11th Business and Statistics Maths Textbook Volume 1

Example 5.34
If f(x) = xn and f l]1g = 5 , then find the value of n.

Solution

f(x) = xn

` f l]xg = nxn–1

f l]1g = n(1)n–1

f l]1g = n

f l]1g = 5 (given)

( n=5

Example 5.35

If y = 1 and u = x2 –9 , find dy .
u2 dx

Solution

y = 1 = u–2
u2

dy = - 2
du u3

dy = - ^ x2 2 9h3 ( a u = x2 – 9)
du -

u = x2 – 9

du = 2x
dx

Now, dy = dy $ du
dx du dx

= - ^ x2 2 9h3 $ 2x
-

= - ^ 4x 9h3 .
x2 -

i e entia a cu us 193

ICT Corner

Expected nal outcomes
Step – 1
Open the Browser and type the URL given (or) Scan
the QR Code.
GeoGebra Work book called “11th BUSINESS
MATHS” will appear. In this several work sheets for
Business Maths are given, Open the worksheet named “Elementary Di erentiation”
SEtleemp e-n2tary Di erentiation and Functions page will open. Click on the function check boxes on
the Right-hand side to see the respective graph. You can move the sliders a, b, c and d to change
the co-e cient and work out the di erentiation. en click the Derivatives check box to see the
answer and respective graphs

St e p 1

St e p 2
St e p 4

B ro w s e in th e lin k
11th Business Maths: https://ggbm.at/qKj9gSTG (or) scan the QR Code
194 11th Std. Business Mathematics

Exercise 5.5

1. Differentiate the following with respect to x.

(i) 3x4 - 2x3 + x + 8 (ii) 5 - 2 + 5 (iii) x + 1 + ex
x4 x3 x 3x
3 + 2x - x2
(iv) x (v) x3 ex (vi) ^x2 - 3x + 2h]x + 1g

(vii) x4 - 3 sin x + cos x (viii) d x+ 1 2
x
n

2. Differentiate the following with respect to x.

(i) 1 ex x (ii) x2 + x + 1 (iii) ex
+ x2 - x + 1 1 + ex

3. Differentiate the following with respect to x.

(i) x sin x (ii) ex sin x (iii) ex ^x + log xh

(iv) sin x cos x (v) x3 ex

4. Differentiate the following with respect to x.

(i) sin2 x (ii) cos2 x (iii) cos3 x

(iv) 1 + x2 (v) ^ax2 + bx + chn (vi) sin(x2)

(vii) 1
1 + x2

5.3.3 Derivative of implicit functions

For the implicit function f(x,y) = 0, differentiate NOTE

each term with respect to x treating y as a function of x If the function f (x, y) = 0 is
an icmopnltiaciint sfubnoctthioxn,atnhdeny.ddxy
and then collect the terms of dy together on left hand
dx
side and remaining terms on the right hand side and then

find dy .
dx

Example 5.36

If ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 then find dy .
dx

Solution

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

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Differentiating both side with respect to x,

2ax + 2h c x dy + ym + 2by dy + 2g + 2f dy =0
dx dx dx

^2ax + 2hy + 2g h + ^2hx + 2by + 2f h dy =0
dx
dy ^ax + hy + gh
dx = - ^hx + by + fh .

Example 5.37

If x3 + y3 = 3axy , then find dy .
dx

Solution

x3 + y3 = 3axy

Differentiating both sides with respect to x,

d _x3 + y3i = d ^3axyh
dx dx

3x2 + 3y2 dy = 3a;x dy + yE
dx dx

x2 + y2 dy = ax dy +ay
dx dx

_ y2 - axi dy = (ay – x2)
dx

dy = ay - x2
dx y2 - ax

Example 5.38

Find dy at (1,1) to the curve 2x2 + 3xy + 5y2 = 10
dx

Solution

2x2 + 3xy + 5y2 = 10
Differentiating both sides with respect to x,

d 72x2 + 3xy + 5y2A = d [10]
dx dx

4x + 3x dy + 3y + 10y dy =0
dx dx

(3x+10y) dy = –3y –4x
dx

196 11th Std. Business Mathematics

dy = - ^3y + 4xh
dx ^3x + 10yh

Now, dy at (1, 1) = - 33++140
dx
= - 173

Example 5.39

If sin y = x sin (a+y), then prove that dy = sin2 ^a + yh .
dx sin a

Solution

sin y = x sin (a+y)

x= sin y
sin^a + yh

Differentiating with respect to y,

dx = sin^a + yhcos y - sin y $ cos^a + yh
dy sin2 ^a + yh

= sin^a + y - yh
sin2 ^a + yh

= sin a
sin2 ^a + yh

dy = sin2 ^a + yh
dx sin a

Exercise 5.6

1. Find dy for the following functions
dx
(i) xy = tan(xy) (ii) x2 - xy + y2 = 7 (iii) x3 + y3 + 3axy = 1

2. If x 1+y+y 1 + x = 0 and x ! y, then prove that dy = - ]x 1 1g2
dx +

3. If 4x + 3y = log^4x - 3yh , then find dy
dx

5.3.4 Logarithmic di erentiation

Some times, the function whose derivative is required involves products, quotients,
and powers. For such cases, differentiation can be carried out more conveniently if we
take logarithms and simplify before differentiation.

i e entia a cu us 197

Example 5.40

Differentiate the following with respect to x.

