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UPSC CDS-II Answer Key {General Knowledge}

## UPSC CDS-II Answer Key ~ Math

1. Three rectangles R1, R2 and R3 have the same area. Their lengths x1, x2 and x3 respectively are such that x1<x2<x3 If V1, v2 and v3 are the volumes of the cylinders formed from the rectangles R1, R2 and R3 respectively by joining the parallel sides along the breadth, then which one of the following is correct?
(a) V3<V2<V1
(b) V1<V3<V2
(c) V1<V2<V3
(d) V3<V1<V2
2. If the surface area of a cube is 13254 cm2, then the length of its diagonal is
(a) 44√2 cm
(b) 44 √3 cm
(c) 47 √2 cm
(d) 47√3 cm
3. The area of a trapezium is 336 cm2. If its parallel sides are in the ratio 5:7 and the perpendicular distance between then is 14 cm, then the smaller of the parallel sides is
(a) 20 cm
(b) 22 cm
(c) 24 cm
(d) 26 cm
4. How many spherical bullets each of 4 cm in diameter can be made out of a cube of lead whose edge is 44 cm?
(a) 2541
(b) 2551
(c) 2561
(d) 2571
5. A river 2.5 m deep and 45 m wide is flowing at the speed of 3.6 km/ hour. The amount of water that runs into the sea per minute is
(a) 6650 m3
(b) 6750 m3
(c) 6850 m3
(d)6950 cm3
6. AB is a line segment of length 2a, with M as mid-point. Semicircles are drawn on one side with AM, MB and AB as diameter as shown in the above figure. A circle with centre O and radius r is drawn such that this circle touches all the three semicircles. The value of r is
(a) 2a/ 3
(b) a2/2
(c) a/3
(d) a/4

7. From an external point P tangents PA and PB are drawn to the circle as shown in the above figure, CD is the tangent to the circle at E. If AP=16 cm, then the perimeter of the triangle PCD is equal to
(a) 24 cm
(b) 28 cm
(c) 30 cm
(d) 32 cm
8. In the above figure, ABCD is a parallel-gram. P is a point on BC such that PB:PC=1:2. DP and AB when both produced meet at Q. If area when both produced meet at Q. If area of triangle BPQ is 20 square unit, the area of triangle DCP is
(a) 20 square unit
(b) 30 square unit
(c) 40 square unit
(d) None of the above
9. Chord CD interests the diameter AB of circle at right angle at a point P in the ratio 1:2. If diameter of circle is d, then CD is equal to
(a) √2d/3
(b) 2d/3
(c) 2√2d/3
(d) 2√3d/3
10. A circle of radius r is inscribed in a regular polygon with n sides (the circle touches all sides of the polygon). If the perimeter of the polygon is P, then the area of the polygon is
(a) (p+n)r
(b) (2p-n)r
(c)pr/2
(d) None of the above
11. The LCM of two numbers is 12 times their HCF. The sum of HCF and LCM is 403. If one of the numbers is 93, then the other number is
(a) 124
(b) 128
(c) 134
(d) 138
12. The value of (0.63+0.37) is
(a) 1
(b) 100/91
(c) 100/99
(d) 1000/999
13. A sum of ₹ 10,000 is deposited for 1 year at the rate of interest 10% compounded half yearly. What will be the interest at the end of one year?
(a) ₹ 1000
(b) ₹ 1025
(c) ₹ 1050
(d) ₹ 1100
14. In a mixture of milk and water of volume 30 litre, the ratio of milk and water is7:3. The quantity of water to be added to the mixture to make the ratio of milk and water 1:2 is
(a) 30
(b) 32
(c) 33
(d) 35
15. The difference of maximum values of the expression (6+5x-x2) and (y-6-y2) for any real values of x and y is
(a) 16
(b) 17
(c) 18
(d) 19
16. A’s salary was increased by 40% and then decreased by 20%. On the whole A’s salary is increased by
(a) 60%
(b) 40%
(c) 20%
(d) 12%
17. A number consists of two digits, whose sum is 7. If the digits are reversed, the number is increased by 27. The product of digits of the number is
(a) 6
(b) 8
(c) 10
(d) 12
18. In a race of 100 m, A beats B by 4 m and A beats C by 2 m. By how many metres (approximately) would C beat B in another 100 m race assuming C and B run with their respective speeds as in the earlier race?
