Final-for-print-Illustrated-OPT-Math-9-2076-2-1

14. (a) If the midpoint of the line segment joining the points (a, b) and (–3, 4) is (5, 1), find the values
of a and b.

(b) Find the coordinates of two points which divide the line joining the points (1, 1) and (10, –2) into
three equal parts.

(c) Find the centroid of ΔABC having the vertices A(1, 3), B(–2, 5) and C(–2, 1).

15. (a) Prove that the median drawn from the base points to the opposite sides of an isosceles triangle

with vertices (–3, –1), (1, 3) and (5, 5) are equal in length.

(b) Find the length of perpendicular drawn from the vertex A(0, 7) of the isosceles triangle ABC
with vertices A(0, 7), B(2, 1) and C(6, 5).

16. (a) Show that the perpendicular drawn from each vertex to opposite side of an equilateral triangle

PQR with vertices (3, 3), (–3, –3) and (–3 3 , 3 3 ) are equal in length.

(b) The points (3, –1), (7, 3) and (–1, 3) are the vertices of right-angled triangle in which the right
angle is at (3, –1). Prove that the midpoint of the hypotenuse of the right-angled triangle is
equidistant from its vertices.

17. (a) Find the value of a if a point (2a, a) lies on the locus of x2 + y2 + 2x + 4y = 4.

(b) If the point (4, 2) lies on the locus having the equation x2 + y2 + ax + 2ay – 4 = 0, then show
that another point (1, 5) also lies on the same locus.

18. (a) If the straight lines x – y + 2 = 0, y = 4 – x and ax + 5y = 14 are concurrent, find the value of a.
(b) In which points do the locii with the equations x + y = 4 and x2 + y2 – 2x = 8 intersect?
19. Find the equation of locus of a point which moves such that its distance from,

(a) y-axis is always 11 units. (b) Its ordinate is always 4.
20. Find the equation of the locus of a point so that,

(a) it is parallel to x-axis and passes through (–2, 3). (b) its ordinate is thrice its abscissa.

21. Find the equation of a locus of a point which always moves,

(a) at a distance of 4 units from the origin. (b) at a distance of 5 units from the point (3, –4).

22. Find the equation of a locus of a point which moves so that,
(a) its distance from y-axis is double of its distance from x- axis.
(b) its distance from x-axis is four times of its distance from y- axis.

23. Find the equation of a locus of a point which moves such that the distance from,

(a) the point (1, –2) is one-third the distance from the point (0, 3).

(b) the point (0, – b) is twice its distance from the y-axis.

24. (a) A(1, –2) and B(–3, 0) are the fixed points. Find the equation of the locus of the moving point

P (x, y) such that,

(i) 2PA = 3PB (ii) PA2 = 1 (iii) PA2 + PB2 = 12 (iv) PA2 + PB2 = AB2
2
(b) Verify whether the point (1, 3) lies on the equation of a point which moves such that its distance

1 more than 4 times the distance from the x-axis is equal to its distance from the y-axis.

25. (a) Find the slope of a straight line which makes an angle of 30o with x-axis in anti-clockwise
direction.

(b) Find the slope of a straight line which makes an angle of 45o with positive y-axis in clockwise
direction.

26. Find the slope of the line segment joining the origin and the following points,

(a) origin and (– 3, 2). (b) (– 3, – 2) and (5, 2).

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27. (a) If the slope of the line passing through the points (4, 5) and (– 2, k) is 2, find the value of k.

(b) If the slope of the line passing through the points (– 2, 1) and (3a, a) is 1, find the value of a.

28. (a) Prove that the following points are collinear A(– 4, 1), B(– 2, 2) and C(4, 5).

(b) If the points (a, 0), (0, (b) and (2, 2) are collinear, prove that 1 + 1 = 1 .
a b 2
29. Find the equation of the straight line with,

(a) the slope (m) = 1 and the y-intercept (c) = – 1.
3
(b) the angle with positive y-axis (θ) = 45o and the y-intercept (c) = 7.

30. (a) Find the equation of the straight lines passing through the origin and making angle 30o with x-axis.

(b) Find the equation of the straight lines parallel to x-axis and y-intercept (c) = 3.

31. Find the equation of the straight lines for the following conditions:

(a) Making an angle of 60o with x-axis and passing through (2, –1).

(b) Find the equation of the straight line having the y-intercept – 3 units passing through the point
(3, 2).

32. (a) Find the equation of the straight line that cuts equal intercepts on the positive axes and passes
through the point (3, 4).

(b) Find the equation of the straight line passing through the point (–3, 1) and cutting equal
intercepts on the axes but in opposite sign.

33. (a) Find the equation of a straight line with the x-intercept = 2 and y-intercept = – 4.

(b) If a straight line cuts the x-axis and y-axis at the points (–3, 0) and (0, 2), find the equation of the line.

(c) Find the equation of the straight line passing through the points (2, 0) and (0, 3) and prove that
the line also passes the point (4, –3).

34. (a) The point (4, –3) bisects the line segment intercepted between the axes. Find the equation of
the line.

(b) The point P(6, –2) divides the line segment AB intercepted between the axes at A and B
respectively in the ratio 2:3. Find the equation of the line AB.

35. (a) Find the equation of the line which passes through the point (–2, 5) and cuts an intercept on the
x-axis twice as long as the intercepted on the y-axis in the magnitude.

(b) Find the equation of the straight line that passes through the point (1, 3) and the sum of whose

intercepts on the axes is 15. Y

36. (a) The length of the perpendicular drawn from the origin on a straight A 3O X
line is 4 units and the perpendicular line is inclined to the x-axis X' 30o
at 30o. Find the equation of the line and prove that the line passes

through the point ( 3, 5). B
(b) Find the equation of the straight line AB from the given figure. Y'

37. (a) If the perpendicular distance of a straight line from the origin is 2 2 units and the slope of the

perpendicular is – 3, find the equation of the line.

(b) Find the equation of a straight line such that the perpendicular distance from the origin to the

straight line is 3 units and the angle made by the perpendicular with x-axis is tan–1( 3 ).

38. (a) Reduce the equation 1 x+ 3 y + 3 = 0 into the slope intercept form.
3 4

(b) Reduce the equation 5x – 2y + 4 = 0 into the double-intercepts form.

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39. (a) Find the x-intercept, y-intercept and the angle with x-axis made by the line having the equation
2x – y + 3 = 0.

(b) Calculate the length of the perpendicular distance from the origin to the straight line

2x – y + 3 = 0. Also find the angle made by the perpendicular with x-axis.

40. (a) Reduce ax + by = 1 into the normal form and prove that a2 + b2 = 1 .
p-2

(b) If the length of perpendicular line drawn from (1, 1) to the line ax + by – c = 0, prove that:

2 + 2 – 1 = 2 .
a b ab c

41. (a) Find the perpendicular distance from the point (2, 5) to the line 2x + 3y – 2 = 0.

(b) Find the distance between the following parallel lines x + 3y = 4 and 2x + 6y = 2.

42. Find the equation of the line passing through the point,

(a) (2, 4) and having inclination 45o. (b) (3, –2) and having the slope 3 .
2
43. (a) Find the equation of the line through the point (2, 3) and having inclination of 60o. Also,

show that the line passes through the point (1, 0).

(b) Find the equation of the line segment passing through the midpoint of the line segment joining
the points (– 2, 3) and (8, 5) and making an angle of 30o with x-axis.

44. Find the equation of the line passing through the following pairs of points:

(a) (2, 3) and (–2, 5) (b) (a + b, b) and (a, b – a)

45. Prove that the following points are collinear and write the equation of the line through them,

(a) (4, 3), (2, 2) and (–2, 0) (b) (3a, b), (–2a, 2b) and (0, b)

(c) Find the value of k if the points are collinear (4, 2), (1, k) and (–5 ,–1).

46. (a) Find the equation of the line passing through the point (–1, 3) and the point of intersection of
the lines x + 2y = 1 and 2x – y = 2.

(b) Find the equation of the line passing through the point (4, 3) and the centroid of triangle having
the vertices (2, 3), (0, 2) and (4, 1).

47. (a) A point P(p, q) lies on the line 6x – y = 1 and Q(q, p) lies on the line 2x – 5y = 5. Find the
equation of the line PQ and length of PQ.

(b) Prove that the line joining the points (4, –1) and (–2, –3) bisects the line segment joining the
point (–2, 0) and (4, –4).

48. Find the area of the triangle having the following vertices:

(a) (4, 1), (1, 4) and (–2, 3) (b) (a, b), (a + b, b) and (–b, a – b)

49. If the area of a triangle having the vertices (2a, 5), (a, 3) and (0, – a) is 1 square unit, find the value of a.

50. (a) The points A(6, 3), B(–3, 5), (4, –2) and P(x, y) are the vertices of a quadrilateral. Prove that:

∆ABC = x 7 – 2
∆PBC +y

11
(b) Prove that: p + q = 2, if the three points (p, 0), (0, q) and (2, 2) are collinear.

pp
(c) If P cos α , 0 , Q 0, sin α and R(x, y) are collinear, then show that xcos α + ysin α = p.