(i) xx (ii) ^log xhcosx

Solution

(i) Let y = xx

Taking logarithm on both sides

logy = x log x

Differentiating with respect to x,

1 $ dy = x$ 1 + log x $ 1
y dx x

dy = y71 + log xA
dx

dy = xx 71 + log xA
dx

(ii) Let y = ^log xhcosx

Taking logarithm on both sides

` log y = cos x log^log xh

Differentiating with respect to x,

1 $ dy = cos x 1 $ 1 + 7log^log xhA]- sin xg
y dx log x x

dy = y; xcloosgxx - sin x log^log xhE
dx

= ^log xhcosx ; xcloosgxx - sin x log^log xhE

Example 5.41

If xy = ex–y, prove that dy = ^1 log x
dx + log xh2

Solution

xy = ex–y

198 11th Std. Business Mathematics

Taking logarithm on both sides,

y log x = (x–y)

y(1+log x) = x

y = 1 + xlog x

Differentiating with respect to x

dy = ^1 + log xh]1 g - xb0 + 1 l = log x
dx x ^1 + log xh2
^1 + log xh2

Example 5.42 ]x - 3g^x2 + 4h
Differentiate: 3x2 + 4x + 5

Solution

Let y= ]x - 3g^x2 + 4h = = ]x - 3g^x2 + 4h 12
3x2 + 4x + 5 3x2 + 4x + 5
G

Taking logarithm on both sides,

log y = 12 7log]x - 3g + log^x2 + 4h - log^3x2 + 4x + 5hA

:a log ab = log a + log b and log a = log a - log bD
b

Differentiating with respect to x,

1 $ dy = 12 ;x 1 3 + x22+x 4 - 3x26+x +4 5E
y dx - 4x +

dy = 12 ]x - 3g^x2 + 4h ;x 1 3 + 2x 4 - 6x +4 5E
dx 3x2 + 4x + 5 - x2 + 3x2 + 4x +

Exercise 5.7

1. Differentiate the following with respect to x,

(i) xsinx (ii) ]sin xgx (iii) ]sin xgtanx (iv) ]x - 1g]x - 2g
]x - 3g^x2 + x + 1h

2. If xm $ yn = ^x + yhm + n , then show that dy = y
dx x

i e entia a cu us 199

5.3.5 Di erentiation of parametric functions

If the variables x and y are functions of another variable namely t, then the functions
are called a parametric functions. The variable t is called the parameter of the function.

If x = f (t) and y = g(t), then

dy = dy/dt
dx dx/dt

5.3.6 Di erentiation of a function with respect to another function

Let u = f(x) and v = g(x) be two functions of x. The derivative of f(x) with respect to
g(x) is given by the formula,

d_ f]xgi = dduv//ddxx
d_ g]xgi

Example 5.43

Find dy if (i) x = at2 , y = 2at (ii) x = a cos i, y = a sin i
dx

Solution

(i) x = at2 y = 2at

dx = 2at dy = 2a
dt dt

` dy = dy
dx dt
dx
dt

= 22aat

= 1
t

(ii) x = a cos i, y = a sin i

dx = - a sin i dy = a cos i
di di

dy = dy
dx dt
dx
dt

= a cos i
-a sin i

= - cot i

200 11th Std. Business Mathematics

Example 5.44

If x = a i and y = a , then prove that dy + y = 0
i dx x

Solution

x = ai y = a
i

dx =a dy = -a
di di i2

dy = dy
dx di
dx
di

= a -a k
i2

a

= - 1
i2

= - y
x

i.e. dy + y =0
dx x

Aliter :

Take xy = ai $ a
i

xy = a2

Differentiating with respect to x,

x dy +y = 0
dx

dy + y =0
dx x

Example 5.45
Differentiate 1 +x2x2 with respect to x2

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Solution

Let u = x2 v = x2
1 + x2

du = ^1 + x2h]2xg - x2 ]2xg dv = 2x
dx ^1 + x2h2 dx

= ^1 +2xx2h2

du = _ du i
dv dx

_ dv i
dx

<^1 2x F
+ x2h2
= 2x

= 1
^1 + x2h2

Exercise 5.8

1. Find dy of the following functions
dx

(i) x = ct, y = c (ii) x = log t, y = sin t
t

(iii) x = a cos3 i, y = a sin3 i (iv) x = a]i - sin ig, y = a]1 - cos ig

2. Differentiate sin3 x with respect to cos3 x
3. Differentiate sin2 x with respect to x2

5.3.7 Successive di erentiation

The process of differentiating the same function again and again is called successive
differentiation.

(i) The derivative of y with respect to x is called the first order derivative and is

denoted by dy (or) y1 (or) f l]xg
dx

(ii) If f l]xg is differentiable, then the derivative of f l]xg with respect to x is called
d2 y
the second order derivative and is denoted by dx2 (or) y2 (or) f ll]xg
(n)(x) is called nth
(iii) Further dn y (or) yn (or) f order derivative of the function
dxn

y = f(x)

202 11th Std. Business Mathematics

(i) If y = f^xh,then d2 y = d c dy m.
(ii) If x = f^t h and y dx2 dx dx d2 y
Remarks dx2 dy gl]t g
= g^t h, then = d c dx m dt = d ) f l]t g 3 $ dt
dt dx dt dx

Example 5.46

Find the second order derivative of the following functions with respect to x,

(i) 3 cos x + 4 sin x (ii) x = at2, y = 2at (iii) x sin x

Solution

(i) Let y = 3 cos x + 4 sin x

y1 = –3 sin x + 4 cos x
y2 = –3 cos x – 4 sin x
y2 = –3 (cos x + 4 sin x)
y2 = –y (or) y2 + y = 0

