the points A and B and O is the reference origin, write
uuur uur uuur
down the relation of OM,OA and OB.
(b) If the point P divides the line segment AB joining the points A and B externally in the ratio m1:m2, and
uuur uur
uuur
O the reference origin, write down the relation of OP,OA& OB along with m1and m2.
UNDErRSTANr DING (U) BASED PROBLEMS
r
rr r
1. (a) If a = 4 2, b = 6 and angle between a and b (θ) 450 , find the value of a.b Ans: 24
rr rr rr Ans: 18
(b) If a = 3 3, b = 4 and angle between a and b (θ) 300 , find the value of a.b Ans: 15
Ans: 18
ur r ur r ur r Ans:4
Ans:5
(c) If p = 3 2, q = 5 2 and angle between p and q (θ) 600 , find p.q .
uuur uuur
(d) In an equilateral triangle ABC, AB = BC = CA = 6cm, find of AB.BC
r
2. (a) r = 6, rr = 12 and θ 600 , find the value of b
If a ab
r rr
(b) If q 6 2,p.q 30 and = 450, find the value of p
(c) If u 4 3, u.v 24 and = 450, find the value of v Ans:6
SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 50
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