-COORDINATE GEOMETRY

1. Distance Formula:

The distance between the points A (X1 , Y1) and B (X2 , Y2) is

given by AB = (X2 – X1 )2 + (Y2 – Y1 )2 units or AB =

(X1–X2)2 + (Y1–Y2)2 units or AB = (difference of the abscissa)2

+ (difference of the ordinates)2

2. (a) Section Formula : The co-ordinates of the point P (X,Y)

which divides the join of two points A(X1 , Y1) and B (X2 , Y2)

internally in the ratio m : n is given by

mx2 + nx1

X = m + n

my2 + ny1

, Y = m + n

(b) Section Formula (K : 1) : if the ratio in which P divides

AB is K : 1 then coordination of point P are

kx2 + x1

X = k + 1

ky2 + y1

, Y = k + 1

3. Mid – Point Formula : if P (x , y) is the mid point of AB, then

x1 + x2

X = 2

y1 + y2

, Y = 2

4. Area of a triangle : The area of triangle whose vertices are A

(X1, Y1), B (X2, Y2) and C (X3, Y3) is given by

1

= X1 (y2 – y3) + X2 (y3 – y1) + X3 (y1 – Y2) unit2 2

Condition of collinearity of three points: If the area of the

triangle formed by three sides is zero or X1 (Y2 – Y3) + X2 (Y3 –

Y1) + X3 (Y1 – Y2) = 0, then three points A (X1 , Y1), B (X2, Y2) and

C (X3, Y3) lie on a straight line and vice versa.

5. Centroid : The point which divides each median in the ratio 2 :

1 is called centroid. It is also the point where the three

medians intersect. If (X , Y) are co-ordinates of centroid of

triangle whose vertices are (X1, Y1), (X2 , Y2) and (X3 , Y3) then

X1 + X2 + X3 y1 + y2 + y3

X = , Y = 3 3

6. To prove that a quadrilateral is a :

i. Parallelogram: Show

that two pairs of opposire

sides are equal.

ii. Rectangle: Show that

opposite sides are equal

and digonals are also

equal.

iii. Rhombus: Show all sides

are equal.

iv. Square: Show that all

sides are equal and

or

or

or

or

Show that digonals bisect each

other.

Show that digonals bisect each

other and are equal.

Show that digonals bisect each

other and two adjacent sides are

equal.

Show that digonals bisect each

other and two adjacent sides are

digonals are also equal. equal and digonals are equals.

QUESTIONS:

DISTANCE FORMULA

1. Formula the distance between the points :

i) (p, -q) ii) (a + b, c + d), (a – b, d – c)

2. Find the distance of the point P (6, -6) from the origin.

3. Show that the point A (4, 4), B (3, 5) and C (-1, 1) are the

vertices of right angled triangle.

4. Prove that the point A (-3, 0), B (1, -3) and C (4, 1) are the

vertices of an isosceles right angled triangle.

5. Show that the points (1, 1), (-1 , -1) and (- √3, √3) are the

vertices of an equilateral triangle.

6. Find a relation between x and y such that the point (x, y) is

equidistant from the points (7, 1) and (3, 5).

7. Show that the points A (2, -2), B(14, 10), C(11, 13) and D (-1, 1)

are the vertices of a rectangle.

8. Show that quadrilateral with the vertices (3, 2), (0, 5), (-3, 2)

and (0, -1) is a square.

9. Do the points (3, 2), (-2, -3) and (2, 3) form a triangle? If so,

name the type of triangle formed.

10. Show that the point (6, 9), (0, 1) and (-6, -7) are collinear.

11. Find the point on the x – axis which is equidistant from (2, -5)

and (-2, 9).

12. Find a point on x – axis which is equidistant from the point (7,

6) and (-3, 4).

13. Show that the points (1, 7), (4, 2), (-1, -1) and (-4, 4) are the

vertices of a square.

14. If the distance of P (x, y) from A (a + b, b – a) and B(a – b, a +

b) are equal. Prove that bx = ay.

15. Find the values of y for which the distance between the point P

(2, 3) and Q (10, y) is 10 units.

16. If the points P (x, y) is equidistant from the points A (5, 1) and

B (-1, 5). Prove that 3x = 2y.

17. The vertices of a triangle are (-2, 0), (2, 3) and (1, -3). Is the

triangle equilateral, isosceles or scalene?

18. Find a point on the y – axis which is equidistant from the points

A (6, 5) and B (-4, 3).

19. If a point (p, q) is equidistant from the point (5, 3) and (-2, -4).

Prove that p + q = 1.

20. If Q (0, 1) is equidistant from P (5, -3) and R (x, 6), find the

values of x. Also, find the distances QR the PR.

21. Find a relation between x and y such that the point (x, y) is

equidistant from the point (3, 6) and (-3, 4).

22. Find the centre of a circle passing through the points (6, -6), (3,

-7) and (3, 3).

23. Show that (2, -1), (3, 4), (-2, 3) and (-3, -2) are the vertices of a

rhombus.

24. Find the circumcentre of the triangle whose vertices are (3, 0),

(-1, -6) and (4, -1).

25. Name the type of quadrilateral formed, if any, by the following

points, and give reasons for your answer:

i) (-1, -2), (1, 0), (-1, 2), (-3, 0) ii) (-3, 5), (3, 1), (0, 3), (-1,

-4) iii) (4, 5), (7, 6), (4, 3), (1, 2).

SECTION FORMULA & MID – POINT FORMULA

26. Find the coordinate of the point which divides the line segment

joining (1, -3) and (-3, 9) in the ratio 1 : 3.

27. Find the coordinates of the point which the divides the join of

(2, 3) and (3, 4) internally in the ratio 3 : 2.

28. Find the coordinates of the point which divides the join of (-1,

7) and (4, -3) in the ratio 2 : 3.

29. Find the coordinates of a point A, Where AB is the diameter of

a circle whose centre is (2, -3) and B is (1, 4).

30. If A and B are (-2, -2) and (2, -4) respectively, find the

coordinates of P such that AP = 3/7 AB and P lies on the line

segment AB.

31. In what ratio does the point (-4, 6) divide the line segment

joining the points A (-6. 10) and B (3, -8) ?

32. Find the coordinates of the point which divide the line segment

joining A(-2, 2) and B (2, 8) into four equal parts.

33. Find coordinates of points of trisection of the line segment

joining the points (3, -2) and (-3, -4).

34. Find the coordinates of the point of trisection of the line

segment joining (4, -1) and (-2, -3).

35. Find the ratio in which the point P (2, -5) divides the line

segment joining the points A (-3, 5) and B (4, -9).

36. Determine the ratio in which the point P (m, 6) divides the join

of A (-4, 3) and B (2, 8). Also, find the value of m.

37. Find the ratio in which the line segment joining the points (6, 4)

and (1, -7) is divided internally by the axis of x.

38. Find the ratio in which the line segment joining the point

A (2, -2) and B (-7, 4).

39. Find the ratio in which the line segment joining the points

(-3, 10) and (6, -8) is divide by (-1, 6).

40. If A(-4, 2), B (2, 0), C (8, 6) and D (a, b) are the vertices of

parallelogram, find a and b.

41. In what ratio is the line segment joining the points A (-2, -3)

and B (3, 7) divide by the y – axis? Also, find the coordinates of

the point of division.

42. The line segment joining A (-2, 9) & B (6, 3) is the diameter of a

circle with centre C. Find the coordinates of C.

43. Find the ratio in which the y – axis divide the line segment

joining the points (5, -6) and (-1, 4). Also, find the point of

intersection.

44. If A(-1, 3) B (1, -1) and C(5, 1) are the vertices of triangle, find

the length of medians.

45. The mid – point of the sides of the triangle are (1, 2), (0, 1) and

(2, -1), find its vertices.

46. The line joining the points (2, 1) & (5, -8) is trisected at the

point P & Q. If point P lies on the line 2x – y + k = 0, find the

value of k.

47. In what ratio x – axis divides the join of (2, -4) and (-3, 6).

48. Determine the ratio in which y – x + 2 = 0 divides the join of

(3, -1) and (8, 9).

49. Centre of the circle is (-2, 5) and one end of the diameter is (2,

3). Find the other end.

50. Find the coordinates of the point which divides the line

segment joining the points (-4, 0) and (0, 6) in four equal parts.

51. The mid – point of the line segment joining (3p, 4) and (-2, 2q)

is (2, 2p + 2). Find the value of p and q.

52. The points (3, -4) and (-6, 2) are the extremities of a diagonal

of a parallelogram. If the third vertex is (-1, -3). Find the

coordinates of the fourth vertex.