51. (a) Find the area of the quadrilateral whose vertices are (2, 2), (–3, 3), (–3, 0) and (3, –2).

(b) Find the area of the quadrilateral whose one vertex is origin and the other vertices are (2, 1),
(5, 4) and (0, 2).

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52. (a) If A, B and C are the midpoints of the sides LM, MN and LN of ∆LMN L

with vertices L(5, 4), M(1, 2) and N(3, 0) respectively, prove that:
1
Area of ∆ABC = 4 × Area of ∆LMN. AC

(b) If A(4, 1), B(6, 3), C(0, 3) and D(–2, 1) are the vertices of a parallelogram

ABCD prove that the area of ABCD = 2 × the area of ∆ABC. MB N

ANSWERS

1. (a) 5 units (b) 2 5 units (c) 2 5 units

2. (a) p2 – 2pq + 2q2 units (b) p2 + q2 units (c) p units

3. (a) 13 units 4. (a) ±4 (b) (5, 0), (–3, 0) (c) (0, 7) 10. (a) (0, 0) (b) 2, 3
13. (a) 4 2
11. (a) (2, 2) (b) (– 1, 2) 12. (a) 1 : 3 (b) 3 : 2
(b) (1, 6)

14. (a) 13, – 2 (b) 2, 0 , 1, – 1 (c) (–1, 3) 15. (b) 4 2 units 17. (a) – 2, 2 (b) – 2
18. (a) – 1 33 5
(b) (4, 0), (1, 3) 19. (a) – x – 11 = 0 (b) y – 4 = 0

20. (a) y – 3 = 0 (b) x – 3y = 0 21. (a) x2 + y2 = 16 (b) x2 + y2 – 6x + 8y = 0

22. (a) x – 2y = 0 (b) 4x – y = 0 23. (a) 4x2 + 4y2 – 9x + 21y + 18 = 0 (b) 3x2 – y2 – 2b – b2 = 0

24. (a) (i) 5x2 + 5y2 + 62x – 16y + 61 = 0 (ii) 2x2 + 2y2 – 4x + 8y + 9 = 0

(iii) x2 + y2 + 2x + 2y + 1 = 0 (iv) x2 + y2 + 2x + 2y + 3 = 0

25. (a) 1 (b) – 1 26. (a) – 2 (b) 1 27. (a) – 7 (b) – 3
3 3 2 2

30. (a) x – 3y – 3 = 0 (b) x – y + 7 = 0 31. (a) x – 3y = 0 (b) y – 3 = 0

32. (a) 3x – y – 1 – 2 3 = 0 (b) 5x – 3y – 9 = 0 33. (a) x + y = 7 (b) x – y + 2 = 0

34. (a) 2x – y – 4 = 0 (b) 2x – 3y + 6 = 0 (c) 3x + 2y = 6

35. (a) 3x – 4y – 24 = 0 (b) x – 2y – 10 = 0 36. (a) x + 2y – 8 = 0 (b) 2x + y – 10 = 0, 3x + 2y – 8 = 0

37. (a) 3x + y – 8 = 0 (b) x – 3y + 6 = 0 38. (a) x – y + 4 = 0 (b) x + 3y – 6 = 0

39. (a) y = – 4 x – 4 xy 40. (a) – 3 , 3, tan-1(2) (b) 3, 153.43o
9 (b) –4/5 + 2 = 1 25
17 units (b) 3 units
42. (a) 43. (a) x – y + 2 = 0 (b) 3x – 2y = 13
13 10

44. (a) 3x – y – 3 = 0 (b) x – 3y + 4 3 – 3 = 0 45. (a) x + 2y = 8 (b) ax – by = a2 + ab – b2

46. (a) x + 2y – 8 = 0 (b) ax – by – a2 + b2 – ab = 0 (c) 1

47. (a) 3x + 2y – 3 = 0 (b) x – 2y + 2 = 0 48. (a) x + y – 6 = 0, 4 2 units

49. (a) 6 sq. units (b) b (a – 2b) sq. units 50. 1 or – 2 52. (a) 37 sq. units (b) 13 sq. units
2 22

PROJECT WORK

Draw a parallelogram in a graph paper and identify the coordinates of the vertices and label it.
Prove the following properties:
(i) The opposite sides are equal.
(ii) The diagonals bisect each other.
(iii) Each diagonal bisects the parallelogram.
(iv) The area of the parallelogram is double of the area of the triangle divided by a diagonal.
(v) The areas of triangles obtained by intersecting both diagonals are equal.

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HOME ASSESSMENT - 5 (TRIGONOMETRY)

1. Convert the following angles into degree measure:

(a) 38° 35' (b) 48° 28' 45" (c) 80g (d) 40g 35 (e) 35g 42 37

2. Change the following angles into grade measure:

(a) 45g 39 (b) 28g 23 38 (c) 20° 30' (d) 49° 36' 56" (e) 25o 25''

3. Express the following angles into sexagesimal system:

(a) 29g (b) 28g 34 (c) 39g 45 (d) 28g 25 35

4. Express the following angles into centesimal system:

(a) 36° (b) 35° 25' (c) 65o 35'' (d) 75o 42' 56''

5. Change the following angles into radian measure:

(a) 36° (b) 75g (c) 420° 30' (d) 120g 45 60

6. Convert the following measure of angles into sexagesimal system. 8πᶜ
2πᶜ 4πᶜ 7πᶜ 9
(a) 3 (b) 5 (c) 6 (d)

7. (a) Divide 63° into two parts such that the ratio of their grade measure is 3:4.

(b) Find the ratio of 50g and 63o.

(c) The sum of two angles is 1c and that of difference is 1°. Find the circular measure of the smaller
angle.

(d) If the sum of angles in the ratio 5:7 is 240°, find the angles in degree and grade measures.

8. (a) If D is the number of degrees and G is the number of grades, prove that D = 9 .
G 10

(b) If α and β denote the number of sexagesimal and centesimal second of any angle respectively,

prove that: α:β = 81:250.

9. (a) If degree and radian measure of any angle are D and C respectively, prove that: D = 2C
90 π
(b) If G, D and R denote the number of grades, degrees and radian respectively of an angle, prove
G D 2R
that: 100 = 90 = π

10. (a) One angle of a right-angled triangle is 54°, find the third angle in grade.

(b) The difference of the acute angles of a right-angled triangle is 1π2ᶜ . Find the angles in degree measure.
(c) The difference of the acute angles of a right-angled triangle is 10g. Find the angles in radian measure.

11. (a) The angles of a triangles are 2x o 50x g 2πx c
3 27 135
, and . Express all of them in degree.

(b) One angle of a triangle is 40g. If the ratio of the remaining two angles is 1:2, find all angles of

the triangle in degree.

(c) The angles of triangle are in the ratio 2:3:4. Find the angles in radian.

12. (a) The angles of a triangle are in arithmetic sequence (AS or AP) and the greatest is double the
least. Express the angles in degrees.

(b) The number of grades in an angle of a triangle is to the number of degrees in the second angle

is to the number of radians in the third angle is in the ratio 280:288:π. Find the angles in degree.

13. (a) One angle of a quadrilateral is 3 of other and the two another angles are 66 2 grades and 3π
8 3 4
radians. Express the angles in degree.

(b) Find each angle of a pentagon in radian if its angles are in the ratio 2:3:4:5:6.

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14. (a) Find the interior angle of the regular heptagon in degree, grade and circular measures.

(b) Find the exterior angle of the regular octagon in degree, grade and radian measure.

15. (a) TThheeienxtetreiroiroarnagnlegilsefoofurretigmuelsarthpeoelyxgteorniorisanπ3gᶜle. Find the number of sides of the polygon. have?
(b) of a polygon. How many sides does the polygon

16. Find the angle in degree, grade and radian formed by the minute hand and hour 11 12 1
10 2
hand of a clock at, 93
84
(a) Half past 3 (b) Quarter past 6 (c) 20 minutes past 9
765
17. Find the perimeter of the following sectors:

(a) N (b)

O 14cm M O 45o E
21cm

F

18. (a) Find the central angle in centesimal measure subtended at the centre of a circle of radius 14 cm
by an arc of 21 cm long.

(b) The arc of the length 35 cm subtends an angle of 60o at the centre of a circle. Find the length of
the radius of the circle.

19. (a) A cow is tied to a pole with a rope of length 2.1 m. If the cow moves such that it describes a
circular path, how far will it move when the rope traces an angle of 72g at the pole?

(b) A metro train is travelling on a curve of 7 km radius at the rate of 200 km per hour. Through
what angle has it turned in 30 minutes?

