The Union Public Service Commission (UPSC) conducted the Combined Defence Services (CDS) (I) Exam on 5 February 2017. The exam consisted of three papers – General Knowledge, English and Elementary Mathematics.
To facilitate CDS Exam aspirants' understanding of the difficulty level and the range of questions being asked in the exam, the CDS (I) 2017 Elementary Mathematics Question Paper (Series A) is given below.
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2. (x + 4) is a factor of which one of the following expressions?
(a) x^{2} ─ 7x + 44
(b) x^{2} + 7x ─ 44
(c) x^{2} ─ 7x ─ 44
(d) x^{2} + 7x + 44
(a) 2
(b) ─ 2
(c) 9
(d) ─ 9
4. Consider the following statements:
1. If a = bc with HCF (b, c) = 1, then HCF (c, bd) = HCF (c, d).
2. If a = bc with HCF (b, c) = 1, then LCM (a, d) = LCM (c, bd).
Which of the above statements is/are correct?
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
(a) 1
(b) 2
(c) 3
(d) 4
6. What is the number of digits in 2^{40}?
(Given that log_{10} 2 = 0.301)
(a) 14
(b) 13
(c) 12
(d) 11
8. What is the remainder when the number (4444)^{4444} is divided by 9?
(a) 4
(b) 6
(c) 7
(d) 8
17. If α and β are the roots of the equation x^{2} + px + q = 0, then what is α^{2} + β^{2} equal to?
(a) p^{2} – 2q
(b) q^{2} – 2p
(c) p^{2} + 2q
(d) q^{2} – q
18. If a^{3} = 335 + b^{3} and a = 5 + b, then what is the value of a + b (given that a > 0 and b > 0)?
(a) 7
(b) 9
(c) 16
(d) 49
19. If 9^{x} 3^{y} = 2187 and 2^{3x} 2^{2y} – 4^{xy} = 0, then what can be the value of (x + y)?
(a) 1
(b) 3
(c) 5
(d) 7
20. The pair of linear equations kx + 3y + 1 = 0 and 2x + y + 3 = 0 intersect each other, if
(a) k = 6
(b) k ≠ 6
(c) k = 0
(d) k ≠ 0
21. The number of prime numbers which are less than 100 is
(a) 24
(b) 25
(c) 26
(d) 27
22. The cost of a diamond varies directly as the square of its weight. A diamond broke into four pieces with their weights in the ratio of 1 : 2 : 3 : 4. If the loss in total value of the diamond was Rs 70,000, what was the price of the original diamond?
(a) Rs 1,00,000
(b) Rs 1,40,000
(c) Rs 1,50,000
(d) Rs 1,75,000
24. If 15 men take 21 days of 8 hours each to do a piece of work, then what is the number of days of 6 hours each that 21 women would take, if 3 women would do as much work as 2 men?
(a) 18
(b) 20
(c) 25
(d) 30
(a) 6
(b) 8
(c) 9
(d) 11
26. A sum of Rs 8,400 was taken as a loan. This is to be paid in two equal instalments. If the rate of interest is 10% per annum, compounded annually, then the value of each instalment is
(a) Rs 4,200
(b) Rs 4,480
(c) Rs 4,840
(d) None of the above
(a) 1 year
(b) 2 years
(c) 3 years
(d) 4 years
28. A and B working together can finish a piece of work in 12 days while B alone can finish it in 30 days. In how many days can A alone finish the work?
(a) 18 days
(b) 20 days
(c) 24 days
(d) 25 days
29. The values of x which satisfy the equation 5^{1 + x} + 5^{1 – x} = 26 are
(a) ─1, 1
(b) 0, 1
(c) 1, 2
(d) ─1, 0
30. If 5 men can do a piece of work in 10 days and 12 women can do the same work in 15 days, the number of days required to complete the work by 5 men and 6 women is
31. A passenger train departs from Delhi at 6 p.m. for Mumbai. At 9 p.m., an express train, whose average speed exceeds that of the passenger train by 15 km/hour leaves Mumbai for Delhi. Two trains meet each other mid-route. At what time do they meet, given that the distance between the cities is 1080 km?
(a) 4 p.m.
(b) 2 a.m.
(c) 12 midnight
(d) 6 a.m.
32. In a class of 49 students, the ratio of girls to boys is 4 : 3. If 4 girls leave the class, the ratio of girls to boys would be
(a) 11 : 7
(b) 8 : 7
(c) 6 : 5
(d) 9 : 8
33. If a + b = 5 and ab = 6, then what is the value of a^{3} + b^{3} ?
(a) 35
(b) 40
(c) 90
(d) 125
34. Rajendra bought a mobile with 25% discount on the selling price. If the mobile cost him Rs 4,875, what is the original selling price of the mobile?
(a) Rs 6,300
(b) Rs 6,400
(c) Rs 6,500
(d) Rs 6,600
35. A 225 m long train is running at a speed of 30 km/hour. How much time does it take to cross a man running at 3 km/hour in the same direction?