53. If three consecutive vertices of a parallelogram ABCD are A (1,

2), B (1, 0) and C (4, 0). Find its fourth vertex D.

54. If two vertices of parallelogram are (3, 2), (-1, 0) and the

diagonals cut at (2, -5). Find other vertices of the

parallelogram.

55. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a

parallelogram taken in order. Find x and y.

56. Determine the ratio in which the line 2x + y – 4 = 0 divides the

line segment joining the points A(2, -2) and B (3, 7).

57. Prove that the coordinates of he point which divides the line –

segment joining the points (x1 , y1) and (x2 , y2) internally in the

ratio m : n are given by

mx2 + nx1 my2 + ny1 X = , Y = m + n m + n

AREA OF THE TRIANGLE

Find the area of triangle whose vertices are

58. (3, 8), (-4, 2) & (5, -1) 59. (2, 4), (-3, 7) & (-4,

5)

60. Find the area of quadrilateral ABCD whose vertices are A (-5,

7), B (-4, -5), C (-1, -6) & D (4, 5).

61. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1,

4) and (-2, -1) taken in order.

62. If (1, -1), (6, -6) and (4, k) are the vertices of a triangle whose

are is 10 sq units, find k.

63. Find the area of the quadrilateral whose vertices are (2, -1),

(3, 4), (-2, 3) and (-3, -2).

64. Find area of triangle, coordinate of mid – points of whose sides

are (2, -1), (3, 2), (5, 9).

65. Show that the points (0, 1), (1, 2) & (-2, -1) are collinear.

66. Find the value of p for which the points A (-1, 3), B (2, p) &

C (5, -1) are collinear.

67. For what value of k are the point A (-3, 12), B (7, 6) & C(k, 9)

collinear?

68. Find a relation between x and y if the points (x, y), (1, 2) and

(7, 0) are collinear.

69. If the points (a, 0), (0, b) & (1, 1) are collinear, show that

1/a + 1/b = 1.

70. Find the value of k if the point A(2,3), B(4, k) and C(6, -3) are

collinear.

71. Show that the points A(a, b + c), B(b, c +a) and C(c, a + b) are

collinear?

CENTROID OF TRIANGLE

72. Find the centroid of the triangle whose vertices are (4, -8), (-9,

7) and (8, 13).

73. Find the third vertex of a triangle, if two vertices are at (-3, 1)

and (0, -2) and the centroid is at the origin.

74. The coordinates of centroid of a triangle are (1, 3) & two of its

vertices are (-7, 6) & (8, 5). Find the third vertex.

75. The vertices of a triangle are (1, 2), (h, -3) and (-4, k). If

centroid of the triangle is (5, -1). Find h and k.

76. If (-2, 3), (4, -3) and (4, 5) are the mid – point of the sides of a

triangle, find the coordinates of its centroid.

MIXED STUFF

77. Show that the following points lie on a straight line :

i) (1, -1), (2, 1), (4, 5) ii) (3a, 0), (0, 3b) and (a, 2b).

78. In each of the following find the value of ‘k’ for which the

points are collinear.

i) (7, -2), (5, 1), (3, k) ii) (8, 1), (k, -4), (2, -5).

79. If (x, y), (1, 2) and (7, 0) are collinear, find the relation between

x & y.

80. Show that (1, -1) is the centre of the circle circumscribing the

triangle whose angular points are (4, 3), (-2, 3) and (6, -1).

81. If a point P (x, y) lies on a circle whose centre is (3, -2) and

radius 3 units, show that x2 + y2 – 6x + 4 + 4 = 0.

82. Determine by distance formulae the following points are

collinear or not: a) (1, 2), (5, 3) and (18, 6) b) (2, 5), (-1, 2)

and (4, 7)

83. An equilateral triangle has one vertex at the point (3, 4) and

another at the point (-2, 3) find the coordinates of the third

vertex.

84. Find the value of x of x if the distance between the points (x, 1)

and (2, 3) be 4 units.

85. Determine the ratio in which the straight line x – y – 2 = 0 the

line segment joining (3, -1) and (8, 9).

86. ABCD is a square each of whose sides is ‘a’ units. If A lies at

the origin, sides AB and AD lies along x – axis and y – axis

respectively, find the coordinates of each vertex of the square.

87. Find the ratio in which the point (-3, p) divides the line

segment joining the points (-5, 4) and (-2, 3). Hence find the

value of p.

88. Prove that the points (2a, 4a) (2a, 6a) & (2a + √3 a, 5a) are the

vertices of an equilateral triangle.

89. The co – ordinates of a point which divides the joint of A (3, 6)

and B internally in the ratio 2 : 3 is c[1/5 , 34/5]. Find the co –

ordinates of B.

90. In what ratio does the point (-1, 1) divides the join of (2, 4) and

(5, 7)?

91. For what value (s) of x, the area of the triangle formed by the

points (5, -1), (x, 4) and (6, 3) is 5.5 square units.

92. a median of triangle divides it into two triangle of equal areas.

Verify this result for ∆ABC whose vertices are A(4, -6), B(3, -2)

and C(5, 2).

93. If the area of a quadrilateral, whose vertices are A, B, C, D

taken in order are (1, 2), (-5, 6), (7, -4) and (k, -2) be zero, find

the value of k.

94. For what value of x will the points (x, 3), (-5, 6) and (-8, 8) be

collinear.

95. Prove that the three points (3a, 0), (0, 3b) and (a, 2b) are

collinear.

96. If (-1, 3), (1, -1) and (5, 1) are the vertices of triangle, find the

length of the median through the first vertex.

97. P, Q and R are three collinear points. P and Q are (3, 4) and

(7, 7) respectively and PR is equal to 10 units; find the

co – ordinates of R.

98. The vertices of a ∆ABC are A (4, 6), B (1, 5) and C (7, 2). A line

is drawn to intersect sides AB and AC at D and E respectively,

such the AD/AB = AE/AC = ¼, Calculate the area of the ∆ABC

and compare it with the are of ∆ABC.

99. Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ∆ABC.

i) The median from A meets BC at D. Find the coordinates of

the point D.

ii) Find the coordinates of the point P on AD such that AP :

PD = 2 : 1

iii) Find the coordinates of points Q and R on medians BE and

CF respectively such that BQ : QE = 2 : 1 and CR : RF =

2 : 1.

100. If A (x1 , y1) , B (x2, y2) and C (x3, y3) are the vertices of ∆ABC,

find the coordinates of the centroid of the triangle.

101. ABCD is a rectangle formed by the points A (-1, -1), B (-1, 4), C

(5, 4) and D (5. -1). P,Q,R and S are the mid – points of AB, BC,

CD and DA respectively. Is the quadrilateral PQRS a square? A

rectangle ? or a rhombus? Justify your answer.

TYPICAL PROBLEM

102. Find the co – ordinates of the point which at the distance of 2

units from (5, 4) & 10 units from 911, -2).

103. The area of a triangle is 5 square units. Two of its vertices are

(2, 1) and (3, -2). The third vertex is (x, y) where y = x + 3.

Find the co – ordinates of the third vertex.

104. Find the area of the triangle formed by the mid – point of sides

of the triangle whose vertices are (2, 1), (-2, 3) and (4, -3).

105. Find the length of the altitude of the triangle, co – ordinates of

whose vertices are (5, 1), (2, 4), and (-1, -1).

106. A, B are the two points (3, 4) and (5, -2). Find the point P such

that PA = PB and the area of ∆PAB equal to 10 square units.

107. A, B, C are the points (-1, 5), (3, 1) and (5, 7) and D, E, F are

the mid – point of BC, CA and AB respectively. Prove that are of

∆ABC is equal to four times the area of ∆DEF.

108. Find the length of the sides of the triangle whose vertices are

(4, 13), (8, 10) and (0.8, 0.4). Show that (2.4, 6.7) is at a

distance 6.5 from each vertex.

109. The co – ordinates of the angular points of a triangle are (x1, y1)

, (x2, y2) and (x3 , y3). The line joining the first two is divided in

the ratio q : p, and the line joining this point of section to the

opposite vertex is then divided in the ratio r : p + q. Find the

co – ordinates of the later point of division.

110. Prove that the co – ordinates x and y, of the mid – point of the

line joining the points (1, 2) and (2, 3) satisfy the equation

x – y +1 = 0.

111. The two opposite vertices of a square are (-1, 2) and (3, 2).

Find the coordinates of the other two vertices.