(ii) x = at2 , y = 2at

dx = 2at dy = 2a
dt dt

dy = ` dy j
dx dt

_ dx i
dt

= 1
t

Now, d2 y = d c dy m
dx2 dx dx

= d c dy m dt
dt dx dx

= d b 1 l $ 21at
dt t

= - 1 $ 21at
t2

= - 1
2at3

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(iii) y = x sin x

dy = x cos x + sin x
dx

d2 y = –x sin x + cos x + cos x
dx2

= 2 cos x – x sin x

Example 5.47
If y = A sin x + B cos x , then prove that y2 + y = 0

Solution
y = A sin x + B cos x

y1 = A cos x – B sin x
y2 = –A sin x –B cos x
y2 = –y
y2 + y = 0

Exercise 5.9

1. Find y2 for the following functions

(i) y = e3x + 2 (ii) y = log x + ax (iii) x = a cos i, y = a sin i

2. If y = 500e7x + 600e-7x , then show that y2 - 49y = 0

3. If y = 2 + log x , then show that xy2 + y1 = 0

4. If y = a cos mx + b sin mx , then show that y2 + m2 y = 0
5. If y = ^x + 1 + x2hm , then show that ^1 + x2hy2 + xy1 - m2 y = 0

6. If y = sin^log xh , then show that x2 y2 + xy1 + y = 0 .

204 11th Std. Business Mathematics

Exercise 5.10

Choose the correct answer

1. If f^xh = x2 - x + 1then f^x + 1h is

(a) x2 (b) x (c) 1 (d) x2 + x + 1
(d) 7
2. Let f^xh = * x2 - 4x if x$ 2 then f(5) is (d) 0
x + 2 if x1 2

(a) –1 (b) 2 (c) 5

3. For f (x) = *x2 - 4x if x$2 then, f(0) is
x + 2 if x<2

(a) 2 (b) 5 (c) –1

4. If f^xh = 11 - x then f(–x) is equal to
+ x

(a ) –f(x) (b) 1 (c) - 1 (d) f(x)
f^xh f^xh

5. The graph of the line y = 3 is (b) Parallel to y-axis
(a) Parallel to x-axis (d) Perpendicular to x-axis
(c) Passing through the origin

6. The graph of y = 2x2 is passing through

(a) (0,0) (b) (2,1) (c) (2,0) (d) (0,2)

7. The graph of y = ex intersect the y-axis at

(a) (0, 0) (b) (1, 0) (c) (0, 1) (d) (1, 1)

8. The minimum value of the function f(x)=| x | is

(a) 0 (b) –1 (c) +1 (d) –∞

9. Which one of the following functions has the property f^xh = f` 1 j
x

(a) f^xh = x2 - 1 (b) f^xh = 1 - x2 (c) f(x) = x (d) f^xh = x2 + 1
x x x

10. If f^xh = 2x and g(x) = 1 then (fg)(x) is
2x
1
(a) 1 (b) 0 (c) 4x (d) 4x

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11. Which of the following function is neither even nor odd?

(a) f^xh = x3 + 5 (b) f^xh = x5 (c) f^xh = x10 (d) f^xh = x2

12. f^xh =- 5 , for all x ! R is a (b) modulus function
(a) an identity function (d) constant function
(c) exponential function

13. The range of f^xh = x , for all x ! R is

(a) (0, ∞) (b) [0, ∞) (c) (–∞, ∞) (d) [1, ∞)

14. The graph of f(x) = ex is identical to that of

(a) f^xh = ax,a 2 1 (b) f^xh = ax,a 1 1

(c) f^xh = ax, 0 1 a 1 1 (d) y = ax + b, a ! 0

15. If f^xh = x2 and g(x) = 2x+1 then (fg)(0) is

(a) 0 (b) 2 (c) 1 (d) 4
(d) θ
16. lim tan i = (d) 0
i (d) –1
i"0

(a) 1 (b) ∞ (c) –∞

17. lim ex - 1 =
x
x"0

(a) e (b) nxn-1 (c) 1

18. For what value of x, f^xh = x + 21 is not continuous?
x -

(a) –2 (b) 1 (c) 2

19. A function f(x) is continuous at x = a if lim f (x) is equal to
x"a
1
(a) f(–a) (b) f` a j (c) 2f(a) (d) f(a)

20. d ` 1 j is equal to
dx x
1 1 1
(a) - x2 (b) - x (c) log x (d) x2

206 11th Std. Business Mathematics

21. d ^5e x - 2 log xh is equal to
dx
2 1
(a) 5ex - x (b) 5ex - 2x (c) 5ex - x (d) 2 log x

22. If y = x and z = 1 then dy =
x dz
1
(a) x2 (b) 1 (c) –x2 (d) - x2

23. If y = e2x then d2 y at x = 0 is
dx2

(a) 4 (b) 9 (c) 2 (d) 0

24. If y = log x then y2 =

(a) 1 (b) - 1 (c) - 2 (d) e2
x x2 x2 (d) ax loge a

25. d ^a xh =
dx
1
(a) x loge a (b) aa (c) x loge a

Miscellaneous Problems

1. If f^xh = 2x1+ 1 , x !- 12 , then show that f(f(x)) = 22xx + 13 , provided x !- 32
+

2. Draw the graph of y = 9–x2

x- x if x ! 0
3. If f(x)= * x if x = 0 then show that lim does not exist.
f^xh
2 x"0

4. Evaluate: lim ^2x - 3h^ x - 1h
2x2 + x - 3
x"1

5. Show that the function f(x) = 2x–| x | is continuous ]\]]]]]]Z][a1(3t1--x-x=xx0) (2 - x) if x <1 at
6. if 1#x #2
Verify the continuity and differentiability of f(x) = if x>2
x = 1 and x = 2

7. If xy = yx , then prove that dy = y d x log y - y n
dx x y log x - x

8. If xy2 = 1, then prove that 2 dy + y3 =0.
dx

9. If y =tanx, then prove that y2 - 2yy1 = 0.

10. If y = 2 sin x + 3 cos x , then show that y2 + y = 0.

i e entia a cu us 207

Summary

z Let A and B be two non empty sets, then a function f from A to B,
associates every element of A to an unique element of B.