20. (a) A 14 cm long pendulum oscillates through an angle of 10o. Find the length of path its extremity
describes between the extreme positions.

(b) In how many degrees does the tip of the 21 cm long minute hand of a clock rotate the distance
of 28 cm?

21. Simplify:

(a) 3cos2 α – 4sin2 α – 4cos2 α + 5sin2 α (b) sin2 α . sec2 α (cos α + sin α)(cos α – sin α)

22. Multiply: (a) (sin α + cos α) (sin α – cos α) (b) (1 + sin α) (1 – sin α) (1 + sin2 α)

23. Factorize:(a) cos A – cos3 A (b) sec3 A – tan3 A (c) 3cos2 θ – 5cos θ + 2

24. (a) Express all trigonometric ratios in terms of sin A.

(b) If 17sin α = 15, find the ratios of sin α, cos α and sec α.

25. Eliminate θ between the following pairs of equations:

(a) a = p(sec θ + tan θ) and b = p(sec θ – tan θ) (b) a cos θ – b sin θ = p and a sin θ + b cos θ = q
26. (a) If cos θ = a , prove that a sin θ = b cos θ.

a2 – b2
(b) If sinA = m and tanA = n, prove that m-2 – n-2 = 1.

27. Prove the following trigonometric identities:

(a) cot A .sin A = cos A (b) sin2 A + cos4 A = cos2 A + sin4 A
(c) tan2 θ – cot2 θ = sec2 θ – cosec2 θ (d) sec2 α + cosec2 α = sec2 α . cosec2 α

28. Show the following trigonometric identities:

(a) (1 – tan B)2 + (1 – cot B)2 = (sec B – cosec B)2

(b) (1 – sin θ + cos θ)2 + (1 + sin θ – cos θ)2 = 4 (1 – sin θ. cos θ)

(c) (sec θ – cos θ) (cosec θ – sin θ) (tan θ + cot θ) = 1

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29. Prove that:

(a) 1 – sin A = 1 sin A (b) 1 + tan α = cot α + 1
cos A . cot A + sin A 1 – tan α cot α – 1

(c) 1 – tan2 α = cos2 α – sin2 α = 1 – 2sin2 α (d) tan A 1 – 1 sin A A = 2 cot A
1 + tan2 α secA – + cos

(e) 1 + sin2 A = 2 sec2 A – 1 (f) 1 1 + 1 1 = 2 cot A . cosec A
1 – sin2 A sec A + sec A +

1 – sin α (h) 1 + sin θ – 1 – sin θ = 2 tan θ
(g) 1 + sin α = (sec α – tan α)2 1 – sin θ 1 + sin θ

30. Prove the following trigonometric identities:

(a) tan A – sec A+ 1 = 1 cos A cot α + cosec α – 1
tan A + sec A– 1 + sin A (b) cot α – cosec α + 1 = cosec α + cot α

(c) 1 – sec A + tan A = sec A + tan A – 1 (d) 1 + sec θ – tan θ +11 + sec θ + tan θ = 2sec θ
1 + sec A – tan A sec A + tan A + 1 1 + sec θ + tan θ + sec θ – tan θ

31. Prove that: (b) 1 – cos4 A = 1 + 2cot2 A
(a) cosec4 θ – cot4 θ = 1 + 2cot2 θ sin4 A

(c) sin3 θ + cos3 θ = 1 – sin θ . cos θ sin4 α + cos4 α tan2 α + cot2 α
sin θ + cos θ (d) sin4 α cos4 α = tan2 α – cot2 α

(e) cos8 β – sin8 β = (cos2 β – sin2 β) (1 – 2sin2 β.cos2 β)

32. Show that:

(a) 1 + cot2 θ = 1 – cot θ 2 (b) sec A – tan A = 1 – 2secA.tanA + 2tan2A
1 + tan2 θ 1 – tan θ sec A + tan A

(c) 1+ cot2 α = sin2 cos2 α α (d) 1 tan α α + 1 cot α α = 1 + tan α + cot α
tan2 α – cot2 α α – cos2 – cot – tan

(e) sec A + tan A – sec A – tan A = 2 (sec A – cosec A)
cosec A + cot A cosec A – cot A

33. Prove the following trigonometric identities:

(a) sin α + sin β + cos α – cos β = 0
cos α + cos β sin α – sin β

(b) (sin α.cos β – cos α.sin β)2 + (cos α.cos β + sin α.sin β)2 = 1
cos2 A – sin2 A

(c) sin A.cos2 A – cos A.sin2 A = cosec A + sec A

(d) sin2 A . sec2 B + tan2 B . cos2 A = sin2 A + tan2 B

34. Find the value of:

(a) sin2 45o + 3 tan2 30o – cot2 30o (b) sin2 60o (2cos2 45o – cosec2 30o – sec2 45o)
2

(c) (sin πc + cos πc ) (sin πc – cos πc ) (d) cot2 πc + cos2 πc – sec2 πc
3 3 6 6 6 2 4

35. If A = 90o, B = 60o, C = 45o, D = 30o and E = 0o, find the value of sin A + tan C – cos B + sin D – cos E.

357

36. Prove the following:

(a) 1 – cos 60o = 1 – cot 60o
1 + cos 30o 1 + cot 60o

(b) sin πc . sin πc + sin πc . sin πc = cos πc . cos πc + cos πc . sin πc
6 4 3 4 4 6 4 6

37. If A = 60o and B = 30o, verify that:

(a) cos (A + B) = cos A . cos B – sin A.sin B

(b) tan (A – B) = tan A – tan B
1 + tan A.tan B

38. Find the value of x if:

(a) 2 cos 45o. sec 60o – 2x sin 30o = 1

(b) xsin πc . cos πc + cot2 πc = xcot2 πc – sin2 πc AX
4 4 6 4 3

39. (a) In the given right–angled triangle ABC, ∠ABC 60o
= 90o, AB = 4 cm and BC = 3 cm, find the side
AC, ∠ACB and ∠BAC. 4cm Y 18cm Z

C 3cm B Q.No. 39 (b)
Q.No. 39 (a)

(b) In the adjoining right–angled triangle XYZ, ∠Y D
= 90o, ∠YXZ = 60o, YZ = 18cm, solve ∆XYZ. 5cm
A

(c) Find the length of BC from the adjoining figure 30 o
if ∆ABC and ∆ACD are right-angled triangles.
45o
BC

40. Determine the values of the following trigonometric ratios:

(a) cot3 315o (b) sec 13πc (c) sin (–1740o)
41. Simplify: 4 sin2 60o – cos2 60o

(a) sin (180o + A) × tan (360o – A) × cosec (90o + A) (b) 3 sin2 210o + tan2135o
cos (90o + A) sec (90o – A) cot (180o – A)

42. Find the values of the following: (b) sin2 πc + sin2 3πc + sin2 5πc + sin2 7πc
4 4 4 4
sin 250o + tan290o
(a) cot 200o + cos340o
43. Prove that:

(a) sin130o . cos165o = – sin50o . cos15o

(b) cos8o + cosec36o + cot70o = sin 82o + sec54o + tan20o

sec (360o + θ) . sin (180o + θ) . cot (90o – θ)
(c) cos (90o + θ) . sec (– θ) . tan (180o – θ) = –1

sin (180o – A) cos (180o – A)
(d) sin (270o + A) cos (90o – A) = 1

44. Show that:

(a) cos πc + cos 3πc + cos 5πc + cos 7πc = 0
8 8 8 8

(b) sin2 πc + sin2 3πc + sin2 5πc + sin2 7πc =2
8 8 8 8

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45. Determine the angle A when: (b) cot 4A – tan 2A = cot 90o (c) cosec 5A = sec 7A
(a) sin A = cos 4A
46. Find the value of x if:

(a) tan2 315o – sec2 60o = x sin 225o . cos 135o . cot2 30o
(b) x cosec (180o + A) . cosecA + tan (90o + A) tan (270o – A) = x cot A . cot (180o – A)

47. If A, B and C are the angles of ∆ABC, then prove that:

(a) sin (A + B) + sin (B + C) + sin (C + A) = sinA + sin B + sin C

(b) cos (A + B) + cos (B + C) + cos (C + A) + cos A + cos B + cos C = 0

48. If A, B, C and D are the angles of a quadrilateral ABCD, then show that:

(a) cos (A +C) – cos (B + D) = 0 (b) sin (A + B) + sin (C + D) = 0

49. Draw the graph of the following trigonometric functions: (0o ≤ θ ≤ 180o)

(a) y = sin θ (b) y = tan θ (c) y = sin 3θ (d) y = cos 2θ

50. Find the value of the following trigonometric ratios without using calculator or trigonometric table:

(a) sin15o (b) cos105o

51. sin πc + A =1
Prove that: 4

cos πc – A
4

52. Prove that: sin (2a – 1)θ . cos (3a + 2)θ + cos (2a – 1)θ . sin (3a + 2)θ = sin (5a + 1)θ

53. Prove that: tan 20o + tan 25o + tan 20o . tan 25o = 1

54. Show that: tan 9A – tan 7A – tan 2A = tan 9A . tan 7A . tan 2A

55. Verify that: tan (α + β) – tan (α – β) = tan2β
1 + tan (α + β) . tan (α – β)

56. Prove that: sin (α – β) . sec α . sec β = tan α – tan β

57. Show that: tan 55o – tan 35o = 2tan 20o

58. If sinA = 3 and sinB = 187, find the values of sin (A – B) and tan (A + B)
5
πc
59. If sin α = 1 and sin β = 1 , prove that α+β= 4
10 5