112. Find the area of the triangle the co – ordinates of whose

angular points are respectively [at1, a/t1] , [at2 , a/t2] and [at3 ,

a/t3]

113. If P and Q are two points whose coordinates are (at2 , 2at) and

[a/t2 , -2a/t] respectively and S is the point (a, 0), show that [1/SP +

1/SQ] is independent of t.

QUESTION FROM BOARD PAPER (S) 11 MARCH 2008

(DELHI & OUTSIDE DELHI BOARD CBSE)

1. For what value of p, the points (-5, 1), (1, p) and (4, -2) are

collinear?

2. For what value of k are the points (1, 1), (3, k) and (-1, 4)

collinear?

OR

Find the area of the ∆ ABC with vertices A (-5, 7), B (-4, -5) and

C (4, 5).

3. For what value of p, are the points (2, 1), (p, -1) and (-1, 3)

collinear?

4. Determine the ratio in which the line 3x + 4y - 9 = 0 divides

the line – segment joining the points (1, 3) and (2, 7).

5. The coordinates of A and B are (1, 2) and (2, 3) respectively. If

P lies on AB. Find the coordinates of P such that AP/PB = 4/3.

6. If the distances of P (x, y) from the points A (3, 6) and B (-3, 4)

are equal, prove that 3x + y = 5.

CONSTRUCTION1. Divide a line segment in the

ratio of a : b (internally)2. Draw a tangent to a circle

when a point is on the circle with using centre.

3. Drew a tangent to a circle when a point is on the circle without using centre.

4. Draw the tangents when a point is out side the circle with using centre.

5. Draw the tangents when a point is outside the circle without using centre.

6. Draw the tangents in such a way the angle between their point of intersection is given.

7. Draw a triangle which is similar to given triangle.

EXTRA CONSTRUCTIONS8. Draw a tangent to a circle

which is parallel to the given line.

9. Draw a tangent to a circle which is perpendicular to the given line.

10. Draw a triangle which is similar to the given triangle (harder cases).

11. Draw a quadrilateral which is similar to given quadrilateral.

QUESTIONS

In each of the following give the justification of the construction also:

1. Draw a line segment of length 7.6 cm and divide it in the ratio

5 : 8. Measure the two parts.

2. Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a

triangle similar to it whose sides are 2/3 of the corresponding

sides of the first triangle.

3. Construct a triangle with sides 5 cm, 6 cm and 7 cm and then

another triangle whose sides are 7/5 of the corresponding

sides of the first triangle.

4. Construct an isosceles triangle whose base is 8 cm & altitude

4 cm and then another triangle whose sides are 1-1/2 times

the corresponding sides of the isosceles triangle.

5. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and /ABC

= 60°. Then construct a triangle whose sides are 3/4 of the

corresponding sides of the triangle ABC.

6. Draw a triangle ABC with side BC = 7 cm, /B = 45° , /A = 105°.

Then, construct a triangle whose sides are 4/3 times the

corresponding sides of ∆ABC.

7. Draw a right triangle in which the sides (other than

hypotenuse) are of length 4 cm and 3 cm. Then construct

another triangle whose sides are 5/3 times the corresponding

sides of the given triangle.

8. Draw a circle of radius 6 cm. From a point 10 cm away from its

centre, construct the pair of tangent to the circle and measure

their lengths.

9. Construct a tangent to a circle of radius 4 cm from a point on

the concentric circle of radius 6 cm and measure its length.

Also verify the measurement by actual calculation.

10. Draw a circle of radius 3 cm. Take two points P and Q on one of

its extended diameter each at a distance of 7 cm from its

centre. Draw tangents to the circle from these two points P and

Q.

11. Draw a pair of tangents to a circle of radius 5 cm which are

inclined to each other at an angle of 60°.

12. Draw a line segment AB of length 8 cm. Taking A as centre,

draw a circle of radius 4 cm and taking B as centre, draw

another circle of radius 3 cm. Construct tangents to each circle

from the centre of the other circle.

13. Let ABC be a right triangle in which AB = 6 cm, BC = 8 cm and

/B = 90°. BD is the perpendicular from B on AC. The circle

through B, C, D is drawn. Construct the tangents from A to this

circle.

14. Draw a circle with the help of a bangle. Take a point outside

the circle. Construct the pair of tangents from this point to the

circle.

15. Draw a triangle ABC with side BC = 7 cm. /B = 45°, /C = 30°.

Then construct a triangle whose sides are 4/3 times the

corresponding sides of ∆ABC.

16. Draw a right triangle ABC in which AC = AB = 4.5 cm and

/A = 90° Draw a triangle similar to ∆ABC with its sides equal to

(5/4)th of the corresponding sides of ∆ABC.

17. Draw a pair of tangents to a circle with centre O of radius 3 cm

from an external point at a distance of 5 cm from the centre.

18. Construct a triangle similar to a given triangle ABC in which

AB = 7 cm, /CAB = 60° and /ABC = 105°, such that each side

of a new triangle is 3/4th of given ∆ABC.

19. Construct a triangle with sides 5 cm, 6 cm and 7 cm and then

another triangle whose sides are 7/5 of the corresponding

sides of the first triangle.

20. Construct a circle whose radius is equal to 4 cm. Let P be a

point whose distance from its centre is 6 cm. Construct two

tangents to it from P.

MIXED STUFF

21. Draw a triangle ABC in which perimeter is 15 cm & the side are

in the ratio of 4 : 2 : 3.

22. Draw a triangle ABC in which AB = 7 cm, BC = 3.2 cm, AD ┴ BC

at D and AD = 4.5 cm.

Also, draw another triangle similar to given triangle such that

its sides are 2/5 of the sides of given triangle.

23. Draw a triangle ABC in which AB + AC = 10 cm, BC = 5 cm &

/C = 75° and then, construct a triangle whose sides are 4/3

times the corresponding sides of ∆ABC.

24. Draw a triangle ABC in which perimeter is 15 cm & the base

angles are 60° & 45°. Then construct another triangle whose

sides are 5/3 times the corresponding sides of the given

triangle.

25. Draw an equilateral triangle ABC whose height 3.2 cm.

Construct a triangle similar a given triangle ABC with its sides

equal to 3/4 of the corresponding sides of the triangle ABC

(i.e., of scale factor 3/4).

26. Draw a quadrilateral ABCD with AB = 3 cm, AD = 2.7 cm,

BD = 3.6 cm, /B = 110° & BC = 4.2 cm. Construct another

quadrilateral A’B’C’D’ similar to a given quadrilateral ABCD so

that BD’ = 4.8 cm.

27. Construct a quadrilateral similar to a given quadrilateral ABCD

in which AB = 6.3 cm, BC = 5.2 cm, CD = 5.6 cm, DA = 7.1 cm

& /B = 60°, whose sides are 4/5th of the corresponding sides of

ABCD.

28. Draw a square about a circle of radius 3.0 cm.

29. Divide 7 cm in the ratio of 2 : 5 internally.

QUESTION FROM BOARD PAPER (S) 11 MARCH 2008 (DELHI &

OUTSIDE DELHI BOARD CBSE)

1. Draw a ∆ ABC with side BC = 6 cm, AB = 5 cm and /ABC = 60.

Construct a ∆ A B’ C’ similar to ∆ ABC such that sides ∆ A B’ C’

are 3/4 of the corresponding sides of ∆ ABC.

2. Construct a ∆ ABC in which AB = 6.5 cm, /B = 60° and BC =

5.5 cm. Also, construct a triangle A B’ C’ similar to ∆ ABC,

whose each sides is 3/2 times the corresponding side of the ∆

ABC.

AREA RELATED TO CIRCLES

Area of Circle : .r2, (r = radius, r = d/2 where d is the diameter of circle.

Circumference : 2r

Area of sector : 0/360 .r2, (0 = central angle/angle of the sector & r =

radius)

Length of an Arc : 0/360. 2r, (0 = central angle/angle of the sector & r =

radius)

Area of segment (Minor) : Ar. of sector – Ar. of ∆, (Ar. of ∆ = 1/2 r2sin0)*

Area of segment (Major) : Area of circle – Area of Minor Segment

: r2 - {(0/360) r2 – 1/2 r2 sin 0}, (0 = central angle &

r = radius)

QUESTIONS :

1. The radii of two circle are 19 cm an 9 cm respectively. Find the

radius of the circle which has circumference equal of the sum

of the circumference of the two circle.

2. The length of a wire which is tied at a boundary of a semi –

Circular Park is 72m. Find the radius of the semi – circular park

and its area.

3. The sum of the radii of two circles is 7cm and the difference of

their circumferences is 8cm. Find the circumferences of the

circles.