z lim f]xg exists + lim f]xg = lim f (x).
x"a x " a- x " a+

z For the function f(x) and a real number a, lim f]xg and f(a) may not be the same.
x"a

z A function f(x) is a continuous function at x = a if only if lim f]xg = f(a).
x"a

z A function f(x) is continuous, if it is continuous at every point of its domain.

z If f(x) and g(x) are continuous on gth!e0irtchoemn mgfonisdaolsmoacinon, tthineunouf s±. g, f $ g, kf (k is
a constant) are continuous and if

z A function f(x) is differentiable at x = c if and only if lim f^xh - f (c) exists finitely
and denoted by f l (c) . x-c
x"c

z L [f l (c)] = lim f (c - h) - f (c) and R [f l (c)] = lim f^c + hh - f (c)
-h h
h"0 h"0

z A function f(x) is said to be differentiable at x = c if and only if

L [f l (c)] = R [f l (c)].

z Every differentiable function is continuous but, the converse is not necessarily
true.

z If y = f(x) then d c dy m is called second order derivative of y with respect to x.
dx dx

z If x = f^t h and y = g(t) then d2 y = d ( gl^t h 2 (or) d2 y = d ( g l^ t h 2 $ dt
dx2 dx f l^t h dx2 dt f l^ t h dx

208 11th Std. Business Mathematics

Absolute constants GLOSSARY
Algebric functions
Arbitrary constants முழுமையான ைாறிலிகள்
Chain rule இயறகணித ோர்புகள்
Closed interval தனனிசமே ைாறிலிகள்
Constant ேஙகிலி விதி
Continuous மூடிய இம்டவவளி
Dependent variable ைாறிலி
Derivative வதா்டர்சேி
Domain ோர்்நத ைாறி
Explict வமகயீடு
Exponential ோர்பகம் / அரஙகம்
Function வவளிபடு
Identity அடுக்கு
Implict ோர்பு
Independent variable ேைனி
Interval உடபடு
Le limit ோரா ைாறி
Limit இம்டவவளி
Logarithmic இ்டக்மக எல்மை
Modulus எல்மை
Neighbourhood ை்டக்மக
Open interval ைடடு
Parametric functions அண்மையகம்
Range திற்நத இம்டவவளி
Right limit துமணயைகு ோர்புகள்
Signum function வீசேகம்
Successive di erentiation வைக்மக எல்மை
Transcendental functions குறிச ோர்பு
Variable வதா்டர் வமகயி்டல்
விஞேிய ோர்புகள்

ைாறி

i e entia a cu us 209

ANSWERS

1. MATRICES AND DETERMINANTS

Exercise 1.1

1.(i) M11 =- 1 M12 = 0 M21 = 20 M22 = 5

A11 =- 1 A12 = 0 A21 =- 20 A22 = 5

(ii) M11 =- 12 M12 = 2 M13 = 23 M21 =- 16 M22 =- 4 M23 = 14

M31 =- 4 M32 =- 6 M33 = 11

A11 =- 12 A12 =- 2 A13 = 23 A21 = 16 A22 =- 4 A23 =- 14

A31 =- 4 A32 = 6 A33 = 11 4. –30 5. 123 , 2 6. 0
2. 20 3. 12

Exercise 1.2

1. 4 -3 2. RTSSSSSSSS--711 -3 -103WWWWWWWWXV 3.(i) 15 =-32 11G (ii) 110 3 -1
(iii) >-1 2 H 10. 1 >1 3H
0
7. 11910SSSSRSSTSS330SSSSSTRSSS1000-321-5010--317WWWWWWXVWW-224WWWWWVWWXW m = 74 (iv) 1511 SSSSRSSSST--51232 - 46 ---11719WWWXWWWVWW
14
- 17

11. p = 2, q =- 3

Exercise 1.3

1. x = 1, y = 1 2.(i) x = 3, y =- 2, z = 1

(ii) x =- 1, y = 2, z = 3 (iii) x = 1, y =- 1, z = 2 3. 17.95, 43.08, 103.85
6. 700, 600, 300
4. 3000, 1000, 2000 5. 13,2,5

210 11th Std. Business Mathematics

Exercise 1.4
1. It is viable 2. It is not viable 3. It is viable
4. A = 250.36 tonnes B = 62.44 tonnes 5. 181.62, 84.32
6. 34.16, 17.31, 7. 42 and 78

Exercise 1.5

1 2 3 4 5 6 7 8 9 10 11 12 13
(b) (d) (b) (b) (c) (c) (c) (c) (c) (d) (c) (b) (a)
14 15 16 17 18 19 20 21 22 23 24 25
(c) (c) (c) (b) (a) (d) (b) (b) (d) (d) (b) (a)

Miscellaneous Problems
6. RTSSSSSSSS502 -301XVWWWWWWWW
1. x = 3. x =- 1 2. 0 0 9. x = 20, y = 30, z = 50
7. 15 =-32 11G 1
1

8. x = 1, y = 2, z = 3

10. `1200 Crores, `1600 Crores

2. ALGEBRA

Exercise 2.1

1. 13 2 - 10 1 2. x 3 2 + x 1 1
x- x- - +

3. 1 1g - 1 2g - 1 2g2 4. 1 1g - 1 1g
9]x - 9]x + 3]x + 2]x - 2]x +

5. -4 2g + 4 1g - 1 1g2 6. x 2 2 + ]x 3 2g2 - ]x 9
9]x + 9]x - 3]x - - - - 2g2

7. -2x7 + 1 + 9 2) 8. 1 2) + 4 (x - 2)
x2 2 (x + 5 (x + 5 (x2 + 1)

9. 3 1) - 3 3) + 4 (x 1 3) 2 10. 5 (1x2-+x4) + 1 1)
16 (x - 16 (x + + 5 (x +

1. 64 2. 10 Exercise 2.2 4. 336 5. 85
3. 3

Exercise 2.3

1. 6 2. 14400 3. 1680 4. 4! 31!32!! 2!
5. (a) 72! (b) 7!
6. Rank of the word CHAT = 9.