60. If A + B = 135o, prove that: (1 – tan A) (1 – tan B) = 2

πc
61. If A + B + C = 2 , show that: tan A . tan B + tan B . tan C + tan C . tan A = 1

62. Prove that: sec 45° + A . sec 45° – A . = 2sec A
2 2

63. Prove that: 4cosec 2α . cot 2α = cosec2 α – sec2 α

64. If sin (A + B) = k sin (A – B) then prove that: (k – 1) tan A = (k + 1) tan B

65. If cos α + sin β = p and sin α + cos β = q then verify that: 2sin(α + β) = p2 + q2 – 2

359

ANSWERS

1. (a) 38.5833o (b) 48.4792o (c) 72o (d) 36.315o (e) 31.8813o
2. (a) 45.39g (d) 55.1284g (e) 27.7855g
3. (a) 26o 6' (b) 28.2338g (c) 22.7777g (d) 25o 25' 41.34''
4. (a) 40g (d) 84g 12 83.95
(b) 25o 30' 21.6'' (c) 35o 6' 14.58''

(b) 39g 35 18.52 (c) 72g 23 30.02

5. (a) πc (b) 3πc (c) 841πc (d) 15057πc
5 8 360 25000

6. (a) 120o (b) 144o (c) 210o (d) 160o

7. (a) 30g, 40g (b) 5:7 (c) 180 + πc (d) 100o, 111 1 g 140o, 155 5g
10. (a) 40g (b) 15o 360 9 9
;

(c) 11πc , 9πc
40 40

11. (a) 24o, 60o, 96o (b) 36o, 48o, 96o (c) 2πc , πc , 4πc
9 3 9
3πc 9πc 3πc 9πc 3πc
12. (a) 40o, 60o, 80o (b) 63o, 72o, 45o 13. (a) 45o, 60o, 120o, 135o (b) 10 , 20 , 5 , 10 , 4

14. (a) 128 4o , 142 6g , 5πc (b) 45o, 50g, πc 15. (a) 6 (b) 10
7 7 7 4

16. (a) 75o, 83 1g , 5πc (b) 97.5o, 108 13g, 1234πc (c) 160o, 177 7g , 8πc
3 12 9 9

17. (a) 114 cm (b) 157.5 cm 18. (a) 95.4545g (b) 33.41 cm

19. (a) 239.98 m (b) 14.29c 20. (a) 2.44 cm (b) 76.3636o

21. (a) sin2 α – cos2 α (b) sin2 α (1 – tan2 α) 22. (a) sin2 α – cos2 α (b) 1– sin4 α

23. (a) cos A . sin2 A (b) (sec A – tan A (sec2 A + sec A.tan A + tan2 A) (c) (cos θ – 1) (3cos θ – 2)

24. (a) Show to your teacher. (b) 15 , 8 , 17
17 17 8

34. (a) –2 (b) – 15 (c) – 1 (d) 1
4 2

35. 1 38. (a) 1 (b) 7 1
2

39. (a) 5cm, 53.13o, 36.87o (b) 30o, 6 3 cm, 12 3 cm (c) 5 2 cm

40. (a) – 1 (b) – 2 (c) – 3
41. (a) tan3A 2
45. (a) 30o or 180o (b) 2 (b) 2
7 42. (a) –1

(b) 15o or 45o (c) 10o or 157.50o

46. (a) – 2 (b) – cot2 A 58. 1835, 77
50. (a) 3 – 1 (b) 1 – 3 36

22 22

PROJECT WORK

Construct a circle by using compass with the radius greater than 3 cm. Also, construct a pair of
congruent central angles in any measure. Find the length of the arc which is subtended by the pair
of angles by using formula. Also, measure the same arcs by using flexible ruler. What do you find?
Write your conclusion.

360

HOME ASSESSMENT - 6 (VECTOR)

1. (a) Find the vectors represented by the directed line segment joining the point (–2, 4) with origin
in column form. Also represent the vector in square grid.

(b) Find the vector represented by the directed line segment joining the points A(3, 1) and
B(–2, 3) in column form. Also show the vector in square grid.

2. (a) Find the magnitude and direction of the vector AB = 3 .
4

(b) Find the magnitude and direction of the displacement from L(–2, 3) to M(1, –1).

3. (a) If a vector AB displaces a point A(2, 1) to the point B(4, –3) and another vector CD displaces a
point C(5, –3) to the point D(7, –7) then prove that AB = CD .

(b) If a vector AB displaces a point A(1, 2) to the point B(0, 3) and another vector MN displaces a
point M(5, –4) to the point N(6, –5) then prove that AB = – MN.

4. (a) If the magnitude and direction of a vector a are 2 units and 30o respectively, find the vector.
b
a
(b) If the magnitude of the vector AB = –4 is 5 units, find the value of a and the direction of AB
.

(c) The direction of the vector CD = x is 30o. Find the magnitude of CD .
3

5. If the vector AB displaces a point A(–2, –3) to B(2, 4) then,

(a) Express AB in the column vector. (b) Calculate the length of AB .

(c) Find the direction of AB . (d) Find the unit vector along the direction of AB .

6. (a) Prove that the vectors AB and CD are parallel vectors where AB displaces A(2, 3) to B(2, 5)
and CD displaces C(–1, –2) to D(–1, 0).

(b) Show that the vectors AB and BC are collinear if AB displaces A(–4, –3) to B(2, –1) and BC

displaces B(2, –1) to C(8, 1). A

7. In the given figure, ABC an equilateral triangle E, F and G are the midpoints

of its sides AB, BC and AC respectively. If AB = 2 a , BC = 2 b and E G

EF = c , find FG , CA , CB and BE in terms of a , b and c .

1 –1 4 B F C
3 4 –5 vector,
8. If a = , b= and c = , find the following vectors as column

(a) a + b + c (b) a + b – c (c) a – b – c (d) b + c – a

9. If p = –1 , q = 2 and r = –3 then find the magnitude, direction and unit vector of the
2 4 2

following composite vectors:

(a) p + q (b) q – r (c) p – q + r

361

10. If a = x1 , b = x2 and c = x3 then prove the following:
y1 y2 y3

(a) ( a + b ) + c = a + ( b + c ) (b) a + o = o + a = a (c) a + ( – a ) = 0

11. (a) If AB shifts A(3, 2) to B(–2, 3) and CD shifts C(4, 1) to D(–1, –4), find DC – AB and | AB + CD |.

(b) If P(3, 4), Q(3, 1) and R(5, 2) are the vertices of ∆PQR, find the unit vector along PQ + QR and

the direction of PQ + QR . B

12. In the given figure, OA = a , OB = b . If 3 AC = 2 AB, find OC. S R
13. In the figure, PQRS is a quadrilateral. Prove that: C

(a) PQ + QS + SR = PQ + QR OA PQ
Q. No. 12 Q. No. 13
(b) PR – QS = 2PQ
14. (a) In the adjoining figure, ABCDEF is a regular octagon. F E B
G A
Prove that:
AB + BC + CD + DE + EF + FG + GH = AH D

(b) In the given diagram, prove that: H CE C

AB + BC + CD + DE + EA = 0 AB D
Q. No. 14(a) Q. No. 14(b)

15. (a) If p = 4 , q = 2 and r = –1 , find the vector 5 p – 2(2 q + 3 r ).
3 –3 2

(b) If a = 4 and b = –3 , find the unit vector along (3 a + 2 b ) and the direction of (3 a + 2 b ).
–3 1

(c) If p = 4 and 2 p – q = 8 , find q.
6 10

16. (a) Prove that the points P(3, 3), Q(–3, –3) and R(–3 3 , 3 3 ) are the vertices of an equilateral
triangle.

(b) Show that the points A(–1, –1), B(1, 0) and C(–2, 1) form a right-angled isosceles triangle.

(c) Prove that the points (– 4, 2), (4, 4), (8, 8) and (0, 6) are the vertices of a parallelogram.

(d) Prove that the quadrilateral with vertices (12, 2), (18, 8), (14, 12) and (8, 6) is a rectangle.

(e) Show that the points (– 4, 2), (–10, 0), (–16, 2) and (–10, 4) are the vertices of a rhombus.