4. How long will a man take to walk one round of a circular park

of radius 84cm, @ 2.4km/hr.

5. A wire when bent in the form of an equilateral triangle

enclosed an area of 121 √3cm2. If the same wire is bent in the

form of a circle, find the area of the circle.

6. A copper wire when bent in the form of a square encloses an

area of 121cm2. If the same wire is bent in the form of the

circle, find the area of the circle.

7. A bicycle wheel makes 5000 revolution in moving 11km. Find

the diameter of the wheel.

8. A park is in the form of a triangle 120m x 100cm. At the centre

of the park there is a circular lawn. The area of the park

excluding the lawn is 8700m2. Find the radius of the circular

lawn.

9. A boy is cycling such that the wheels of the cycle are making

140 revolution per minute. If the diameter of each wheel is

60cm, calculate the speed with which the boy is cycling.

10. How long will boy take to cover a round of a circular park of

420m in diameter, walking at a speed of 3 km/hr.

11. The perimeter of a certain sector of a circle of radius 6cm is

20cm. Find the central angle of the sector.

12. In Fig. 1, there are three semicircles A, B and C having

diameter 3cm each and another semicircle E having a circle D

with diameter 4.5cm are shown ; Calculate:

i) the area of he shaded region.

ii) the cost of painting the shaded region at the rate of 25

paise per cm2, to the nearest rupee.

13. PQRS is a diameter of a circle of radius 6 cm. The length PQ,

QR and RS are equal. Semi circles are drawn on PQ and QS as

diameter, Find the perimeter of the shaded region (Fig 2).

14. Find the are of the shaded region n the given Fig 3.

15. The diameter of a wheel of a bus is 90 cm which makes 315

revolutions per minute. Determine its speed in km/hr.

16. The diameter of the front an rear wheels of a tractor are 80 cm

and 2 m respectively. Find the number of revolutions that rear

wheel will make to cover the distance which the front wheel

covers in 1400 revolutions.

17. In an equilateral triangle of side 24 cm a circle is inscribed

touching its sides. Find the area of the remaining portion of the

triangle.

18. An oval shaped meeting table made of wood has its

dimensions as shown in Fig. 4. Find the cost of polishing it at

Rs. 3.50 per sq. m. (Use = 3.14).

19. The radii of two circles are 8 cm and 6 cm respectively. Find

the radius of he circle having are equal to the sum of the areas

of the two circles.

20. The wheels of a car are of diameter 80 cm each. How many

complete revolution does each wheel make in 10 minutes

when the car is traveling at a speed of 66 km per hour?

21. The length of an arc subtending an angle of 72° at the centre is

22cm. Find the area of the circle.

22. The minute hand of a clock is 10cm long. Find the area on the

face of he clock described by the minute hand between 9 A.M.

and 9.35 A.M.

23. The length of the minute hand of a clock is 14 cm. Find the

area swept by the minute hand in 1 hr an in 1min.

24. A rope by which a cow is tethered is increased from 12m to

23m. How much additional ground has it to browse over?

25. A circular disc of 6cm radius is divide into three sectors with

central angles 120°, 150° and 90°. What part of the whole

circle is the sector with central angle 120°? Also give the ratio

of the areas of the three sectors.

26. The min-hand of a clock is 3.5cm long. Find the area swept by

it in 20 min.

27. In a circle of radius 21 cm, an arc subtends an angle of 60° at

the centre. Find :

i) the length of the arc ii) area of the sector formed by the

arc

iii) area of the segment formed by the corresponding chord.

28. A chord of a circle of radius 15 cm subtends an angle of 60° at

the centre. Find the areas of the corresponding minor and

major segments of the circle. (Use = 3.14 and √3 = 1.73)

29. The measure of the minor arc of a circle is 1/5 of the measure

of the corresponding major arc. If the radius of the circle is

10.5cm, find the area of the sector corresponding to the major

arc.

30. If the perimeter and area of a circle are numerically equal, find

the radius of the circle.

31. The quadrants shown in the Fig 5. are each of radius 7cm.

Calculate the area of the shaded portion.

32. A horse is tied to a peg at one corner of a square shaped grass

field of side 15m by means of a 5m long rope. Find

i. the area of that part of the field in which the horse can graze.

ii. the increase in the grazing area if the rope were 10m long

instead of 5m. (Use = 3.14).

33. A car has tow wipers which do not overlap. Each wiper has a

blade of length 25cm sweeping through an angle of 115°. Find

the total area cleaned at each sweep of the blades.

34. A brooch made with silver wire in the form of a circle with

diameter 35 mm. The wire is also used in making 5 diameters

which divide the circle into 10 equal sectors as shown in Fig. 6.

Find:

i) the total length of the silver wire required.

ii) the area of each sector of the brooch.

35. An umbrella has 8 ribs which are equally spaced (Fig. 7).

Assuming umbrella to be a flat circle of radius 45 cm, find the

area between the consecutive ribs of the umbrella.

36. A round table cover has six equal designs as shown in Fig. 8. if

the radius of the cover is 28 cm, find the cost of making the

designs at the rate of Rs 0.35 per cm2. (Use √3 = 1.7)

37. A chord of a circle of radius 10 cm subtends a right angle at

the centre. Find the area of the corresponding.

i) minor segment ii) major sector. (use = 3.14)

38. Four equal circles, each of radius = a, touch each other. Show

that the area between them is nearly 6/7 a2.

39. To warn ships for underwater rocks, a lighthouse spreads a red

coloured light over a sector of angle 80° to a distance of 16.5

km. Find the area of the sea over which the ships are warned.

(Use = 3.14).

40. In Fig. 9, two circular flower beds have been shown on two

sides of a square lawn ABCD of side 6 m. If the centre of each

circular flower bed is the point of intersection O of the

diagonals of the square lawn, find the sum of the areas of the

lawn and the flower beds.

41. Find the area of the shaded region in Fig. 10, where ABCD is a

square of side 14 cm.

42. Find the area of the shaded design in Fig. 11, where ABCD is a

square of side 10cm and semicircles are drawn with each side

of the square as diameter. (Use = 3.14)

43. Find the area of the shaded region in Fig. 12, if PQ = 24 cm,

PR = 7 cm and O is the centre of the circle.

44. Find the area of the shaded region in Fig. 13, if ABCD is a

square of side 14 cm and APD and BPC are semicircles.

45. Find the area of the shaded region in Fig. 14, if radii of the two

concentric circles with centre O are 7 cm and 14 cm

respectively and /AOC = 40°.

46. Find the area of the shaded region in Fig. 15, where a circular

arc of radius 6 cm has been drawn with vertex O of an

equilateral triangle OAB of side 12 cm as centre.

47. In Fig. 16, ABCD is a square of side 14 cm. With centres A, B, C

and D four circles are drawn such that each circle touch

externally tow of the remaining three circles. Find the area of

the shaded region.

48. In a circular table cover of radius 32 cm, a design is formed

leaving an equilateral triangle ABC in the middle as shown in

Fig.17. Find the area of the design (shaded region).

49. From each corner of a square of side 4 cm a quadrant of a

circle of radius 1 cm is cut and also a circle of diameter 2 cm is

cut as shown in Fig.18. Find the area of the remaining portion

of square.

50. Fig. 19, depicts a racing track whose left and right ends are

semicircular.

The distance between the two inner parallel line segments is

60 m and they are each 106 m long. If the track is 10 m wide,

Find :

i) The distance around the track along its inner edge.

ii) The area of the track.

51. In Fig. 20, AB and CD are two diameters of a circle (with centre

O) perpendicular to each other and OD is the diameter of the

smaller circle. If OA = 7cm, find the area of the shaded region.

52. The area of an equilateral triangle ABC is 17320.5 cm2. With

each vertex of the triangle as centre, a circle is drawn with

radius equal to half the length of the side of the triangle (Fig.

21). Find the area of the shaded region. (Use = 3.14 and √3

= 1.73205).

53. On a square handkerchief, nine circular designs each of radius

7 cm are made (Fig. 22). Find the area of the remaining portion

of the handkerchief.

54. In Fig. 23, OACB is a quadrant of a circle with centre O and

radius 3.5 cm. If OD = 2 cm, Find the area of the i) quadrant

OACB, ii) shaded region.

55. In Fig. 24, a square OABC is inscribed in a quadrant OPBQ. If

OA = 20 cm, find the area of the shaded region. (Use =

3.14).

56. AB and CD are respectively arcs of two concentric circles of

radii 21 cm and 7 cm with centre O (Fig. 25). If /AOB = 30°,

find the area of the shaded region.

