SA2 REVISION
QUESTIONS CLASS X
2015-2016
1.
A letter of
English alphabet is chosen at random. Determine the probability that the chosen
letter is a consonant.
2.
The tops of two
towers of height x and y, standing on level ground, subtend angle of 30° and
60° respectively at the centre of the line joining their feet, then find x:y
3.
If
Sn, denotes the sum of first n terms of an A.P., prove that S12 =
3(S8 – S4).
4.
A solid wooden
toy is in the form of a hemisphere surmounted by a cone of same radius. The
radius of hemisphere is 3.5 cm and the total wood used in making of toy is 166
cm3.
Find the height of the toy. Also, find the cost of painting the hemispherical part of the toy at the rate of Rs
10 per cm2.
5.
From a cuboidal solid metallic block, of
dimensions 15 cm × 10 cm × 5 cm, a cylindrical hole of diameter 7 cm is drilled
out. Find the surface area of the
remaining block.
6.
The numerator of
a fraction is 3 less than its denominator. If 2 is added to both the numerator
and the denominator, then the sum of the new fraction and the original fraction is 29/20 Find the original fraction.
7.
The cost of
fencing a circular field at the rate of Rs18 per metre is Rs 3,960. Find the
cost of ploughing the field at the rate
of Rs 0.70 per m2.
8.
A man standing on the top of a multi-storey
building, which is 30 m high, observes the angle of elevation of the top of a
tower as 60° and the angle of depression of the base of the tower as 30°. Find
the horizontal distance between the
building and the tower. Also, find the height of the tower.
9.
An
aeroplane, when 3000 m high, passes vertically above another plane at an
instant when the angles of elevation of the two aeroplanes from the same
point on the ground are 60° and 45° respectively. Find the vertical distance between the two aeroplanes.
10. From a point P on the ground, the angle of elevation
of the top of a 10 m tall building is
30°. A flag is hosted at the top of the building and the angle of
elevation of the
top of the flagstaff from P is 45°. Find the length of the flagstaff. (Take √3 = 1.732)
11. From a window, 60 m high above the ground, of a
house in a street, the angles of elevation and depression of the top and foot
of another house on the opposite side of
the street are 60° and 45° respectively. Show that the height of the opposite house is 60 (1 + √3) metres.
12. Two men on either side of a cliff, 60 m high,
observe the angles of elevation of the top of the cliff to be 45° and 60°
respectively. Find the distance between two men.
13. Due to some technical problem, an aeroplane started
late by one hour from its starting point. The pilot decided to increase the
speed of the aeroplane by 100 km/h from its usual speed, to cover a journey of
1200 km in time.
Read the above passage and answer
the following questions:
(i) Find the usual speed of the
aeroplane.
(ii) What value (quality) of the
pilot is represented in the question?
14. If (-2,-1), (a,0), (4,b) and (1,2)
are the vertices of a parallelogram, then find the value of a and b
15. A
boy standing on a horizontal plane finds a bird flying at a distance of 100 m
from him at an elevation of 30 °. A girl standing on the roof of 20 meter high
building find the angle of elevation of the same bird to be 45°.Both the boy
and the girl are on opposite sides of the bird. Find the distance of bird from
the girl.
16. Find the sum of all natural numbers between
250 and 1000, which are exactly divisible by 3.
17. Draw a line
segment AB = 7.5 cm. Find a point P on it which divides it in the ratio 2 : 7.
18. Draw a line
segment of length 7.6 cm and divide it into the ratio 5 : 8. Measure the two
parts.
19. Three sides PQ,
QR and PR of ∆PQR are 5 cm, 6 cm and 7 cm respectively. Construct the ∆PQR.
Construct a ∆PQ′R′ such that each of its sides is2/3 of corresponding
sides of ∆PQR.
20. Draw a parallelogram ABCD in which BC = 5 cm, AB = 3
cm and ∠ABC
= 60°. Divide it into triangles BCD
and ABD by the diagonal BD and Construct the ∆BD′C′similar to ∆BDC with scale factor 4/3 .Draw the
linesegment D′A′ parallel
to DA, where A′ lies on extended side BA. Is A′BC′D′a parallelogram?