A nsw ers 211

Exercise 2.4

1. 4 2. 8C4 + 8C3 = 9C4 = 126 3. 210 cards
4. 20
8. 11 5. 25200 6. 15 7. 13! # 9! # 9!
10.(i) 186 7! 6!

9.(i) 1365 (ii) 1001 (iii) 364

(ii) 186

Exercise 2.6

1.(i) 16a4 - 96a3 b + 216a2 b2 - 216ab3 + 81b4

(ii) x7 + 7x6 + 21yx2 5 + 35yx3 4 + 35yx4 3 + 21yx5 2 + 7x + 1
y y6 y7

(iii) x6 + 6x3 + 15 + 20 + 15 + 6 + 1
x3 x6 x9 x12

2.(i) 10, 40,60,401 (ii) 995009990004999 3. 11440 x9 y4

4.(i) 11C5 x , 11C5 (ii) 8C14 (681) x12 (iii) -10C5 ^6h5
x x5

5.(i) 9C6 26 (ii) -32 (15C5) (iii) 7920.
36

Exercise 2.7

1 2 3 4 5 6 7 8 9 10 11 12 13
(d) (c) (d) (c) (a) (c) (d) (b) (b) (c) (b) (d) (b)
14 15 16 17 18 19 20 21 22 23 24 25
(a) (c) (a) (c) (a) (c) (a) (b) (c) (d) (b) (a)

Miscellaneous Problems

1. x 3 1 + x 2 3 2. x 3 1 - x 2 2
- + - -

3. x 1 2 + x 5 3 - ]x 3 3g2 4. 3 2 + x22-x - 1 1
+ - - x- x +

5.(i) 7 (ii) 336 (iii) 84

6.(a) 720 (b) 7776 (c) 120 (d) 6

7. 74 ways 8. 85 ways 9. 210 10. 544

212 11th Std. Business Mathematics

3. ANALYTICAL GEOMETRY

Exercise 3.1

1. x2 - 2x - 6y + 10 = 0 2. x2 + y2 - 6x + 4y - 3 = 0
3. 3x2 + 3y2 - 4x - 14y + 15 = 0 4. b 125 , 0l

5. 2x - 3y + 21 = 0

Exercise 3.2

1. Angle between the line is 45c 2. 4 units

4. a = 5 5. y = 1500x + 100000 , cost of 95 TV sets ` 2,42,500

Exercise 3.3

1. a = 6, c = 6 2. 2x - y + 2 = 0 and 6x - 2y + 1 = 0

3. 2x + 3y - 1 = 0 and 2x + 3y - 2 = 0 4. i = tan-1 (7)

Exercise 3.4

1.(i) x2 + y2 - 6x - 10y + 9 = 0 (ii) x2 + y2 - 4 = 0

2.(i) C(0, 0 ), r = 4 (ii) C(11 , 2 ), r = 10
(iii) Cb -52 , 54 l, r = 2
(iv) Cb 32 , 32 l, r = 5
3. x2 + y2 + 6x + 4y - 3 = 0 2
5. x2 + y2 - 5x - 2y + 1 = 0
7. x2 + y2 - 16x + 4y - 32 = 0 4. x2 + y2 - 4x - 6y + 13 = 0
9. x2 + y2 = 9
6. x2 + y2 - x - y = 0

8. x2 + y2 - 2x - 12y + 27 = 0

1. x + 2 = 0 Exercise 3.5
3. 20 units
2. R lies inside, P lies outside and Q lies on the circle.
4. p =! 20.

Exercise 3.6

1. 9x2 + 16y2 + 24xy + 34x + 112y + 121 = 0
2. k = 14 , Length of latus rectum = 1, Focus Fb 14 , 0l

A nsw ers 213

3. Vertex Focus Equation of Length of
Axis V(1, 4) F(3, 4) directrix latus rectum
x+1=0
y=4 4 units

4. Axis Vertex Focus Equation of Length of
y=0 V(0, 0) F(5, 0) directrix latusrectum
Problem x=0 V(0, 0) F(0, 2)
x=0 V(0, 0) F(0, –4) x=–5 20 units
(a)
y2 = 20x y = –2 8 units

(b) y=4 16 units
x2 = 8y

(c)
x2 =- 16y

5. (x - 15)2 = 5 (y - 55). e output and the average cost at the vertex are 15 kg. and ` 55.

6. when x = 5 months

Exercise 3.7

1 2 3 4 5 6 7 8 9 10 11 12 13
(b) (c) (c) (c) (b) (c) (a) (c) (a) (b) (c) (a) (d)
14 15 16 17 18 19 20 21 22 23 24 25
(a) (a) (d) (a) (c) (b) (b) (a) (d) (b) (b) (b)

Miscellaneous problems

1. 2x + y - 7 = 0 2. y = 6x + 3000 4. p = 1, p = 2 6. a = 9, b = 8

7. (–1, –2) is on the line, (1, 0) is above the line and (–3, –4) is below the line

8. (–2, –7) 9. y2 =- 92 x

10. Axis : y = 2, Vertex : (1, 2), Focus : (2, 2),
Equation of directrix : x = 0 and Length of latus rectum : 4 units

4. TRIGONOMETRY

1.(i) r (ii) 56r Exercise 4.1 (iv) -196r
3 (iii) 43r

214 11th Std. Business Mathematics

2.(i) 22c30l (ii) 405c30l (iii) -171c48l (iv) 110c

3.(i) 1st quadrant (ii) 3rd quadrant (iii) 2nd quadrant.