57. In Fig. 26. ABC is a quadrant of a circle of radius 14 cm and a

semicircle is drawn with BC as diameter. Find the area of the

shaded region.

58. Calculate the area of the designed region in Fig. 27 common

between the two quadrants of circles of radius 8 cm each.

MIXED STUFF

59. A fence is to be erected around a circular field. The cost of

fencing at the rate of Rs. 2 per meter is Rs. 2640. Find the cost

of ploughing the field at Rs. 0.50 per m2.

60. A car is moving with the speed of 22m/sec. Find the diameter

of the wheel if it performs 700 revolution per second.

61. A path of 4 m width runs round a circular grassy plot whose

circumferences is 163-3/7 m. Find:

i. the area of the path

ii. the cost of gaveling the path at the rate of Rs. 150 per square

meter.

62. The perimeter of a sector of a circle of radius 5.7m is

27.2m. Find the area of the sector.

63. The area of an equilateral triangle is 49√3 cm2. Taking

each angular point as centre, a circle is described with

radius equal to half the length of the side of the triangle.

Find the area of the triangle not included in the circle.

64. A chord of a circle of radius 12 cm subtends as angle of

120° at the centre. Find the area of the corresponding

segment of the circle.

(Use = 3.14 and √3 = 1.73)

65. In Fig.28, shows a kite in which BCD is in the shape of

quadrant of a circle of radius 42 cm. ABCD is a square and

∆CEF is an isosceles right angled triangle whose equal

sides are 6 cm long. Find the area of the shaded region.

SURFACE AREA & VOLUME

BASIC QUESTIONS

1. The volume of a wall 5 times as high as it is broad & 8

times as long as it is high, is 18225 cu m, find the breadth of

the wall.

2. The cost of a wood of Rs. 1500 per cu. m. A certain cube

of that wood was bought for Rs. 768. Find the edge of the

cube.

3. The 3 coterminous edges of a rectangular solid are 36

cm, 75 cm & 80 cm respectively. Find the edge of a cube which

will be of the same capacity.

4. If the radius of the base of rt. circular cylinder is halved,

keeping the height same, What is the ratio of volume of the

reduced cylinder to that of original?

5. Two right circular cones X & Y are made, X having there

times the radius of Y and Y having half the volume of X.

Calculate the ratio of heights of X & Y.

6. A semicircular thin sheet of paper of diameter 28 cm is

bent & an open conical cup is made. Find the capacity of the

cone.

7. A cone of maximum volume is cut out of a cuboid 20cm

long & having a cross section a square of side 10 cm. Calculate

the volume of cone.

8. A metallic cylinder has radius 3cm & height 5cm. It is

made of a metal A. To reduce its weight, a conical hole is

drilled in the cylinder & it is completely filled with a lighter

metal B. The conical hole has radius of 3/2 cm & its depth is

8/9 cm. Calculate the ratio of the volume of the metal A to the

volume of metal B in the solid.

9. Find the canvas required for a 3m high conical tent in

which a boy of 1.5m height may just stand straight at a

distance of 2m from the centre.

10. The difference between outer & the inner surface area of

a cylinder 14 cm long is 88 sq. cm. Find the outer & inner radii

of the cylinder, given that the volume of metal used is 176 cu.

cm.

11. The curved surface of a cylinder is 1000 sq. cm. A wire

of diameter 5mm is wound round it so as to cover it

completely. Find the length of the wire.

12. A cooper wire of 0.2 cm in diameter is evenly wound

about a cylinder whose length is 12 cm & diameter 10 cm so as

to cover curved surface. Find the weight of the wire if relative

density (R.D.) of copper is 8.88.

13. A right angled triangle, whose remaining angles are 60°

and 30° revolves about the hypotenuse which is 84 cm long.

Find volume of double cone so formed.

14. The areas of the three adjacent faces of a cuboid are x,

y & z. If the volume is V, prove that V2 = x. y. z.

15. The V be the volume of a cuboid of dimensions are a, b,

c, & S is the surface area, then prove that [1/v] = [2/S] [[1/a]

+ [1/b] + [1/c]].

16. The h, c & V are the height, the curved surface area &

volume of a cone respectively. Prove that 3Vh3 – c2h2 + 9V2 =

0.

17. 2 cubes each of volume 64 cm3 are joined end to end.

Find the surface area of the resulting cuboid. Otherwise, take

= 22/7.

18. A sector of a circle of radius 6 cm has an angle of 120°.

It is rolled up so that the two bounding radii are joined together

to form a cone. Find (a) radius of the cone (b) the total surface

area of the cone (c) the volume of the cone.

19. A rectangular strip of 28cm by 7cm is rolling along

28cm/along 7cm. Find the volume of cylinder so formed. (in

each case).

20. A rectangle, whose sides are 3cm and 4cm containing

right angle is revoling about 4cm/3cm. Find the volume of

cylinder so formed. (in each case)

21. A right triangle, whose sides are 3 cm and 4 cm (other

than hypotenuse) is made to revolve about its hypotenuse.

Find the volume and surface are of the double cone so formed.

CONVERSION OF SOLID

22. Selvi’s house has an overhead tank in the shape of a cylinder.

This is filled by pumping water from a sump (an underground

tank) which is in the shape of a cuboid. The sump has

dimensions 1.57m x 1.44m x 95cm. The overhead tank has its

radius 60 cm and height 95 cm. Find the height of the water

left in the sump after the overhead tank has been completely

filled with water from the sump which had been full. Compare

the capacity of the tank with that of the sump. (Use = 3.14)

23. A metallic sphere of radius 4.2 cm is melted and recast into the

shape of a cylinder of radius 6cm. Find the height of the

cylinder.

24. Metallic spheres of radii 6 cm, 8 cm and 10 cm, respectively,

are melted to from a single solid sphere. Find the radius of the

resulting sphere.

25. A cylindrical bucket, 32cm high and with radius of base 18 cm,

is filled with sand. This bucket is emptied on the ground and a

conical heap of sand is formed. If the height of the conical heap

is 24 cm, find the radius and slant height of the heap.

26. What length of a solid cylinder 2cm in diameter must be taken

to recast into a hollow cylinder of external diameter 20cm,

0.25cm thickness & 15m long?

27. A cylindrical bucket 28 cm in diameter & 72 cm high is full of

water. The water is emptied into a rectangle tank 66cm long, &

28 cm wide. Find the height of the water level in the tank.

28. Eight metallic spheres each of radius 2mm are melted & cast

into a single sphere. Calculate the radius of the new single

sphere.

29. A hollow metallic cylindrical tube has an internal radius 3cm &

height 21cm. The thickness of the tube is 0.5 cm. The tube is

melted & cast into a right circular cone of height 7cm. Find the

radius of the cone.

30. A copper rod of diameter 1 cm and length 8 cm is drawn into a

wire of length 18 m of uniform thickness. Find the thickness of

the wire.

31. An soil funnel made of tin sheet consists of a 10 cm long

cylindrical portion attached to a frustum of a cone. If the total

height is 22 cm, diameter of the cylindrical portion is 8 cm and

the diameter of the top of the funnel is 18 cm, find the area of

the tin sheet required to make the funnel (Fig. 1)

COMBINATION OF SOLID

32. A wooden toy rocket is in the shape of a cone mounted on a

cylinder, as shown in Fig. 2. The height of the entire rocket is

26 cm, while the height of the conical part is 6 cm. The base of

the conical portion has a diameter of 5 cm, while the base

diameter of the cylinder portion is 3 cm. If the conical portion is

to be painted black and the cylindrical portion white, find the

area of the rocket painted with each of these colours. (Take

= 3.14).

33. A right circular cone has been placed upon circular cylinder.

The base of the cone fully coincides with cylinder & covers the

base of cylinder. If the area of the base of the cylinder is 154

sq. cm, height of the cylinder is 10 cm & volume of entire solid

is 1848 cu. cm, calculate the total height of the solid.

34. A canvas tent is in the shape of a cylinder surmounted by a

conical roof. The common diameter of cone & cylinder is 14m.

The height of cylinder is 8m & height of conical roof is 4m.

Find the area of canvas used to make the tent.

35. An open cylindrical vessel of internal diameter 49cm & height

64cm stands on a horizontal platform. Inside this is placed a

solid metallic right circular cone whose base has diameter of

10-1/2 cm & whose height is 12cm. Calculate the volume of

water required to fill the tank.