21. Draw a pair of tangents to a circle of radius 5 cm
which are inclined to each other at an angle of 60°.
22. Draw a circle of radius 3 cm. From a point P, 6 cm
away from its centre, construct a pair of tangents to the circle. Measure the
lengths of the tangents.
23.
Construct
a pair of tangents to a circle of
radius 4 cm inclined at an angle of 45°.
24. Show that the points A(3, 1), B(12, –2) and C(0, 2)
cannot be the vertices of a triangle.
25. If the points
A(– 6, 10), B(– 4, 6) and C(3, –8 )are collinear, then show that AB = 2/9 AC
26. In November 2009, the number of visitors to a zoo
increased daily by 20. If a total of 12300 people visited the zoo in that
month, find the number of visitors on
1st November 2009.
27. The sum of the third and seventh term of an A.P. is
6 and their product is 8. Find the sum of the first sixteen terms of the A.P.
28. Show that the
sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
29. Find the sum of all multiples of 9 lying between 300
and 700.
30. The
product of Tanay’s age (in years) five years ago and his age ten year later is
16. Determine Tanay’s present age.
31. From
a station, two trains start at the same time. One train moves in west direction
and other in North direction. First train moves 5 km/hour faster than the
second train. If after two hours, distance between the two trains is 50 km,
find the average speed of each train.
32. The
radius and slant height of a right circular cone are in the ratio of 7 : 13 and
its curved surface area is 286 cm2. Find its radius.
33. Find
the common difference of an A.P. whose first term is 1/2 and the 8th term is 17/6. Also write its
4th term.
34. If
the equation kx2 - 2 kx + 6 = 0 has equal roots, then find the value of k.
35. A
metallic sphere of total volume π
is melted and recast into the shape of a
right circular cylinder of radius 0.5 cm. What is the height of cylinder ?
36. An
observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of
the top of the chimney from his eyes is 45°. Find the height of the chimney.
37. 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.
38. Prove
that the diagonals of a rectangle with vertices (0, 0), (a, 0), (a, b) and (0,
b) bisect each other and are equal.
39. A
card is drawn at random from a well – shuffled deck of 52 playing cards. Find
the probability that the card drawn is : (i) either a spade or an ace (ii) a
black king
40. An
open container made up of a metal sheet is in the form of a frustum of a cone
of height 7 cm with radii of its lower and upper circular ends as 4 cm and 10
cm respectively. Find the cost of oil which can completely fill the container
at the rate of Rs. 50/litre.
41. From
the top of a tower the angle of depression of an object on the horizontal
ground is found to be 60°.
On descending 20 m vertically downwards from the top of the tower, the angle of
depression of the object is found to be 30°. Find the height of the tower.
42. Find
the roots of the equation x2- 2(a2+b2)x + (a2-b2 ) = 0
43. Divide
29 into two parts so that the sum of the squares of the two parts is 425.
44. Find
the sum of all two digit natural numbers which when divided by 3 yield 1 as
remainder.
45. The
line segment joining the points (3, - 4) and (1, 2) is trisected at the points P and Q. If the
co-ordinates of P(p,-2) and Q are (5/3,
q) respectively, find the values of p
and q.
46. There
are three consecutive positive integers such that the sum of the square of the
first and the product of the other two is 154. Find the integers.
47. The difference of the ages of Sohrab and
his father is 30 years. If the difference of the squares of their ages is 1560,
find their ages.
48. For what value of k are the points
(1,1), (3,k) and (-1,4) collinear?
49. Two circles with centres X and Y touch externally at P. If tangents AT and BT meet the common tangent at T, then prove that AT = BT. (FIG.1)
49. Two circles with centres X and Y touch externally at P. If tangents AT and BT meet the common tangent at T, then prove that AT = BT. (FIG.1)
50. A cone, a hemisphere and a
cylinder stand on equal bases and have the same height. Show that their volumes
are in the ratio 1:2 :3 .