4.(i) -3 (ii) -21 (iii) 2 (iv) 1
2 10. -27 3

(v) 2

Exercise 4.2

1.(i) 22 (ii) -e 3 + 1 o (iii) 3 -1
3 -1 2 2 3 +1

2.(i) sin 92c (ii) 3 (iii) cos 80c (iv) 3
3.(i) -6353 2 (iii) 1363 14. 2
(ii) -6156 2 -1

6. 121 , a + b lies in 1st quadrant. 7. ! 2 - 1

9.(i) 193 (ii) - 2812987 10. - 14245 , - 14147

Exercise 4.3

1.(i) 12 bcos A4 - cos A2 l (ii) 12 b- sin 2A + 3 l
(iii) 12 bsin 4A - sin 23A l 2
2.(i) 2 sin 32A cos A2
(iv) 12 ]cos 7i + cos ig
(iii) 2 sin 2i cos 4i
(ii) 2 cos 3A cos A

(iv) 2 sin 32i sin i
2

Exercise 4.4

1.(i) -r (ii) -r (iii) r (iv) r
6 4 6 4
4. x = 16 5. x = 12
6.(i) 54 1 7. -6257
(ii) 10 10. r + 2x
4

Exercise 4.5

1 2 3 4 5 6 7 8 9 10 11 12 13
(b) (a) (c) (b) (b) (d) (a) (d) (a) (c) (c) (d) (b)
14 15 16 17 18 19 20 21 22 23 24 25
(c) (b) (a) (c) (b) (c) (c) (b) (c) (a) (b) (d)

A nsw ers 215

Miscellaneous Problems

4. 3 and -1 6.(i) 6+ 2 (ii) 2 – 3
10 10 4

7. 15 + 2 2 9. 5363 10. r - x
12 4

5. DIFFERENTIAL CALCULUS

1.(i) Odd function Exercise 5.1
(ii) Even function

(iii) Neither even nor odd function (iv) Even function

(v) Neither even nor odd function 2. k = 0 5. 13 - x , 3xx-+11
+ x

6.(i) e (ii) 0 (iii) 3e (iv) 0

7.(i) y (0, 16) (ii) y

y=2–x, x<2
(–1, 15) (1, 15) (–2, 4)
(–1, 3)
(–2, 12) (2, 12) (0, 2) y=x-2, x>2 (5, 3)
(4, 2)
(–1, 1)
(–3, 17) (3, 17) (3, 1)

y=16–x2 x′ (2, 0) x

(–4, 0) (4, 0) y′
x′ x

(iii) y′ (iv) y x
y
x′ x′ y=e2x
(–1, –1) (3, 9)
y=x2, x>0 (0, 1)
(–2, –4) (2, 4) O
y=–x2, x<0 (1, 1) x y′
(–3, –9) (0, 0)

y′

216 11th Std. Business Mathematics

(v) y (vi) y

x′ y=e –2x y=1, x>0

(0, 1) x ′ y=0, x=0 x
xO
O
y=–1, x<0
y′

1.(i) 130 (ii) 0 Exercise 5.2 y′
(v) 1156 a- 214 (vi) 9 (iii) 12
(iv) 1
2. a = ± 1 3. n = 7 4. 258 5. f(–2) = 0

Exercise 5.3
1.(a) Not a continuous function at x = 2 (b) continuous function at x = 3

Exercise 5.4

(i) 2x (ii) -e-x (iii) x 1 1
+

Exercise 5.5

1.(i) 12x3 - 6x2 + 1 (ii) - 20 + 6 - 5 (iii) 1 -3 1 + ex
x5 x4 x2 2x 3 x4

(iv) - 3 +x2x2 (v) x2ex ^x + 3h (vi) 3x2 - 4x - 1

(vii) 4x3 - 3 cos x - sin x (viii) 1 - 1
x2
2^x2 + 1h
2.(i) xex (ii) ^x2 - x + 1h2 (iii) ex
(1 + x)2 ^1 + exh2

3.(i) x cos x + sin x (ii) ex ^sin x + cos xh

(iii) ex `1 + 1 + x + log xj (iv) cos 2x (v) exx2 ^x + 3h
x

4.(i) sin 2x (ii) - sin 2x (iii) -23 cos x sin 2x (iv) x
1 + x2

(v) n^ax2 + bx + chn - 1 ^2ax + bh (vi) 2x^cos^x2hh (vii) -x
^1 + x2h 1 + x2

A nsw ers 217

Exercise 5.6

1.(i) - y (ii) y - 2x (iii) – (x2 + ay) 3. 34 c 1 - 4x + 3y m
x 2y - x (y2 + ay) 1 + 4x - 3x

1.(i) xsin x 8 sin x + cos x log xB Exercise 5.7
x (ii) ^sin xhx 6x cot x + log^sin xh@

(iii) ^sin xhtanx 61 + sec2 x log^sin xh@

(iv) 12 ^x - 1h^x - 2h $ x 1 1 + 1 2 - 1 3 - x22+x + 1 1 .
^x - 3h^x2 + x + 1h - x- x- x +

1.(i) - 1 (ii) t cost Exercise 5.8 (iv) cot i
t2 3. sin2x2x (iii) - tan i 2

2. – tan x

Exercise 5.9

1.(i) 9y (ii) - 1 + ax ^log ah2 (iii) - 1 cosec3 i
x2 a

Exercise: 5.10

1 2 3 4 5 6 7 8 9 10 11 12 13
(d) (c) (a) (b) (a) (a) (c) (a) (d) (a) (a) (d) (b)
14 15 16 17 18 19 20 21 22 23 24 25
(a) (a) (a) (c) (b) (d) (a) (a) (c) (a) (b) (d)

Miscellaneous Problems

2.

y (0, 9)

(–1, 8) (1, 8) 4. - 110
6.(i) continuous at
(–2, 5) (2, 5) = 1; differentiable at x = 1