36. A godown building is in the form as shown in Fig. 3 The vertical

cross section parallel to the width side of the building is a

rectangle of size 7m x 3m mounted by a semicircle of radius

3.5m. Find the volume of the godown and the total internal

surface area excluding the floor.

37. A vessel is in the form of a hollow hemisphere mounted by a

hollow cylinder. The diameter of the hemisphere is 14 cm and

the total height of the vessel is 13 cm. Find the inner surface

area of the vessel.

38. A toy is in the form of a cone of radius 3.5 cm mounted on a

hemisphere of same radius. The total height of the toy is

15.5 cm. Find the total surface area of the toy.

39. A cubical block of side 7 cm is surmounted by a hemisphere.

What is the greatest diameter the hemisphere can have? Find

the surface area of the solid.

40. A hemisphere depression is cut out from one face of a cubical

wooden block such that the diameter ∫ of the hemisphere is

equal to the edge of the cube. Determine the surface area of

the remaining solid.

41. A medicine capsule is in the shape of a cylinder with two

hemispheres stuck to each of its ends (see Fig. 4). The length

of the entire capsule is 14 mm and the diameter of the capsule

is 5 mm. Find its surface area.

42. A tent is in the shape of a cylinder surmounted by a conical

top. If the height and diameter of the cylindrical part are 2.1 m

and 4 m respectively, and the slant height of the top is 2.8 m,

find the area of the canvas used for making the tent. Also, find

the cost of the canvas of the tent at the rate of Rs. 500 per m2.

43. From a solid cylinder whose height is 2.4 cm and diameter

1.4 cm, a conical cavity of the same height and same diameter

is hollowed out. Find the total surface area of the remaining

solid to the nearest cm2.

44. A wooden article was made by scooping out a hemisphere from

each end of a solid cylinder, as shown in Fig. 5. If the height of

the cylinder is 10 cm, and its base is of radius 3.5 cm, find the

total surface area of the article.

45. A solid toy is in the form of a hemisphere surmounted by a

right circular cone. The height of the cone is 2 cm and the

diameter of the base is 4 cm. Determine the volume of the toy.

If a right circular cylinder circumscribes the toy, find the

difference of the volumes of the cylinder and the toy. (Take =

3.14).

46. A solid is in the shape of a cone standing on a hemisphere with

both their radii being equal to 1 cm and the height of the cone

is equal to its radius. Find the volume of the solid in terms of .

47. Anubhav, an engineering students, was asked to make a model

shaped like a cylinder with two cones attached at its two ends

by using a thin aluminum sheet. The diameter of the model is 3

cm and its length is 12 cm. If each cone has a height of 2 cm,

find the volume of air contained in the model that Anubhav

made. (Assume the outer and inner dimensions of the model to

be nearly the same.)

48. A gulab jamun, contains sugar syrup up to about 30% of its

volume. Find approximately how much syrup would be found in

45 gulab jamuns, each shaped like a cylinder with two

hemisphere ends with length 5 cm and diameter 2.8 cm.

49. A pen stand made of wood is in the shape of a cuboid with four

conical depressions to hold pens. The dimensions of the cuboid

are 15 cm by 10 cm by 3.5 cm. The radius of each of the

depressions is 0.5 cm and the depth is 1.4 cm. Find the volume

of wood in the entire stand.

50. A solid iron pole consists of a cylinder of height 220 cm and

base diameter 24 cm, which is surmounted by another cylinder

of height 60 cm and radius 8 cm. Find the mass of the pole,

given that 1 cm3 of iron has approximately 8g mass. (Use =

3.14)

51. A solid consisting of a right circular cone of height 120 cm and

radius 60 cm standing on a hemisphere of radius 60 cm is

placed upright in a right circular cylinder full of water such that

it touches the bottom. Find the volume of water left in the

cylinder, if the radius of the cylinder is 60 cm and its height is

180 cm.

52. A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm

in diameter; the diameter of the spherical part is 8.5 cm. By

measuring the amount of water it holds, a child finds its

volume to be 345 cm3. Check whether she is correct, taking the

above as the inside measurements, and = 3.14.

FRUSTUM

53. The height of a cone is 30cm. A small cone is cut off at the top

by a plane to its base. If its volume be (1/27)th of the volume

of the given cone, at what height above the base is the section

made.

54. A hollow cone is cut by a plane parallel to the base and the

upper portion is removed. If the curved surface of he

remainder is 8/9 of the curved surface of the whole cone, find

the ratio of the line segment into which the cone’s altitude is

divide by the plane.

55. A drinking glass is in the shape of a frustum of a cone of height

14 cm. The diameters of its two circular ends are 4 cm and 2

cm. Find the capacity of the glass.

56. The slant height of a frustum of a cone is 4 cm and the

perimeters (circumference) of its circular ends are 18 cm and

6 cm. Find the curved surface area of the frustum.

57. A fez, the cap used by the Turks, is shaped like the frustum of

a cone. If its radius on the open side is 10 cm, radius at the

upper base is 4 cm and its slant height is 15 cm, find the area

of material used for making it.

58. A container, opened from the top and made up of a metal

sheet, is in the form of a frustum of a cone of height 16 cm

with radii of its lower and upper ends a 8 cm and 20 cm

respectively. Find the cost of the milk which can completely fill

the container, at the rate of Rs. 20 per liter. Also find the cost

of metal sheet used to make the container, if it costs Rs 8 per

100 cm2. (Take = 3.14).

59. A metallic right circular cone 20 cm high and whose vertical

angle is 60° is cut into two parts at the middle of its height by

a plane parallel to its base. If the frustum so obtained be drawn

into a wire of diameter 1/16 cm, find the length of the wire.

60. A cone of height d is cut by a parallel at a distance d/3 from

the base. Show that the volume of the frustum produced is

70.37% of the original cone.

61. If the height of a frustum of a cone is twice the mean

proportional between the radii of its base, show that the slant

height equals the sum of their radii.

SPECIAL CASES : WATER FLOW

62. The water of a river is flowing with a speed of 30km/hr. If the

average breadth & depth of the river are 10.5m & 2.4m

respectively, calculate the volume of water which is flowing

every hour in the river.

63. Water is flowing at a rate of 6400 It/sec in a tank 40m long &

32m wide. How many cm per sec the water is rising in the

tank?

64. A rectangle water reservoir is 18m by 16m at the base. Water

flows into it through a pipe whose cross section is 4cm X 3cm

at a rate of 12m per sec. Find the height to which the water will

rise in the reservoir in 20 minutes.

65. Water flows through a pipe (cylindrical) of internal diameter

7cm at 5m/sec. Find the time taken in minutes, the pipe would

take to fill an empty rectangular tank 4m X 3m x 2.31m.

66. Water flows through a cylindrical pipe of internal diameter 7cm

at 36km/hr. Calculate the time in minutes it would take to fill

cylindrical tank, the radius of whose base is 35cm & height 1m.

67. A hemispherical tank full of water is emptied by a pipe at the

rate of 3-4/7 litres per second. How much time will it take to

empty half the tank, if it is 3m in diameter? (Take = 22/7).

68. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a

speed of 10 km/h. How much area will it irrigate in 30 minutes,

if 8 cm of standing water is needed?

69. A farmer connects a pipe of internal diameter 20 cm from a

canal into a cylindrical tank in her field, which is 10 m in

diameter and 2 m deep. If water flows through the pipe at the

rate of 3 km/h, in how much time will the tank be filled?

EARTH DUG OUT

70. A 20 m deep well with diameter 7 m is dug and the earth from

digging is evenly spread out to form a platform 22 m by 14 m.

Find the height of the platform.

71. Find the volume of earth dug out to make a well 3.5m deep &

2m in diameter. Find the cost of plastering its inner curved

surface at the rate of Rs. 75per m2.

72. A field is 250m long & 30m broad. A tank 30m long, 10m broad

& 6m deep is dug in the field & the earth taken out of it, is

spread evenly over the field. Find how much the level of field is

raised.

73. A well of diameter 3m is dug 14m deep. The earth taken out of

it has been spread evenly all around it in the shape of a

circular ring of width 4 m to form an embankment. Find the

height of the embankment.

CALCULATING “n

74. A wall of dimensions 40cm by 75cm by 5m is to be built by

using bricks of dimensions 25cm by 5cm by 10cm. Find the

number of bricks required.

75. How many bricks will be required to build a wall 30m long,

30cm thick & 4m high with a provision of 1 door being 3.0m X

1.2m, each brick being 25cm x 20cm x 8cm when (1/9)th of the

wall is filled with mortar?