(a) 2
(b) 2.04
(c) 2.08
(d) 3.2
19. In an election 10% of the voters on the voter list did not cast their vote and 60 voters cast their ballot papers blank. There were only two candidates. The winner was supported by 47% of total voters in the voter list and he got 308 votes more than his rival. The number of voters on the voter list is
(a) 3600
(b) 6200
(c) 3028
(d) 6400
20. Consider all positive two digit numbers each of which when divided by 7 leaved a remainder 3. What is their sum?
(a) 661
(b) 666
(c) 676
(d) 777
21. A pipe with square cross-section is supplying water to a cistern which was initially empty. The area of cross-section is 4 cm2 and the nozzle velocity of water is 40 m/s. The dimensions of the cistern are 10 m x 8m x 6m. Then the cistern will be full in
(a) 9.5 hours
(b) 9 hours
(c) 8 hours 20 minutes
(d) 8 hours
22. A hollow cylindrical drum has internal diameter of 30 cm and a height of 1 m. What is the maximum number of cylindrical boxes of diameter 10 cm and height 10 cm each can be packed in the drum?
(a) 60
(b) 70
(c) 80
(d) 90
23. The speeds of three buses are in the ratio 2:3:4. The time taken by these buses to travel the same distance will be in the ratio
(a) 2:3:4
(b) 4:3:2
(c) 4:3:6
(d) 6:4:3
24. A square and an equilateral triangle have equal perimeter. If the diagonal of the square is 12Ö2 cm, then the area of the triangle is
(a) 24√2 cm2
(b) 24√3 cm2
(c) 48√3 cm2
(d) 64√3 cm2
25. A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the radius of the wheel is 30 cm, the speed of the cycle is
(a) 15.5 km/hour
(b) 15.84 km/hour
(c) 16 km/ hour
(d) 16.36 km/hour
26. Three are 437 fruit plants in an orchard planted in rows. The distance between any two adjacent rows is 2 cm and the distance between any two adjacent plants is 2m. Each row has the same number of plants. There is 1 m clearance on all sides of the orchard. What is the cost of fencing the area at the rate of ₹ 100 per metre?
(a) ₹ 15,600
(b) ₹ 16,800
(c) ₹ 18,200

27. ABCD is a parallelogram with AB and AD as adjacent sides, If ∠A=60o and AB =2AD, then the diagonal BD will be equal to
28. The point O is equidistant from the three sides of a triangle ABC. Consider the following statements:
∠OAC+∠OCB+VoBA=90o
2. ÐBOC=2ÐBAC
3. The perpendiculars drawn from any point on OA to AB and AC are always equal
Which of the above statements are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
29. Consider the following statements:
If the height of a cylinder is doubled the area of the curved surface is doubled.
2. If the radius of a hemispherical solid is doubled, its total surface area becomes fourfold.
Which of the above statements is/are correct?
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
30. A large water tank has the shape of a cube. If 128 m3 of water pumped out, the water level goes down by m. Then the maximum capacity of the tank is
(a) 512 m3
(b) 480 m3
(c) 324 m3
(d) 256 m3
31. A clock is started at noon. By 10 minutes past 5, through what angle, the hour hand moves?
(a) 160o
(b) 145o
(c) 150o
(d) 155o
32. The minimum value of cos2x+cos2y-cos2z is
(a) -1
(b) 0
(c) 1
(d) 2
33. The value of 32 cot2(p/4)-8sec2(p/3)+8cos3(p/6) is equal to
(a) √3
(b) 2√3
(c) 3
(d) 3√3
34. If x=91/216, then the value of 3-((1/(1-x)1/3) is
(a) 9/5
(b) 5/9
(c) 4/9
(d) 4/5
35. How many right angled triangles can be formed by joining the vertices of a cuboid?
(a) 24
(b) 28
(c) 32
(d) None of the above
36. The average weight of students in a class is 43 kg. Four new students are admitted to the class whose weights are 42 kg, 36.5 kg and 42.5 kg respectively. Now the average weight of the students of the class is 42.5 kg. The number of students in the beginning was
(a)10
(b) 15
(c) 20
(d) 25
37. Four years ago, the average age of A and B was 18 years. Now the average age of A, B and C is 24 years. After 8 years, the age of C will be
(a) 32 years
(b) 28 years
(c) 36 years
(d) 40 years
38. If a variable takes discrete value a+4, a-3.5, a-2.5, a-3, a-2, a+0.5, a+5 and a-0.5 where a.0, then the median of the data set is
(a) a-2.5
(b) a-1.25
(c) a-1.5
(d) a-0.75
39. If each of n numbers x1=i(i=1, 2, 3, ….n) is replaced by (i+1) x1, then the new mean is
(a) (n+3)/2
(b) n(n+1)/2
(c) (n+1)(n+2)/3n
(d) (n+1)(n+2)/3
40. The weighted arithmetic mean of first 10 natural numbers whose weights are equal to the corresponding number is equal to
(a) 7
(b) 14
(c) 35
(d) 38.5
41. The number of values of x satisfying (x+(100/x))>50, where x is a natural number less than or equal to 100 is
(a) 51
(b) 53
(c) 55
(d) 57
42. What is the sum of digits of the least multiple of 13, which when divided by 6, 8 and 12 leaves 5, 7 and 11 respectively as the remainders/
(a) 5
(b) 6
(c) 7
(d) 8
43. A milkman claims to sell milk at its cost price only. Still he is making a profit of 20% since he has mixed some amount of water in the milk. What is the percentage of milk in the mixture?