AC

17. (a) In the adjoining figure, P, Q and R are the midpoints D

of the sides AB, BC and CA of ∆ABC respectively, P RE F
then prove that: AQ + BR + CP = 0 A B

(b) In the given figure, E and F are the midpoints of BQC

the sides AD and BC of the quadrilateral ABCD Q. No. 17(a) Q. No. 17(b)

respectively, show that: AB + DC= 2EF A

(c) If M and N are the midpoints of AB and AC of MN
∆ABC respectively. Prove that BC = 2 MN and

BC // MN . BC

362

(d) In the figure, ABCDEF is a regular hexagon. ED ED
Prove that: AB + AC + AD + EA + FA = 4AB. F OC
FC
(e) In the given figure, O is the centre of a regular AB
hexagon ABCDEF. AB Q. No. 17 (e)
Q. No. 17 (d)
Prove that: AB + AC + AD + AE + AF = 6 AO SR
S
(f) In the figure, PQRST is a regular pentagon. Prove TR PQ
that: PQ + PT + QR + SR + TS = 2 PR . Q. No. 17 (g)
PQ
(g) In the adjoining figure, PR and QS are the Q. No. 17 (f)
diagonals of the parallelogram PQRS. Show that:

PR + QS = 2QR and PR – QS = 2 PQ .

ANSWERS

1. –2 –5 (a) 5 units, tan–1 4 or 53.13o (b) 5 units, 126.87o
(a) 4 (b) 2 2. 3

4. 3 (b) ± 3, 126.87o or 306.87o (c) 6 units
(a) 1

5. (a) 4 (b) 65 units (c) 60.26o (d) 4/ 65
7 7/ 65

7. a , – 2 c , – 2 b, – a

8. 4 (b) –4 (c) –2 2
(a) 2 12 4 (d) –4

9. (a) 37 units, tan–16, 1/ 37 (b) 29 units, tan–1 2 , 5/ 29 (c) 6 units, 0o, 1
11. (a) 6/ 37 5 2/ 29 0

10 1/ 2 315o 12. 13 2
4 ,2 29units ( )(b) – 1/ 2 , a + 3 b

15. (a) 18 (b) –7 0
15 6/85 , 6 (c) 2
– 7/85

PROJECT WORK

Draw a right-angled triangle in a graph paper with identifying the coordinates of the vertices and
label it. Prove the following properties:
(i) Identify the right-angled triangle as scalene or isosceles.
(ii) Establish the Pythagoras theorem.
(iii) Midpoint of the hypotenuse of the right-angled triangle is equidistant from its vertices.
(iv) Hypotenuse of the right-angled triangle is the longest side.

363

HOME ASSESSMENT - 7 (TRANSFORMATION)

1. (a) Translate a point (8, – 2) by the given translation vector –1 . Write down its image.
2

(b) Translate a line segment PQ joining the points P(– 2, 0) and Q(3, – 4). Write down the
–1
coordinates of the image by the translation vector 3 .

2. (a) Plot the vertices A(1, 2), B(5, 1), C(4, 4) and D(– 1, 5) of the quadrilateral ABCD on the
2
coordinate plane and find the coordinates of the image under the translation vector –4 . Also
draw the image on the same coordinate plane.

(b) Find the coordinates of the vertices of the image of the rhombus ABCD having the vertices
–3
A(4, 2), B(–2, 0), C(0, –6) and D(6, –4) under the translation vector 2 and draw this
translation on the graph paper.

3. (a) The vertices of a square ABCD are A(3, 4), B(6, 1), C(9, 4) and D(6, 7). Translate the square
ABCD by the vector ACand then draw the above transformation on the square grid.

(b) Find the coordinates of the vertices of the image of ∆ABC with vertices A(– 3, 5), B(– 2, 0)

and C(2, 4) under the translation (x, y) → (x – 2, y + 3).

4. (a) If the image of the point (2, 4) is (– 2, 1) under the translation vector T = a , find T and then
find the image of the point (– 4, – 5) by the same translation. b

(b) The vertices of the rectangle OPQR are O(0, 0), P(– 4, 2), Q(– 1, 7) and R(3, 5). If the image of O is
(– 4, 4) under translation, find the coordinates of other vertices of the image of the rectangle
OPQR. Draw this translation on the same graph.

(c) If the image of the line segment AB joining the points A(2x – y, 3) and B(4, 2) are A'B' joining
the points A' (3, 4) and B'(1, 2x + 1) under the same translation vector, find the values of x and y.

5. Reflect the point P(–3, 2) in the following reflecting axes and find the image,

(a) x-axis (b) line x = 0 (c) y = x (d) x + y = 0 (e) x = 2 (f) y + 3 = 0

6. (a) The vertices of ∆ABC are A(1, 1), B(5, 2) and C(3, 5). Find the coordinates of the vertices
of the image of ∆ABC under the reflection on the line y = 0. Draw the reflection in the same
graph.

(b) The points P(– 4, 6), Q(0, 4), R(2, 6) and S(0, 8) are the vertices of the parallelogram PQRS. Reflect
the vertices of PQRS in the y-axis by using graph and write down the coordinates of the vertices
of the image parallelogram.

7. (a) Find the coordinates of the point A which reflects into A'(– 2, 5) in the line y = 0.

(b) If the image of T(4a + b, 4) is T'(2, a + b) under the reflection in the x-axis, find the values of a and b.

8. (a) The points P(0, 1), Q(2, 4) and R(4, 2) are the vertices of ∆PQR. Reflect the vertices of ∆PQR
in the line x = y and write down the coordinates of the vertices of the image ∆P'Q'R'. Also,
draw this reflection on the same graph.

(b) Find the coordinates of the vertices of the image of ∆CDE with vertices C(2, 4), D(–2, 3) and
E(0, –3) under the reflection in the line x = 3. Draw this transformation on the graph.

9. (a) If L(0, 2), M(8, 2), N(12, – 2) and O(4, – 2) are the vertices of the parallelogram. Then, reflect
it in the line x + y = 0 and write down the co-ordinates of the image parallelogram L'M'N'Y'.
Also, show this reflection in the same graph.

364

(b) Draw the parallelogram ABCD with vertices A(– 4, 0), B(0, – 4), C(12, 2) and D(8, 6) on
the graph paper. Reflect the parallelogram ABCD in the line y – 1 = 0 and write down the
coordinates of the vertices of the image of the parallelogram ABCD. Also, draw the image on
the same graph paper.

10. (a) Write down the coordinates of the image of the point (– 3, 4) under reflecting about the straight
line joining the points (2, – 2) and (2, 4).

(b) The coordinates of the image of a point P is P'(– 4, – 2) under the reflection about the line
y + 3 = 0. Find the coordinates of the point P.

11. (a) The image of A(2, q) when reflected in the line y = 2 is A'(3p – 1, 3). Find the values of p and q.

(b) The image of P(2 – a, 3 – b) is P'(2b – 4, 2a – 3) under the reflection in the line y = –x. Find
the coordinates of P and P'.

(c) The vertices of ∆RST are R(2, 4), S(3, 1) and T(– 6, 2). If T'(6, 2) is the image of T(– 6, 2)
under the reflection, find the remaining vertices of the image ∆R'S'T'.

12. Find the image of the point P(– 2, 1) through the following angles of rotation about the centre as
origin.

(a) 90o (b) – 90o (c) 180o (d) 270o (e) 360o

13. Rotate the line segment AB joining the points A(2, – 3) and B(0, 4) taking (1, 2) as centre of
rotation and the following angles of rotation:

(a) Positive quarter turn (b) Negative quarter turn

(c) Half turn (d) Full turn

14. (a) The vertices of ∆ABC are A(2, 1) B(– 4, 2) and C(0, 3). Rotate ∆ABC under the rotation about
the origin through 90o and find its image. Draw this rotation in the graph.

(b) Draw the square PQRS having the vertices P(1, 2), Q(– 1, 4), R(– 3, 2) and S(– 1, 0) in the
graph and rotate it about the centre as A(2, 3) through the quarter turn in clockwise direction.
Write down the coordinates of the vertices of the image square P'Q'R'S' and also draw the
image in the same graph.

15. (a) The images of the points P(3, – y) and Q(x, – 4) are P'(1, – 3) and Q'(– 4, 2) under the rotation
through – 90o about (0, 0). Find the values of x and y.

(b) If A'(2, 5) and B'(2, 3) are the image of A(x, y) and B(– 3, 2) under the rotation about (0, 0)

through the certain angle then find the coordinates of A.

16. Find the coordinates of the images of the points A(– 2, 3) under the following conditions, where
E indicates enlargement.

(a) E[(0, 0); 2] (b) E[(1, 1); 2] (c) E[(1, – 2); – 3]

17. (a) The vertices of ∆ABC are A(1, 1), B(4, 0) and C(3, 5). Enlarge ∆ABC about the centre as
origin with scale factor 2. Find the coordinates of the vertices of image of ∆ABC and draw
∆ABC and its image on the same graph.