76. A barrel of fountain pen, cylindrical in shape is 7cm long &

5mmm diameter. A full barrel of ink in the pen will be used up

when writing 330 words on an average. How many words

would use up a bottle of ink with (1/2)th litre?

77. A right circular cylinder having diameter 21cm and height

38cm is full of ice-cream. The ice-cream is to be filled in cones

of height 12cm and diameter 7cm having hemispherical top.

Find the number of such cones which can be filled with ice-

cream.

78. A vessel is in the form of an inverted cone. Its height is 8cm

and the radius of its top, which is open, is 5cm. It is filled with

water up to the brim. When lead shots, each of which is a

sphere of radius 0.5 cm are dropped into the vessel, one-fourth

of the water flows out. Find the number of lead shots dropped

in the vessel.

79. A container shaped like a right circular cylinder having

diameter 12 cm and height 15 cm is full of ice cream. The ice

cream is to be filled into cones of height 12 cm and diameter 6

cm, having a hemispherical shape on the top. Find the number

of such cones which can be filled with ice cream.

80. How many silver coins, 1.75 cm in diameter and of thickness

2 mm, must be melted to form a cuboid of dimensions 5.5 cm x

10cm x 3.5 cm?

MIXED STUFF

81. A copper wire, 3 mm in diameter, is wound about a cylinder

whose length is 12 cm, and diameter 10 cm, so as to cover the

curved surface of the cylinder. Find the length and mass of the

wire, assuming the density of copper to be 8.88 g per cm3.

82. A sphere and a cube have the same surface. Show that the

ratio of the volume of the sphere to that of the cube is √6 : √

83. Volume of a cuboid is 900cm3 & its height is 12cm. The cross

section is a rectangle with the length & breadth in 3 : 1. Find

the perimeter of cross section.

84. A copper wire of diameter 6 mm is evenly wrapped on a

cylinder of length 18 cm and diameter 49 cm, to cover the

whole surface. Find the length and the volume of the wire. If

the specific gravity of the wire be 88.8 gm/cm3, find the weight

of the wire.

85. If the diameter of the cross-section of a wire is decreased by

5%, how much percent will the length be increased so that the

volume remains the same?

86. The decorative block is made of two solids – a cube and a

hemisphere. The base of the block is a cube with edge 5 cm,

and the hemisphere fixed on the top has a diameter of 4.2 cm.

Find the total surface area of the block. (Take = 2/7)

87. That radius of the base of right circular cone is R. It is cut by a

plane parallel to the base at a height h from the base and the

distance of boundary of upper surface from the midpoint of

base of frustum is 1/3 √ 9h2 + R2 , show that volume of frustum

is 13/27R2h.

88. Derive that formula for the curved surface area and total

surface area of the frustum of a cone, using the symbols as

explained.

89. Derive that formula for the volume of the frustum of a cone,

using the symbols as explained.

90. A tank measures 2m long, 1.6m wide & 1m in depth. Water is

there upto 0.4m height. Bricks measuring 25m x 14cm x 10cm

are put into tank so that the water may come up to the top.

End brick absorbs water equal to (1/7)th of its own volume. How

many bricks will be needed?

QUESTION FROM BOARD PAPER(S) 11 MARCH 2008 (DELHI & OUTSIDE DELHI BOARD CBSE)

1. A gulab Jamun, when ready for eating, contains sugar syrup of

about 30% of its volume. Find approximately how much syrup

would be found in 45 such gulab jamuns, each shaped like a

cylinder with two hemispherical ends, if the complete length of

each of them is 5 cm and its diameter is 2.8 cm.

OR

A container shaped like right circular cylinder having diameter

12cm and height 15 cm is full of ice-cream. This ice-cream is to

be filled into cones of height 12 cm and diameter 6 cm, having

a hemispherical shape on the top. Find the number of such

cones which can be filled with ice-cream.

2. A bucket made up of a metal sheet is in the form of a frustum

of a cone of height 16 cm with diameters of its lower and upper

ends as 16 cm and 40 cm respectively. Find the volume of the

bucket. Also, find the cost of the bucket if the cost of metal

sheet used is Rs. 20 per 100 cm2.

OR

A farmer connects a pipe of internal diameter 20 cm from a

canal into a cylindrical tank in his field which is 10 m in

diameter and 2 m deep. If water flows through the pope at the

rate of 6 km/h., in how much time will the tank be filled?

3. A tent consists of frustum of a cone, surmounted by a cone. If

the diameters of the upper and lower circular ends of the

frustum be 14 m and 26 m respectively. The height of the

frustum be 8 m and the slant height of the surmounted conical

portion be 12m, find the area of canvas required to make the

tent. (Assume that the radii of the upper circular end of the

frustum and the base of surmounted conical portion are equal).

STATISTICS

1. Find the mean of the following frequency distribution:

Classes 0 -10 10 - 20 20 – 30 30 - 40 40 – 50

Frequenc

y

7 8 12 13 10

2. The following table gives the literacy rate (in percentage) of 35

cities. Find the mean literacy rate.

Literacy Rate (in %)

45-55 55 – 65 65 – 75 75 - 85 85 – 95

Number of Cities

3 10 11 8 3

3. To find the out the concentration of SO2 in the air (in parts per million,

i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:

Concentration of SO2 (in

pm)

0.00-0.04 0.04–0.08 0.08–0.12 0.12-0.16 0.16–0.20 0.20-0.24

Frequency 4 9 9 2 4 2

Find the mean concentration of SO2 in the air.

4. The following distribution shows the daily pocket allowance of

children of a locality. The mean pocket allowance is Rs. 18.

Find the missing frequency f.

Daily Pocket allowance (in Rs)

11-13 13-15 15-17 17-19 19-21 21-23 23-25

Number of Children

7 6 9 13 f 5 4

5. The following frequency distribution gives the city-wise

teacher-student ratio in senior secondary schools of 50 cities.

Find the mean of this date:

Number of students per teacher

15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55

Number of cities

10 8 9 15 3 2 1 2

6. A class teacher has the following absentee record of 40

students of a class for the whole term. Find the mean number

of days a student was absent.

Number of days

0-6 6-10 10-14 14-20 20-28 28-38 38-40

Number of Students

11 10 7 4 4 3 1

Age (in year)

5-14 15-24 25-34 35-44 45-54 55-64 65-74

Number of cases

8 10 12 8 15 17 10

7. The following table shows that age distribution of cases of

Jaundice reported during a year in a particular city. Find the

mean using suitable method.

8. The mean of the following frequency distribution is 50. Find the

missing frequencies f1 and f2.

Classes 0-20 20-40 40-60 60-80 80-100 Total

Frequenc

y

17 f2 32 f2 19 120

9. The following table gives the enrolment in schools in 2008.

Find the mean enrolment per school.

Enrolment 200-249 250-299 300-349 350-399 400-449 450-499

Number of children

198 312 537 429 378 146

10. The distribution below shows the number of wickets taken by

bowlers in one-day cricket matches. Find the mean number of

wickets by choosing a suitable method. What does the mean

signify?

Number of

Wickets

20-60 60-100 100-150 150-250 250-350 350-450

Number of

bowlers

7 5 16 12 2 3

11. The table below gives the percentage distribution of female

teachers in the primary schools or rural areas of various States

and Union Territories (UT) of India. Find the mean percentage

of female teachers by all the three methods.

Percentage of

female teachers

15-25 25-35 35-45 45-55 55-65 65-75 75-85

Number of States/UT

6 11 7 4 4 2 1

12. A student noted the number of cars passing through a spot on

a road for 100 periods each of 3 minutes and summarised it in

the table given below. Find the made of the data:

Number of car

0-10 10-

20

20-

30

30-

40

40-

50

50-

60

60-

70

70-

80

Frequenc

y

7 14 13 12 20 11 15 8

13. The given distribution shows the number of runs scored by

some top batsman of the world in one-day international cricket

matches.

Runs Scored (in thousand)

3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11

No. of batsman

4 18 9 7 6 3 1 1

14. The following data give the information on the observed

lifetimes (in hours) of 225 electrical components:

Lifetimes (in hours)

0-20 20-40 40-60 60-80 80-100 100-

120

Frequenc

y

10 35 52 61 38 29

15. The following data gives the distribution of total monthly household expenditure of 200

families of a village. Find the modal monthly expenditure of the families. Also, find the

mean monthly expenditure:

Expenditure (in Rs.

hundreds)

10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50

No. of Families

24 40 33 28 30 22 16 7

16. The following distribution gives the state-wise teacher –

student ration in higher secondary schools of India. Find the

mode and mean of this data. Interpret the two measures.