51. The
vertices of a triangle are (-2,
0), (2, 3) and (1,-3). Is the triangle equilateral, isosceles or scalene ?
52. Find
five numbers in A.P. whose sum is 12½ and the ratio of first to the last is 2 :
3.
53. How
many spherical bullets can be made out of a solid cube of lead whose edge
measures 44 cm, each bullet being 4 cm in
diameter.
54. If
the coordinates of the vertices of a triangle are (6, 7), (4, -5) and (x, 2x) and its
area is 3 sq. cm, find the
value of x.
55. Four
points A(6, 3), B(-3,
5), C(4, -2) and D(x, 3x) are given in such way that area of triangle DBC / area of triangle
ABC = 1/2
56. A
plane left 30 minutes later than the schedule time. In order to reach its
destination 1500 km away in time, it has to increase the speed by 250 km/hr.
Find its usual speed.
57. Some
students planned a picnic. The budget for food was Rs. 480. But 8 of these failed
to go and thus cost of food for each member increased by Rs. 10. How many
students attended the picnic
58. In
an A.P. 6th term is half the 4th term, and the 3rd term is 15. How many terms
are needed to give a sum that is equal to 66.
59. A
container made up of metal sheet is in the form of a frustum of a cone of
height 16 cm with radii of its lower and upper ends as 8 and 20 cm
respectively. Find the cost of milk which can completely fill the container at
the rate of Rs. 20 per litre. (Use π = 3.14)
60. A
card is drawn at random from a well shuffled deck of playing cards. Find the
probability that the card drawn is (i) a king or a jack (ii) a non ace (iii)
neither an ace nor a king
61. Prove
that the intercept of a tangent between two parallel tangents to a circle
subtends a right angle at the centre.
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62. In
which of the following cases construction of a triangle is possible, when its
sides are (A) 2.5 cm, 3.5 cm, 6
cm (B) 2.5 cm, 3 cm, 6 cm (C) 2.5 cm, 4.5 cm, 6.5 cm (D) 2.5 cm, 2.5cm, 6 cm
63. Find
the first four terms of an A.P. whose first term is 3x-y and common difference is x-y
64. Find
the coordinates of the point which divides the line segment joining the
points (1,3) and (2,7) in the ratio of 3 : 4.
65. If
there are 2 children in a family, find the probability that there is at least
one boy in the family.
66. Two
dice are thrown together. Find the probability that a multiple of 2 occurs on
one dice and a multiple of 3 occurs on the other.
67. Cards
marked with numbers 5, 6, 7, ……., 30 are placed in a box and mixed thoroughly
and one card is drawn at random from the box. What is the probability that
the number on the card is (i) a prime number ? (ii) a multiple of 3 or 5 ?
(iii) neither divisible by 5 nor by 10 ?
68. In
a flight of 600 km, an aircraft was slowed down due to bad weather. The
average speed for the trip was decreased by 200 km/hr. and the time of flight
increased by 30 minutes. Find the duration of flight.
69. In
the FIG.2, PQRS is a diameter of a circle of radius 6 cm, such that
the lengths PQ, QR and RS are equal. Semicircles are drawn on PQ and QS as
diameters. Find the area of the shaded region. (Use π = 22/ 7 )
70. Three
cubes of a metal whose edges are in the ratio 3 : 4 : 5 are melted and
converted into a single cube whose edge is 12cm. Find the edges of the three
cubes.
71. Solve
the quadratic equation 9x2 -
15x- 6 = 0 by the method of
completing the square
72. Find
the middle terms in the AP 20, 16, 12, ………….., (- 176).
73. A
circle touches the side BC of a ∆ ABC at point P and touches AB
and AC when produced at Q and R respectively. Show that AQ = 1/2 perimeter of ∆ ABC
74. A
boy is cycling such that the wheels of the cycle are making 140 revolutions
per minute. If the diameter of the wheel is 60 cm, calculate the speed per
hour which the boy is cycling.