(a) 200/3%
(b) 75%
(c) 80%
(d) 250/3%
44. What is equal to? (a) 3
(b) √13-1/2
(c) √13+1/2
(d) 0
45. The largest natural number which divides every natural number of the form (n3-n)(n-2), where n is a natural number greater than 2 is
(a) 6
(b) 12
(c) 24
(d) 48
46. The average of m numbers is n4 and the average of n numbers is m4. The average of (m+n) numbers is
(a) mn
(b) m2+n2
(c) mn(m2+n2)
(d) mn(m2+n2-mn)
47. The digit in the units place of the resulting number of the expression (234)100+(234)101 is
(a) 6
(b) 4
(c) 2
(d) 0
48. If x=√3+√2, then the value of x3+x+(1/x)+(1/x3) is
(a) 10√3
(b) 20√3
(c) 10 √2
(d) 20√2
49. The value of (1/1×4)+(1/4×7)+(1/7×10)+…….+(1/(16×19) is
(a) 5/19
(b) 6/19
(c) 8/19
(d) 9/19
50. Two trains are moving in the same direction at 1.5 km/minute and 60km/ hour respectively. A man in the faster train observes that it takes 27 seconds to cross the slower train. The length of the slower train is
(a) 225 m
(b) 230 m
(c) 240 m
(d) 250 m
51. A tin of oil was 4/5 full. When 6 bottles of oil were taken out from this tin and 4 bottles of oil were poured into it, it was 3/4 full. Oil of how many bottles can the tin contain? (All bottles are of equal volume)
(a) 35
(b) 40
(c) 45
(d) 50
52. A number, when divided by 7 leaves a remainder 3 and the resulting quotient when divided by 11 leaves a remainder 6. If the same number when divided by 11 leaves a remainder m and the resulting quotient when divided by 7 leaves a remainder n. What are the values of m and n respectively?
(a) 1 and 4
(b) 4 and 1
(c) 3 and 6
(d) 6 and 3
53. Consider the following in respect of the equation y=(√(x-1)2)/(x-1)
y=1 if x>1
2.y=-1 if x<1
3. Y exists for all values of x
Which of the above statements is/are correct?
(a) 1 only
(b) 2 only
(c) 1 and 2 only
(d) 1, 2 and 3
54. If (x+1) is the HCF of Ax2+Bx+C and Bx2+Ax+C where A≠B, then the value of C is
(a) A
(b) B
(c) A-B
(d) 0
55. The seven digit number 876p37q is divisible by 225. The values of P and q can be respectively
(a) 9,0
(b) 0,0
(c) 0, 5
(d) 9, 5
56. Two trains, one is of 121 m in length at the speed of 40 km/hour and the other is of 99 m in the length at the speed of 32 km/hour are running in opposite directions. In how much time will they be completely clear from each other from the moment they meet?
(a) 10 s
(b) 11 s
(c) 16 s
(d) 21 s
57. Three athletes run a 4 km race. Their speeds are in the ratio 16:15:11. When the winner wins the race, then the distance between the athlete in the second position to the athlete in the third position is
(a) 1000 m
(b) 800 m
(c) 750 m
(d) 600 m
58. The sum of first 47 terms of the series (1/4)+(1/5)-(1/6)- (1/4)- (1/5)+ (1/6)+ (1/4)+ (1/5)- (1/6)…… is
(a) 0
(b) -1/6
(c) 1/6
(d) 9/20
59. Which one of the following is correct?
(a) (x+2) is a factor of x4-6x3+12x2-24x+32
(b) (x+2) is a factor of x4+6x3-12x2+24x-32
(c) (x-2) is a factor of x4-6x3+12x2-24x+32
(d) (x-2) is a factor of x4+6x3-12x2+24x-32
60. 20% of a number when added to 20 becomes the number itself, then the number is
(a) 20
(b) 25
(c) 50
(d) 80
61. Consider the following statements:
(1+tan2q)/(1+cot2q)=((1-tanq)/(1-cotq))2 is true for all 0<q<p/2, q≠p/4,
2. cotq=1/tanq is true for q=45o only.
Which of the above statement is/are correct/
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
62. If x=a cosθ and y=b cotθ, then (ax-1 – by-1) (ax-1 + by-1) is equal to
(a) 0
(b) 1
(c) tan2θ
(d) sin2θ
63. cosθ/(1-sinθ) is equal to (where θ≠π/2)
(a) tanq-1/tanq+1
(b) 1+sinq/cosq
(c) tanq+1/tanq-1
(d) 1+cosq/sinq
64. If tan(x+40)o tan(x+20)o tan(3x)otan(70-x)o tan(50-x)o =1, then the value of x is equal to
(a) 30
(b) 20
(c) 15
(d) 10