Number of students per teacher

15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55

Number of Students/UT

3 8 9 10 3 0 0 2

17. Find the median for the following data:

Class interval

0-20 20-40 40-60 60-80 80-100 100-

120

Frequenc

y

9 16 24 15 4 2

18. The distribution below gives the weights of 30 students of a

class. Find the median weight of the students.

Weight (in kg)

40-45 45-50 50-55 55-60 60-65 65-70 70-75

Number of

Students

2 3 8 6 6 3 2

19. The following table gives the distribution of the life time of 400

neon lamps:

Life Time (in Hrs)

1500-2000 2000-2500 2500-3000 3000-3500 3500-4000 4000-4500 4500-5000

No. of Lamps

14 56 60 86 74 62 48

Find the median life time of a lamp.

20. A survey regarding the heights (in cm) of 50 girls of class X of a school was conducted

and the following data was obtained:

Height (in cm)

less than140

less than145

less than150

less than155

less than160

less than165

No. of girls

2 13 17 31 42 50

Determine the median height.

21. A life insurance agent found the following data for distribution of ages of 100 policy

holders. Calculate the median age if policies are given only to persons having age 18

years onwards but less than 60 years.

Age (in

years)

Below20

Below25

Below30

Below35

Below40

Below45

Below50

Below55

Below60

No. of policy Holde

r

2 6 24 45 78 89 92 98 100

22. If the median of the distribution given below is 28.5, find the value of x and y.

Class Interval

0-10 10-20 20-30 30-40 40-50 50-60 Total

Frequenc

y

5 X 20 15 Y 5 60

23. If the median of the following frequency distribution is 46, find

the missing frequencies.

Class Interval

10-20 20-30 30-40 40-50 50-60 60-70 70-80 Total

Frequenc

y

12 30 F1 65 f2 25 18 229

24. The following table gives production yield per hectare of wheat

of 100 farms of a village.

Production yield (in kg/ha)

50-55 55-60 60-65 65-70 70-75 75-80

Number of farms

2 8 12 24 38 16

Change the distribution to a “more than” type cumulative

frequency distribution and draw its give.

25. Computer the median for each of the following data:

26.

During the medical check – up of

35 students of a class, their weights were recorded as follows:

Weight(in kg)

Less than38

Less than40

Less than42

Less than44

Less than46

Less than48

Less than50

Lessthan52

Marks No. of Students

More than 150

More than 140

More than 130

More than 120

More than 110

More than 100

More than 90

More than 80

0

12

27

60

105

124

141

150

Marks No. of Students

Less than 10

Less than 30

Less than 50

Less than 70

Less than 90

Less than 110

Less than 130

Less than 150

0

10

25

43

65

87

96

100

No. of Stu.

0 3 5 9 14 28 32 35

Draw a “less than” type ogive for the given data. Hence, obtain

the median weight from the graph and verify the result by

using the formula.

27. Age (in years) of 100 persons of a small locality were recoded and data is presented

below:

Age (in years)

0-10 10-20 20-30 30-40 40-50 50-60 60-70

Number of

Persons

5 15 20 23 17 11 9

Draw “less than” ogive and “more than” ogive simultaneously

on the same graph and find the median of the data from the

graph. Also, verify your result by using the formula.

28. Draw “more than” ogive for the following distribution:

Class Interval

0-10 10-20 20-30 30-40 40-50 50-60

Frequency

Students

5 3 10 6 4 2

MIXED STUFF

29. Consider the following distribution of daily wages of 50 workers

of a factory.

Daily Wages (in 100-120 120-140 140-160 160-180 180-200

Rs)

Number of Workers

12 14 8 6 10

Find the mean daily wages of the workers of the factory by

using an appropriate method.

30. Thirty women were examined in a hospital by a doctor and the

number of heart beats per minute were recorded and

summarised as follows. Find the mean heart beats per minute

for those women.

Number of heart

beats per minute

65-68 68-71 71-74 74-77 77-80 80-83 83-86

Number of

women

2 4 3 8 7 4 2

31. Find the mode of the following frequency distribution:

Class Interval

25-30 30-35 35-40 40-45 45-50 50-55 55-60

Frequenc

y

12 16 8 10 8 2 4

32. Calculate the model monthly income of the employee of a

factory from the frequency distribution given below:

Daily Income

(in Thousand

)

0-5 5-10 10-15 15-20 20-25 25-30

No. of Employee

s

90 10 100 80 70 10

33. The length of 40 leaves of a plant are measured correct to the

nearest millimentre and the data obtained is represented in the

following table:

Length (in mm)

118-126 127-135 136-144 145-153 154-162 163-171 172-180

Number of

Leaves

3 5 9 12 5 4 2

Find the median length of the leaves.

34. The following distribution gives the daily income of 50 workers

of a factory.

Daily Income

100-120 120-140 140-160 160-180 180-200

Number of

Workers

12 14 8 6 10

Convert the distribution given above a “less than” type

cumulative frequency distribution and draw its ogive.

35. Draw ‘less than’ ogive for the following frequency distribution :-

Mark 0-20 20-40 40-60 60-80 80-100

Number of

Students

7 12 23 18 10

Also, find the median from the ogive and verify that by using

the formula.

36. The marks of 200 students in a test were recorded as follows:

Marks 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89

Number of

Students

7 11 20 46 24 37 15 7

Draw “more than” type cumulative frequency table.

Draw “more than” ogive and use it to find:

i) the media; ii) the number of students who scored more than 35%

marks.

37. The frequency distribution of the weight of 120 birds is recorded below:

Weight (ingrams

)

141-150 151-160 161-170 171-180 181-190 Total

Number of Birds

6 28 48 30 8 120

Find the median weight of a bird.

38. Construction of a dam was completed in 220 days and data about the number of daily

workers employed is as below:

Number of

workers (per day)

20-25 25-30 30-35 35-40 40-45 45-50

Number of days

20 26 70 40 30 34

Find the median of the data and interpret the result.

39. The following frequency distribution gives the monthly

consumption of 100 consumers of a locality. Find the median,

mean and mode of the data and compare them.

Monthly consumption

(in units)

70-90 90-110 110-130 130-150 150-170 170-190

Number of Consumers

8 12 14 26 24 16

40. Draw “Less than” ogive for the following frequency distribution:

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Number of Students

5 3 4 3 3 4 7 9 7 5

41. Construct the “less than” ogive for the following distribution.

From the ogive estimate, find

i) the median earnings;

ii) the number of employees having earnings less than Rs.

800

Wages (In Rs) Cumulative Frequency

Less than 500

Less than 600

Less than 700

Less than 800

Less than 900

Less than 1000

2

10

18

32

45

50

42. Draw “more than” ogive for the following frequency

distribution. Use your ogive to estimate:

i) the median;

ii) the number of students who obtained more than 75% mark

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

Number of Students

5 9 16 22 26 18 11 6 4 3

QUESTION FROM BOARD PAPER(S) 11 MARCH 2008 (DELHI & OUTSIDE DELHI BOARD CBSE)

Area Related to Circles

1. In Fig. Find the perimeter of shaded region where ADC, AEB

and BFC are semi – circle on diameters AC, AB and BC

respectively.

OR

Find the area of the shaded region in Fig., where ABCD is a

square of side 14 cm.

2. Find the perimeter of Fig., where AED is a semi – circle and

ABCD is a rectangle.

3. In Fig., ABC is a right – angled triangle right – angled at A.

Semicircles are drawn on AB, AC and BC as diameters. Find the

area of the shaded region.

4. In Fig. O is the centre of a circle. The area of sector OAPB is

5/18 of the area of the circle.

Find x

Statistics

1. Which measure of central tendency is given by the x –

coordinate of the point of intersection of the “more than ogive”

and “less than ogive” ?

2. Find the median class of the following data:

Marks Obtained

0-10 10-20 20-30 30-40 40-50 50-60

Frequenc

y

8 10 12 22 30 18

3. 100 surnames were randomly picked up from a local telephone

directory and the distribution of number of letters of the

English alphabet in the surnames was obtained as follows:

Number of Letters

1-4 4-7 7-10 10-13 13-16 16-19

Number of

Surnames

6 30 40 16 4 4

Determine the median and mean number of letters in the

surnames. Also find the modal size of surnames.

4. A survey regarding the heights (in cm) of 50 girls of Class X of

a School was conducted and the following data was obtained.

Number of

Letters

120-130 130-

140

140-

150

150-

160

160-

170

Total

Number of girls

2 8 12 20 8 50

Find the mean, median and mode of the above data.