(b) Transform the unit square OPQRS with vertices O(0, 0), P(2, 0), Q(2, 2) and R(0, 2) by the
enlargement E[(– 1, – 3), – 2]. Write down the coordinates of the vertices of the image square
P'Q'R'S' and represent the transformation on the graph.

18. (a) If the point A(a, – 1) moves to the point B(3, b) by the enlargement of E[(1, 2); 3] then find the
values of a and b.

(b) An enlargement maps the points P(2, – 1) onto P'(4, – 2) and the point Q(– 2, 4) onto Q'(– 4, 8).
Find the centre and the scale factor of the enlargement.

365

19. (a) An enlargement maps point A(1, 3) onto A'(2, 6) and B(– 2, 2) onto B'(– 4, 4). Find the centre
and the scale factor of the enlargement.

(b) Show that the enlargement of a point (2, 3) with centre (0, 0) and scale factor 3 is equivalent
4
to the transformation of the same point by the vector 6 .

ANSWERS

1. (a) (7, 0) (b) P'(– 3, 3), Q'(2, – 7)

2. (a) A'(3, –2), B'(7, –3), C'(6, 0), D'(1, 1) (b) A'(1, 4), B'(–5, 2), C'(–3, –4), D'(3, –2)

3. (a) A'(9, 4), B'(12, 1), C'(15, 4), D'(12, 7) (b) A'(–5, 8), B'(–4, 3), C'(0, 7)

4. (a) –4 , (–8, –8) (b) P'(–8, 6), Q'(–5, 11), R'(–1, 9) (c) 1, – 4
–3 (e) (7, 2) (f) (–3, – 8)

5. (a) (–3, –2) (b) (3, 2) (c) (2, –3) (d) (–2, 3)

6. (a) A'(1, –1), B'(5, –2), C'(3, –5) (b) P'(4, 6), Q'(0, 4), R'(–2, 6), S'(0, 8)

7. (a) A(–2, –5) (b) 2, –6

8. (a) P'(1, 0), Q'(4, 2), R'(2, 4) (b) C'(4, 4), D'(8, 3), E'(6, – 3)

9. (a) L'(–2, 0), M'(–2, –8), N'(2, –12), O'(2, –4) (b) A'(–4, 2), B'(0, 6), C'(12, 0), D'(8, –4)

10. (a) (7, 4) (b) (–4, –4)

11. (a) 1, 3 (b) P(1, 2), P'(–2, –1) (c) R'(–2, 4), S'(–3, 1)

12. (a) (–1, –2) (b) (1, 2) (c) (2, –1) (d) (1, 2) (e) (–2, 1)

13. (a) A'(6, 3), B'(– 1, 2) (b) A'(– 4, 1), B'(3, 3) (c) A'(0, 7), B'(2, 0) (d) A'(2, – 3), B'(0, 4)

14. (a) A'(–1, 2), B'(–2, –4), C'(–3, 0) (b) P'(1, 3), Q'(3, 6), R'(1, 7), S'(– 1, 6)

15. (a) – 2, –1 (b) A(–5, 2)

16. (a) A'(–4, 6) (b) A'(–5, 5) (c) A'(10, –17)

17. (a) A'(2, 2), B'(8, 0), C'(6, 10) (b) O'(–3, –9), P'(–7, –9), Q'(–7, –13), R'(–3, –13)
19. (a) Centre (0, 0), k = 2
18. (a) 52, – 7 (b) (0, 0), 2

PROJECT WORK

Draw a triangle ABC in any shape in the first quadrant of a graph paper with identifying the coor-
dinates of its vertices and follow the following activities in the same graph paper:

(i) Reflect the triangle in the x-axis and write the coordinates of the vertices of the image ∆A1B1C1.

(ii) Rotate the image ∆A1B1C1 about the origin through + 90o and name the image ∆A2B2C2.

(iii) Enlarge the image ∆A2B2C2 with centre (0, 0) and the scale factor 2 and write the coordinates of the
vertices of the image ∆A3B3C3.

(iv) Translate the image ∆A3B3C3 by the vector –2 and write the coordinates of the vertices of the
image ∆A4B4C4. 5

366

HOME ASSESSMENT - 8 (STATISTICS)

1. Define the following terms: (b) Quartiles
(a) Median (d) Third quartile
(c) First quartile (f) Fourth decile
(e) Deciles (h) Percentiles
(g) Ninth decile (j) 75th percentile
(i) 24th percentile

2. (a) Into how equal parts is divided the data by the quartiles ?
(b) Into how equal parts do the deciles divide a data ?
(c) Into how equal parts do the percentiles divide a data ?
(d) Into how equal parts do the median divide a data ?

3. (a) How many median are there in a data ?
(b) How many quartiles are there in a data ?
(c) How many deciles are there in a data ?
(d) How many percentiles are there in a data ?

4. What are the first (lower) quartile and third (upper) quartile from the data given below ?
(a) 1, 4, 5, 7, 8, 10, 23, 25, 28, 32, 40
(b) 4, 7, 2, 6, 10, 6, 14
(c) 7, 8, 3, 6, 5, 4, 6, 1, 7, 3
(d) 12, 5, 22, 30, 7, 36, 14, 42, 15, 53, 25, 65
(e) 4, 8, 12, 18
(f) 25, 36, 57, 85, 91

5. (a) If the first quartile of the ascending ordered data x – 2, x + 1, 6, 7, 2x + 1, 22, 33 is 5, find the
value of x.

(b) If the third quartile of the ascending ordered data 12, 15, a + 1, 2a – 15, 25, 2a – 6, 34, 36 is
34, find the value of a and its first quartile.

6. Calculate the quartiles from the following data:

(a) X 2 4 6 8 10 (b) X 10 28 15 5 12

f 5 4 3 2 4 f 4 10 8 12 6

7. Find the quartiles from the data given below:

(a) Height (in inches) 58 59 60 61 62 63 64 65 66 Total
No. of Students 15 20 32 35 33 x 20 10 8 195

(b) X 012345678
cf 1 10 36 95 167 219 248 255 256

367

8. Find the fourth, seventh and ninth deciles from the data given below:

(a) 650, 500, 800, 600, 625, 400, 700, 750, 800, 925, 850, 550

(b) 26, 45, 12, 14, 25, 26, 26, 25

9. Find the 3rd, 6th and 8th deciles from the following data:

(a) X 6 12 18 24 30
3 5 2 7 5
f

(b) X 26 12 15 23 29
68947
f

10. Find the values of D1, D2 and D5 from the data given below:

(a) Length (in cm) 20 5 30 15 25 45 35 40 10 Total

No. of plants 10 25 23 24 5 27 a 25 20 177

(b) X 8 9 12 15 16 20 25 29 32

cf 3 8 14 16 20 27 32 42 50

11. Find the thirty second, sixty seventh and eighty fifth percentiles from the data given below:
(a) 3, 5, 2, 7, 6, 4, 9
(b) 24,23, 29,25, 27, 26, 22,21
12. Find the 38th, 74th and 91st percentiles from the following data:

(a) Scores 21 26 31 39 42
Frequency 35275

(b) X 5 7 10 20 30 56
f 5 6 4 8 9 12

13. Find the values of P9, P32 and P73 from the data given below:

(a) Length (in cm) 41 42 46 49 50 51 52 54 61 Total

No. of plants 6 5 8 10 p 11 8 5 3 68

(b) X 120 130 150 170 180 210 240 260 280

cf 6 15 19 22 27 32 35 42 50

14. Find the inter-quartile range, quartile deviation and coefficient of quartile deviation from the
following data:

(a) Share stock (Rs.) : 135, 275, 350, 360, 420, 405, 490

(b) Height (ft.) : 3.8, 4.5, 4.9, 5.0, 5.1, 5.2, 5.6, 5.7

368

15. Calculate the quartile deviation and its coefficient from the given data:

(a) Obtained marks 35 42 49 59 65 72
3 1
No. of Students 2 3 7 4

(b) Length (cm) 9.2 12.1 14.5 16.2 17.3 18.1 18.5
No. of Mobiles 34322 33

16. Find the quartile deviation and its coefficient from the given data:

(a) Wages (Rs.) 7000 7500 8000 8500 9000 9500 10000
15 5
No. of Workers 11 12 15 23 30

(b) Height (cm) 23 28 35 45 30 38 40 50

No. of Plants (f) 8 2 3 5 8 9 10 12

17. State which is more variability by using the coefficient quartile deviation.
The daily records of a week of sales of books in Bhotahity (X1) and Kirtipur (X2) are given below:

Week Sunday Monday Tuesday Wednesday Thursday Friday

No. of books X1 1280 1100 1350 1225 1175 1215

No. of books X2 1150 800 1170 1105 1225 1185

State which has more variation by using the coefficient of quartile deviation.