75. A
motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24km
upstream than to return downstream to the same spot. Find the speed of
stream.
76. The
sum of n,2n, 3n terms of an A.P. are S1, S2, S3respectively. Prove that S3 = 3(S2 - S1)
77. An
oil funnel of tin sheet consists of a cylindrical portion 10cm long attached
frustum of a cone. If the total height be 22cm, diameter of the cylindrical
portion be 8cm and the the diameter of the top of the funnel be 18cm, find
the area of the tin required to make the funnel.
78. Two
pillars of equal height are on either side of a road, which is 100m wide. The
angles of elevation of the top of the pillars are 60° and 30° at a point on the road
between the pillars. Find the position of the point between the pillars and
the height of each pillar (√3 = 1.732)
79. A
solid wooden toy is in the shape of a right circular cone mounted on a
hemisphere. If the radius of the hemisphere is 4.2cm and the total height of
the toy is 10.2cm, find the volume of the wooden toy.
80. The
diameter of a copper solid sphere is 6 cm. The sphere is melted and is drawn
into a wire of uniform circular cross
section. If the length of the wire is 36 m, find its radius.
81. By
a reduction of Rs. 1 per kg in the price of sugar, Mohan can buy one kg sugar
more for Rs. 56. Find the original price of sugar per kg.
82. In
a leap year, find the probability that there are 53 tuesdays in the year.
83. Two
tangents PA and PB are drawn from an external point P to a circle with centre
O. Prove that AOBP is a cyclic quadrilateral.
84. Prove
that the points (2,3) (-4,-6) and (1, 3/2 ) do not form
a triangle.
85. The
time taken by Ram to cover 150km in one direction was 150 minutes more than
the time in the return journey. If he returned at a speed of 10km/hr more
than the speed of going. What was the speed per hour in each direction ?
86. Which
point on y–axis is
equidistant from points A(5,-2)
and B(-3,2)
87. If
the perimeter of a sector of a circle of radius 5.7m is 27.2m, then find the
area of the sector.
88. There
are three children in a family. Find the probability of that there is at most
one girl in the family.
89. In
given FIG. 3, the height of a
solid cylinder is 15 cm and diameter of the base is 7 cm. Two equal conical
holes each of radius 3 cm and height 4 cm are cut off as shown in the figure.
Find the surface area of the remaining solid.
 
90. In
the given FIG 4, find the area of
the shaded region and also its perimeter, if the lengths of AB and BC (in cm)
is 28 and 21 respectively and BCD is a quadrant of a circle. AEC is a
semicircle on AC as diameter.
91. Each
year a tree grows 5 cm less than it did the preceding year. If it grew by 1 m
in the first year, in how many years will it have ceased growing ?
92. A
milk tanker cylindrical in shape having diameter 2 m and length 4.2 m.
supplies milk to the two booths in the ratio 3 : 2. One of the milk booths
has cuboidal vessel having base area 3.96 sq.m. and the other has a
cylindrical vessel having radius 1 m. Find the level of milk in each of the
vessels.
93. A
coin is tossed three times. Find the probability of getting exactly two
tails.
94.
For what value(s) of k will the quadratic equation (2k+1) x2
+ 2(k+3)x + (k+5) = 0 have real and
equal roots ?
95. Find
two consecutive odd positive integers, sum of whose squares is 290.
96. The
sum of first 6 terms of an A.P. is 42. The ratio of its 10th term to its 30th
term is 1:3. Find the first term and the thirteenth term of the A.P.
97. Draw
a triangle ABC in which BC= 6.5
cm, AB = 4.5 cm and ∟ ABC = 60°. Construct a triangle similar to this
triangle whose sides are ¾ of the corresponding sides of ABC.
98. AB
is a diameter of a circle, where the coordinates of centre C are (1,- 3). Find the coordinates of B,
if coordinates of A are (4, - 1).
99. Find
the middle term of the A.P. -11, - 7, - 3, ……. , 45.
100. Two parallel
lines touch the circle at points A and B separately. If the area of the
circle
is 25π cm2 , then find AB
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