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-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.