18. (a) Find the value of the third quartile if the values of the first quartile and inter quartile range are
45 and 80 respectively. Also, find the QD and CQD.

(b) If the third quartile and quartile deviation of the distribution are 250 and 110 respectively, then
find the value of its first quartile and the coefficient of quartile deviation.

19. Calculate the mean deviation from mean and its coefficient from the following data:
(a) 21, 25, 27, 35, 39, 42.

(b) Age 345678

No of students 3 3 2 7 4 1

20. Compute the mean deviation from median and its coefficient from the data given below:
(a) 20, 35, 30, 25, 50, 45, 40.

(b) X 7 14 21 24 27 30 35
cf 3 5 8 12 15 20 24

21. Find the standard deviation and its coefficient from the following data: 17 18
(a) 230, 240, 250, 260, 270, 280. 1 3

(b) X 5 10 15 16
f 2342

369

22. Find the variance and its coefficient of variation from the following data: 63
8
(a) 75, 86, 97, 102, 117, 128, 135.

(b) X 24 29 36 44 57
f 6 12 7 9 10

ANSWERS

4. (a) 5, 28 (b) 4, 10 (c) 3, 7 (d) 12.5, 40.5 (e) 5, 16.5 (f) 30.5, 88

5. (a) 4 (b) 20, 16.5 6. (a) 2,10 (b) 5, 28 7. (a) 60, 63 (b) 3, 5

8. (a) 630, 800, 902.5 (b) 25, 26, 45 9. (a) 12, 24, 30 (b) 15, 23, 29

10. (a) 5, 10, 30 (b) 9, 12, 16 11. (a) 3.56, 6.36, 8.6 (b) 22.88, 26.03, 28.3

12. (a) 31, 42, 42 (b) 20, 30, 56 13. (a) 42, 49, 51 (b) 120, 150, 260

14. (a) 130, 65, 0.19 (b) 0.7, 0.35, 0.07 15. (a) 5, 0.093 (b) 3, 0.199

16. (a) 500, 0.059 (b) 7.5, 0.2 17. X2
18. (a) 125, 40, 0.471 (b) 30, 0.786

19. (a) 7.167, 0.227 (b) 1.26, 0.231

20. (a) 8.571, 0.245 (b) 6.79, 0.251

21. (a) 17.078, 0.067 (b) 4.287, 0.317

22. (a) 414.72, 19.26% (b) 192.70, 32.60%

PROJECT WORK

Record the ages (in months) of your classmates and make frequency distribution table. Find the
following dispersions:

i. Quartiles ii. 3rd Decile

iii. 7th Decile iv. 12th percentile

v. 38th percentile vi. 60th percentile

vii. 74th percentile viii. 82nd percentile

ix. Inter-quartile range x. Quartile deviation

xi. Mean deviation from mean xii. Mean deviation from median

xiii. Standard deviation and its coefficient xiv. Variance and coefficient of variation

370

Sample Model Question

Subject: Opt. Mathematics F.M.: 100
Time: 3 hrs.

Candidates are required to give their answers in their own words as far as practicable. The figures in the
margin indicate full marks.

Attempt all questions: Group 'A' [5 × (1+ 1) = 10]


1. (a) Define function.

(b) What is the additive inverse of the polynomial 3x2 – x + 1?

2. (a) What is the limit of a sequence 190, 1900, 10900, ..................?
2 0
(b) Name the type of the matrix 0 2 .

3. (a) Write down the distance formula for two points (x1, y1) and (x2, y2). C
(b) What are the x-intercept and y-intercept of the straight line ax + by + c = 0?

4. (a) How many degrees are there in the angle 90g?

(b) List the Pythagorean relations of the trigonometric ratios.

5. (a) Write the triangle law of vector addition for the given triangle ABC.

(b) What is reflecting axis? A B

Group 'B' [2 × (2 + 2) + 3 × (2 + 2 + 2) = 26]

6. (a) Find the values of a and b if (2a + 5, 3) = (7, a – 2b).

(b) If A = {1, 2, 3} and B = {1, 4, 9, 16}, find the relation "is square root of" from A to B in the set
of ordered pairs and set builder form.

(c) If f(x) = 2x2 – 5 + 3x and g(x) = 2 – 4x + 5x2 + x3, find 5 f(x) – 3g(x) and its degree.

7. (a) Define identity matrix. For what value of x, the matrix 2 – x 0 is an identity matrix.
01
2x – 2 14
(b) If P = x + y 1 and Q = 2 3 and P + Q = QT, find the values of x and y.

8. (a) Find the equation of a straight line with slope –3 and y-intercept 4 units.

(b) If the points (p, 0), (0, q) and (2, 2) lie on the same line, prove that 1 + 1 = 1 .
p q 2

9. (a) Find the angle in grade formed by the minute hand and hour hand of a clock at half past 2.

(b) Prove that: 1 sinα – 1 sinα = – 2cotα.
+ cosα – cosα

(c) Find the value of x if xsec245o – cot260o = x.sin260o.

10. (a) Find the magnitude and direction of the vector AB which displaces A(4, 2 3 ) to B(3, 3 3 ).

A

(b) In the figure, OA = a, OB = b , and OC = c. If 4BC = 3AB, C
find OC in terms of a and b .

(c) Find the seventh decile from the given data: B O

Marks 25 30 35 40 45 50

No. of Students 3 4 8 2 7 1

371

Group 'C' [11 × 4 = 44]

11. If f(x – 3) = f(x) – f(3), show that f(a) = 0 and f(–3) = –f(3)

12. If A = {1, 4, 5} and B = {2, 3}, find A × B and B × A. Also, show them in mapping diagram and tree
diagram.
lim x2 – 36
13. Find the limit by using tables: x→6 x–6

14. If A = 2 4 0 and B = –2 –3 1 , find the matrix CT such that A – B + C is a null matrix.
–1 2 3 0 2 4
15. Find the equation of a straight line having the y-intercept –3 and passing through the point (4, 1).

16. Prove that: 1 tan A A + 1 cot A = 1 + tanA + cotA.
– cos – tan A
πc 3πc 5πc 7πc
17. Prove that: cos2 16 + cos2 16 + cos2 16 + cos2 16 =2

18. If tan α = k.tan b, verify that: (k + 1).sin(α – b) = (k – 1).sin(α + b).

19. Determine the vertices of the images ∆A'B'C' formed when ∆ABC with vertices A(2, 1), B(4, 6) and
C(5, 1) is reflected in the line y + 1 = 0.

20. Find the quartile derivation and its coefficient from the data given below:

Marks 15 20 25 30 35 40 45 50

No. of Students 7 3 8 7 6 2 9 4

21. Calculate the mean deviation from mean of the data given below: 25, 20, 28, 32, 15, 17, 38

Group 'D' [4 × 5 = 20]

22. Study the given pattern of the figures and copy them.

,,, , ____, ____, _____

(a) Add one more figure in the same pattern.
(b) Find the nth term of the sequence.
(c) Find the number of points in the 12th term of the same sequence.

23. R(2, 3) and S(– 1, 4) are two fixed points. Find the equation of the locus of a moving point P(x, y)
such that PR2 + PS2 = RS2. Identify the point (– 2, – 3) lies on the same locus or not?

24. In the triangle ABC, P, Q and R are the midpoints of the A
sides BC, CA and AB respectively. Prove that: RQ

AP + BQ + CR = 0

25. Find the coordinates of the vertices of the image of the BPC
parallelogram PQRS with vertices P(1, 0), Q(2, 2), R(–1, 3) and

S(–2, 1) under the translation by PR . Represent this translation

on the same graph.
Best of Luck

Letter Grading System

Marks 100-90 90-80 80-70 70-60 60-50 50-40 40-30 30-20 20-0
D+ D E
Grade A+ A B+ B C+ C
Below Average Insufficient Poor
Remarks Outstanding Excellent Very good Good Above average Average

372

Unit Test – 1 (Algebra) Unit Test – 2 (Limit)

Time : 45 mins. Group 'A' FM: 25 Time : 40 mins Group 'A' FM: 20
Attempt all questions: [1 × (1 + 1) = 2] Attempt all questions: [2 × (2 + 2) = 8]


1. (a) What are the domain of the function f ={(2, 3, (3, 5), (4, 7)}? 1. (a) Find the 8th term of the sequence 5.9, 5.99, 5.999, ............ Also find
the approximately last term when its number of terms are increased.
(b) What is the degree of polynomial?

Group 'B' [1 × (2 + 2 + 2) = 6] (b) Find the sequence of perimeters of the

2. (a) If (3x – 1, 2x + 1) = (5, 2y – 1), find the values of x and y. triangles formed by joining the midpoints

(b) The function f(x) is defined by f(x) = {(x, y): y = 3x2 + 1}. If the of the successive sides of the given rhombus
image of x is 49, find its pre-image.
PQRS having the side 8 cm. 8 cm

(c) What should be added to p(x) = 3x2 – x + 4 to get q(x) = 4x2 + 3x + 4. 2. (a) Find the sum of the every number of the terms of the infinite series

Group 'C' [3 × 4 = 12] having the nth term tn = (–21)n.