(–3, 0) y=9–x2 (3, 0) (ii) continuous at x = 2; not differentiable at x = 2
x′ x

y′

218 11th Std. Business Mathematics

Absolute constants GLOSSARY
Adjoint Matrix
Algebric functions முழுமையான ைாறிலிகள்
Allied angle சேர்ப்பு அணி
Analysis இயறகணித ோர்புகள்
Angle துமணக் சகாணஙகள்
Arbitrary constants பகுப்பாய்வு
Binomial சகாணம்
Centre தனனிசமே ைாறிலிகள்
Chain rule ஈருறுப்பு
Chord மையம்
Circle ேஙகிலி விதி
Circular Permutation நாண்
Closed interval வட்டம்
Coe cient வட்ட வாிமே ைாறறம்
Cofactor மூடிய இம்டவவளி
Combination வகழு
Componendo and dividendo. இமணக் காரணி
Compound angle சேர்வு
Concurrent Line கூட்டல் ைறறும் கழிததல் விகித ேைம்
Conics கூடடுக் சகாணம்
Constant ஒரு புள்ளி வழிக் சகாடு்
Continuous கூம்பு வவடடிகள்
Degree measure ைாறிலி
Dependent variable வதா்டர்சேி
Derivative பாமக அளமவ
Determinant ோர்்நத ைாறி
Diameter வமகயீடு
Digonal Matrix அணிக்சகாமவ
Directix விட்டம்
Domain மூமை விட்ட அணி
Equation இயக்குவமர
Explict ோர்பகம் / அரஙகம்
Exponential ேைனபாடு
Factorial வவளிபடு
Focal distance அடுக்கு
Focus காரணீயப் வபருக்கம்
Function குவியததூரம்
Identity குவியம்
Implict ோர்பு
Independent term ேைனி (அல்ைது) முறவறாருமை
Independent variable உடபடு
Input ோரா உறுப்பு
Interval ோரா ைாறி
Inverse function உள்ளீடு
Inverse Matrix இம்டவவளி
Latus rectum சநர்ைாறு ோர்பு
Le limit சநர்ைாறு அணி
Length of the arc. வேவவகைம்
இ்டக்மக எல்மை
வில்லின நீளம் 219

G l ossa ry

Length of the tangent வதாடுசகாடடின நீளம்
Limit எல்மை
Linear factor ஒரு படிக்காரணி
Locus நியைப்பாமத அல்ைது இயஙகுவமர
Logarithmic ை்டக்மக
Mathematical Induction கணிதத வதாகுததறிதல்
Middle term மைய உறுப்பு
Minors ேிறறணிக் சகாமவகள்
Modulus ைடடு
Multiple angle ை்டஙகு சகாணம்
Multiplication Principle of counting எண்ணுதலின வபருக்கல் வகாள்மக
Neighborhood அண்மையகம்
Non-Singular Matrix பூசேியைறறக் சகாமவ அணி
Open interval திற்நத இம்டவவளி
Origin ஆதி
Output வவளியீடு
Pair of straight line இரடம்ட சநர்சகாடு
Parabola பரவமளயம்
Parallel line இமண சகாடு
Parameter துமணயைகு
Parametric functions துமணயைகு ோர்புகள்
Partial fraction பகுதிப் பினனஙகள்
Pascals’s Triangle பாஸகலின முக்சகாணம்
Permutations வாிமே ைாறறஙகள்
Perpendicular line வேஙகுதது சகாடு
Point of concurrency ஒருஙகிமணவுப் புள்ளி
Point of intersection வவடடும் புள்ளி
Principle of counting எண்ணுதலின வகாள்மக
Quadrants கால் பகுதிகள்
Radian measure சரடியன அளவு / ஆமரயன அளவு
Radius ஆரம்
Range வீசேகம் / வீசசு
Rational Expression விகித முறு சகாமவ
Right limit வைக்மக எல்மை
Scalar திமேயிலி
Signum function குறிச ோர்பு
Singular Matrix பூசேியக் சகாமவ அணி
Straight line சநர்க் சகாடு
Successive di erentiation வதா்டர் வமகயி்டல்
Tangent வதாடுசகாடு
Transcendental functions விஞேிய ோர்புகள்
Transformation formulae உருைாறறு சூததிரஙகள்
Transpose of a Matrix நிமர நிரல் ைாறறு அணி
Triangular Matrix முக்சகாண அணி
Trignometric identities திாிசகாணைிதி முறவறாருமைகள்
Trignometry Ratios திாிசகாணைிதி விகிதஙகள்
Variable ைாறி
Vertex முமன

220 11th Std. Business Mathematics

Books for Reference

1. Introduction to Matrices, S.P.Gupta , S.Chand & Company
2. Matrices, Shanthi Narayanan,S.Chand & Company
3. Matrices and Determinants,P.N.Arora, S.Chand & Company
4. Topics in Algebra, I.N.Herstein, Vikas Publishing Company
5. Algebra A Complete Course , R.D.Sharma, Sultan Chand & Sons
6. Analytical Geometry, T.K.Manicavachagon Pillay, S.Narayanan, S.Viswanathan Publishers
7. Analytical Geometry,P.K.Mittal, Shanthi Narayanan, Durai Pandiyan, S.Chand & Company
8. Trigonometry, R.D.Sharma, Sulatan Chand & Sons
9. A Text Book of Trigonometry, M.D Raisingania and Aggarwal
10. Trigonometry, D.C.Sharma, V.K.Kapoor, Sulatan Chand & Sons
11. Trignonometry, S.Arumugam , S.Narayanan, T.K.Manicavachagon Pillay, New Gama