3. If A = {1, 2, 3} and B = {0, 4}, find the Cartesian product of A and B. (b) Find the values of the function f(x) = x4 – 16 at x = – 1.9, – 1.999, –
Represent it in the set-builder, table and tree diagram. x + 2

4. Write down the name of the types of the following relations. Also represent 2.01, – 2.001.
them in mapping diagrams.
Group 'C' [3 × 4 = 12]

(a) R1 = {(2, 5), (5, 8), (8, 11), (2, 11)} (b) R2 = {(0, 0), (2, 2), (4, 4)} 3. A glass contains 150 ml of milk. A boy drinks half of it in each drink. Find
5. If a function f(x + 5) = f(x) + f(5), x ∈ R, show that f(0) = 0 and f(– 5) = – f(5). the sequence of the amount of drinking he finishes the glass of milk. What
amount of milk is in the glass at last? Also, find the sum of milk that is
Group 'D' [1 × 5 = 5] drunk by him.

6. Study the following figure sequence and copy them:

4. What is the limit of a function? Estimate the limit numerically for the
x3 + 2x2 – x – 2
, , , , _______, _______, ................ function f(n) = x+1 at x = – 1.

(a) Draw one next figure in the same sequence. 5. Find the value of x lim f(n) x lim 3 where f(n) 4x2 – 30, x < 3
(b) Find its nth term. →3 → x2 – 9, x > 3
(c) Find the number of balls in the 15th figure. x–3

Best of Luck

Best of Luck

373

374 Unit Test – 3 (Matrix) Unit Test –4 (Coordinate Geometry)

Time : 45 mins. FM: 25

Time : 40 mins. FM: 20 Attempt all questions:
[1 × (1 + 1) = 2]
Attempt all questions: Group 'A' [2 × (1 + 1) = 4]

Group 'A' 1. (a) Write the distance formula.

1. (a) What is the order of the matrix [1 2 4 –5]? (b) What are the coordinates of the midpoint of the line segment joining
(b) List any two properties of matrix addition. the points (4, – 2) and (2, 8)?

Group 'B' [1 × (2 + 2 + 2) = 6] 2. (a) Name three standard forms of the equation of a straight line.

2. (a) Construct a 2 × 2 matrix whose elements are aij= 2i – 3. (b) What is the x-intercept made by the straight line having equation
ax + by + c = 0?

(b) For what values of x, y and z, the matrix 3x – 2 2y – 1 is an Group 'B' [2 × (2 + 2) = 8]
3 – 2z 1
identity matrix?
3. (a) Find the coordinates of the points on the x-axis whose distance from
(c) If A = [1 2] and B = 1 2 , is AB compatible? If yes, find AB. the point (8, 3) is 5 units.
0 1
(b) The coordinates of the points A, B and P are (3, 0), (4, 0) and (x, y).
If 2AP = BP, find the equation of the locus of the point P(x, y).

Group 'C' [3 × 4 = 12] 4. (a) If the slope of line passing through the points P(4, b) and Q(8, 10) is

3. If A = 2 –1 ,B= 4 5 and C = 3 4 , prove that: 3 , find the value of b.
3 0 –7 8 6 –5 4

(A + B) – C = A + (B – C). (b) Find the value of k if the points (2, -1), (k, 3) and (– 4, – 9) are
collinear.
1 3
4. If I is a 2 × 2 unit matrix and 4A – 3I = 2 –4 0 , find the matrix A. Group 'C' [2 × 4 = 8]

5. If P = 2 –3 and Q = –1 2 , prove that (AB)T = BTAT. 5. Prove that the points (1, 1), (2, 3), (5, 1) and (6, 3) are the vertices of a
1 4 3 –2
parallelogram.

6. Reduce ax + by = 1 into perpendicular form and prove that a2 + b2 = 1 .
p2

Best of Luck Group 'D' [1 × 5 = 5]

7. Prove that the area of the parallelogram PQRS with vertices P(–1, 1),
Q(4, 0), R(7, 2) and S(2, 3) is double of the area of ∆PQS.

Best of Luck

Unit Test – 5 (Trigonometry) Unit Test – 6 (Vector)

Time : 45 mins. FM: 25 Time : 40 mins. FM: 20

Attempt all questions: Group 'A' [1 × (1 + 1 + 1) = 3] Attempt all questions: Group 'A' [1 × (1 + 1) = 2]
SR

1. (a) What is one degree of an angle? 1. (a) Define direction of a vector.
(b) Write the Pythagorean relations of trigonometric ratios.
(c) What are the values of sin 30° and cos 60°? sr

(b) Write the parallelogram law of vector Pp Q
addition for the given parallelogram PQRS.
Group 'B' [1 × (2 + 2 + 2) = 6]

2. (a) An arc subtends a central angle of 45o in a circle of radius 7 cm. Find Group 'B' [2 × (2 + 2) = 8]
the length of the arc. [π = 22]
7 2. (a) Find the magnitude and direction of the vector AB in B

(b) Prove that: tan A + cot A = cosec A . sec A the adjoining square grid.

(c) Find the value of sin 15° without using calculator or trigonometric A
table.
(b) Calculate the unit vector along the direction of PQ = 4 .
Group 'C' [4 × 4 = 16] –3
3. (a) If a = 2 and b = – 1 , find (2 a –3 b).
3πc T S
4 32
3. One angle of a triangle is . If the ratio of the remaining two angles of

the triangle is 4:5 then find the angles in grade. (b) In the adjoining figure, PQRSTU is a regular U R

4. Prove that: 1 + cos A + 1 – cos A = 2 cosec A hexagon, show that: PR + PS + TP + UP = 3PQ PQ
1 – cos A 1 + cos A .

5. Find the value of x if x sec(90o + θ) . cosec(180o – θ) tan(90o + θ). cot(180o – θ) Group 'C' [2 × 5 = 10]
= x tan(270o – θ) . tan(90o + θ).
4. In the given figure, ABC is an equilateral triangle and P, A

sin 2A Q and R are the midpoints of its sides AB, BC and AC P R
cos2 A – sin2 respectively. If AB = 2 c , BC = 2 a and PQ = b , find
5. Prove that : tan (A + B) + tan (A – B) = B
QR , CA , CB and BP in terms of a , b and c .
B QC

Best of Luck 5. Prove that the medians of a triangle are concurrent.

Best of Luck

375

376 Unit Test – 7 (Transformation) Unit Test –8 (Statistics)

Time : 40 mins. FM: 20 Time : 40 mins. FM: 20

Attempt all questions: Group 'A' [1 × (1 + 1) = 2] Attempt all questions: Group 'A' [1 × (1 + 1) = 2]


1. (a) Name the types of isometric transformation. 1. (a) If 2, 2x + 1, 3x + 1, 4x, 17, 20, 25 are in acceding order and the first
quartile of the data is 7, find the value of x.
(b) What are the coordinates of the image of an object P(x, y) under
reflection in the line x – a = 0? (b) Find the seventy first percentile of the data: 28, 38, 18, 25, 48, 32, 55,
62.

Group 'B' [2 × 4 = 8]

2. The quadrilateral ABCD has the vertices A(8, 8), B(6, 8), C(6, 6) and D(7, Group 'B' [4 × 4 = 16]
7). Find the coordinates of the image of quadrilateral A'B'C'D' under the
translation by the vector T = – 2 . 2. Calculate Q1, D6 and P39 from the given table:
–3 x 12 14 8 10 16 20 18
f 7965256
3. A(2, 1), B(–4, 5) and C(–1, 4) are the vertices of the ∆ABC. Find the
coordinates of the vertices of the image ∆A'B'C' when it is enlarged with 3. Find the quartile deviation and its coefficient from the data given below:
origin as centre and scale factor of 2.
Wages per day ($) 4 4.7 5 6 6.5 7
Group 'C' [2 × 5 = 10] No. of Staffs 342 315

4. Determine the vertices of the image ∆A'B'C' formed by reflecting ∆ABC 4. Find the mean deviation from mean in the data given below: 74
with vertices A(6, 3), B(–3, 5) and C(4, –2) in the line y + 3 = 0. Also, draw 4
both triangles on the same graph paper. Marks obtained 42 48 52 55 67
No. of students 5 6 8 12 14
5. The vertices of a square ABCD are A(2, 0), B(5, 2), C(3, 5) and D(0, 3).
Rotate the square ABCD about the centre (– 2, 3) through the angle 270° 5. Find the standard deviation and the coefficient of variation from the given
in anti-clockwise direction in the graph. Write down the coordinates of the data:
vertices of the image square A'B'C'D'.
x 10 20 30 40 50 60
cf 3 7 16 21 28 32

Best of Luck

Best of Luck