Publications, S.Viswanathan Printers and Publishers Pvt. Ltd.
12. Calculus, Mohamd Arif, S.Narayanan, T.K.Manicavachagon Pillay, S.Viswanathan Printers

and Publishers Pvt. Ltd.
13. Di erential and Integral Calculus, N.Piskunov, Mir Publishers, Moscow
14. Di erential and Integral Calculus,Schamum’s Outline Series,Frank Ayres
15. Calculus (Volume I & II ), Tom.M.Apostol, John Wiley Publications
16. Calculus : An Historical Approach, W.M, Priestly (Springer)
17. Calculus with Analytic Geometry (Second Edition) George F.Simmons, e Mcgraw Hill
18. Application of Di erentiation, S.Narayanan, T.K.Manicavachagon Pillay, , S.Viswanathan

Printers and Publishers Pvt. Ltd.
19. Application of Di erentiation,P.N.Arora, S.Arora, S.Chand & Company
20. Financial Mathematics, O.P.Malhotra, S.K.Gupta, Anubhuti Gangal, S.Chand & Company
21. Financial Mathematics ,Kashyap Trivedi, Chirag Trivedi, Pearson India Education Services

Pvt. Ltd
22. Descriptive Statistics, Richard I.Levin, David S.Rubin, Prentice Hall Inc,Englewood,

N.J.U.S.A
23. Statistical Methods, S.K.Gupta, Prentice Hall Inc,Englewood, N.J.U.S.A
24. Descriptive Statistics, Anderson, Sweenas, Williams, Library of Congress Cataloging in

Publication Data
25. Correlation and Regression Analysis, Dr.S.P.Gupta, P.K. Gupta, Dr.Manmohan, Sultan

Chand & Sons
26. Correlation and Regression Analysis, John.S.Croucher, Mc Graw-Hill Australia Pvt Limited
27. Operations Research, Dr.S.P.Gupta, P.K. Gupta, Dr.Manmohan, Sultan Chand & Sons
28. Operations Research, A.Ravindran, James J.Solberg, Willey Student Edition
29. Operations Research, Nita H.Shah, Ravi.M.Gor, Hardik Soni, Kindle Edition
30. Operations Research, Frederick S.Hilton, Gerald J.Lieberman,Mc Graw Hill Education
31. Business Mathematics, HSC First & Second Year, Tamilnadu Text Book Corporation,

Reprint 2017
32. Mathematics, HSC First & Second Year, Tamilnadu Text Book Corporation, Reprint 2017

Books for Reference 221

Business Mathematics and Statistics -Higher Secondary First Year
List of Authors and Reviewers

Chairperson Authors
Mr.N. Ramesh, Mr.T.P. Swaminathan
Associate Professor (Retd), Post Graduate Teacher
Department of Mathematics, MMA Govt. Hr. Sec. School
Government Arts College(Men), Nandanam, Pallavaram, Chennai-43
Chennai-600 0035
Reviewers Mr.H. Venkatesh,
Dr. M.R.Sreenivasan Post Graduate Teacher,
Professor & HOD Sir Ramaswami Mudaliar Hr. Sec. School,
Department of Statistics Ambattur, Chennai 600 053.
University of Madras,Chennai-600 005
Dr. D.Arivudainambi Mr. B. Mariappan
Associate Professor Post Graduate Teacher
Department of Mathematics Arignar Anna MPL Boys Hr Sec. School,
Anna University,Chennai-25 Chengalpattu, Kanchipuram- Dist
Content Experts
Dr. Venu Prakash, Mr.S.R. Mohamed Mohindeen Sulaiman
Associate Professor & HOD, Post Graduate Teacher
Department of Statistics MMA Govt. Hr. Sec. School
Presidency College, Chennai-600 005. Pallavaram, Chennai-43
Dr. R. irumalaisamy
Associate Professor, Mr.T. Raja Sekar
Department of Mathematics, Post Graduate Teacher
Government Arts College(Men), Govt. Boys Hr. Sec. School,
Nandanam, Chennai-600 0035 Chrompet, Chennai-44
Dr. S. J. Venkatesan
Associate Professor, Mrs.A. Suganya
Department of Mathematics, Post Graduate Teacher
Government Arts College(Men), GHSS, Kovilambakkam,
Nandanam, Chennai-600 0035 Kancheepuram District
Mrs. M. ilagam
Assistant Professor, Mr.V. Ganesan
Department of Statistics Post Graduate Teacher
Presidency College, Chennai-600 005. Got. Boys Hr. Sec. School
Text Book Group In-Charge Nanganallur, Chennai-114
Mr.A. Ravi Kumar Content Readers
Deputy Director Mr. James Kulandairaj
State Council of Educational Post Graduate Teacher
Research and Training, St. Joseph HSS, Chenglepattu,
Chennai-600 006 Kanchipuram Dist.
Academic Co-ordinator
Mr. S. Babu Mrs. Beaulah Sugunascely
Assistant Professor Post Graduate Teacher
State Council of Educational P.G.Carley HSS, Tambaram,
Research and Training, Chennai-600 006 Kanchipuram Dist.
Mrs.S. Subhashini
Art and Design Team Post Graduate Teacher
GGHSS, Kundrathur,
CanhdieCf rCeoat-iovredHineaatodr Kanchipuram Dist.
Mr.K. Saravanan
Srinivasan Natarajan Grace Mat. Hr. Sec. School,
Porur, Chennai-116.
ICT Co-ordinator
Mr. D. Vasuraj
BT Assistant (Mathematics),
PUMS, Kosapur, Puzhal Block , Tiruvallur District

Layout Designer is book has been printed on 80 G.S.M.
Elegant Maplitho paper.
Joy Graphics, Printed by o set at:
Chennai

In House QC
QC - Gopu Rasuvel

-Tamilkumaran
- Jerald wilson
- Ragu

Co-ordination
Ramesh Munisamy

Typing
P. ulasi

NOTES

NOTES