(a) 40 seconds
(b) 30 seconds
(c) 25 seconds
(d) 15 seconds
36. Which one among the following is the largest?
37. The difference between the simple and the compound interest on a certain sum of money at 4% per annum in 2 years is Rs 10. What is the sum?
(a) Rs 5,000
(b) Rs 6,000
(c) Rs 6,250
(d) Rs 7,500
38. If a% of a + b% of b = 2% of ab, then what percent of a is b?
(a) 50%
(b) 75%
(c) 100%
(d) Cannot be determined
40. Sunil wants to spend Rs 200 on two types of sweets, costing Rs 7 and Rs 10 respectively. What is the maximum number of sweets he can get so that no money is left over?
(a) 25
(b) 26
(c) 27
(d) 28
41. What is the LCM of x^{3} + 8, x^{2} + 5x + 6 and x^{3} + 4x^{2} + 4x ?
(a) x(x + 2)^{2} (x + 3) (x^{2} – 2x + 4)
(b) x(x – 2)^{2} (x – 3) (x^{2} + 2x + 4)
(c) (x + 2)^{2} (x + 3) (x^{2} – 2x + 4)
(d) (x – 2)^{2} (x – 3) (x^{2} – 2x + 4)
42. The HCF of two expressions p and q is 1. What is the reciprocal of their LCM?
(a) p + q
(b) p – q
(c) pq
(d) (pq) ^{─1}
44. A thief is spotted by a policeman from a distance of 100 m. When the policeman starts the chase, the thief also starts running. If the speed of the thief is 8 km/hour and that of the policeman is 10 km/hour, then how far will the thief have to run before he is overtaken?
(a) 200 m
(b) 300 m
(c) 400 m
(d) 500 m
45. Aman and Alok attempted to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in roots (4, 3) . Alok made a mistake in writing down the coefficient of x to get roots (3, 2). The correct roots of the equation are
(a) ─4, ─3
(b) 6, 1
(c) 4, 3
(d) ─6, ─1
46. Consider the following statements:
1. Of two consecutive integers, one is even.
2. Square of an odd integer is of the form 8n + 1.
Which of the above statements is/are correct?
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
47. The system of equations 2x + 4y = 6 and 4x + 8y = 8 is
(a) Consistent with a unique solution
(b) Consistent with infinitely many solutions
(c) Inconsistent
(d) None of the above
48. (N^{p}^{─1} ─1) is a multiple of p, if N is prime to p and p is a
(a) Prime number
(b) Rational number
(c) Real number
(d) Composite number
49. The ratio of two numbers is 1 : 5 and their product is 320. What is the difference between the squares of these two numbers?
(a) 1024
(b) 1256
(c) 1536
(d) 1640
50. 25 kg of alloy X is mixed with 125 kg of alloy Y. If the amount of lead and tin in the alloy X is in the ratio 1 : 2 and the amount of lead and tin in the alloy Y is in the ratio 2 : 3, then what is the ratio of lead to tin in the mixture?
(a) 1 : 2
(b) 2 : 3
(c) 3 : 5
(d) 7 : 11
51. The mean of 5 numbers is 15. If one more number is included, the mean of the 6 numbers becomes 17. What is the included number?
(a) 24
(b) 25
(c) 26
(d) 27
52. The mean marks obtained by 300 students in a subject are 60. The mean of top 100 students was found to be 80 and the mean of last 100 students was found to be 50. The mean marks of the remaining 100 students are
(a) 70
(b) 65
(c) 60
(d) 50
53. Consider the following distribution:
Class |
Frequency |
0 – 20 |
17 |
24 – 40 |
28 |
40 – 60 |
32 |
60 – 80 |
f |
80 – 100 |
19 |
If the mean of the above distribution is 50, what is the value of f?
(a) 24
(b) 34
(c) 56
(d) 96
54. In a pie diagram, there are four slices with angles 150°, 90°, 60° and 60°. A new pie diagram is formed by deleting one of the slices having angle 60° in the given pie diagram. In the new pie diagram
(a) The largest slice has angle 150°
(b) The smallest slice has angle 70°
(c) The largest slice has angle 180°
(d) The smallest slice has angle 90°
55. In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is
(a) 120
(b) 220
(c) 280
(d) 370
57. An individual purchases three qualities of pencils. The relevant data is given below:
Quality |
Price per pencil (in Rs) |
Money spent (in Rs) |
A |
1.00 |
50 |
B |
1.50 |
x |
C |
2.00 |
20 |
It is known that the average price per pencil is Rs 1.25. What is the value of x?
(a) Rs 10
(b) Rs 30
(c) Rs 40
(d) Rs 60
58. Consider the following frequency distribution:
x |
Frequency |
Cumulative frequency |
1 |
8 |
8 |
2 |
10 |
18 |
3 |
f_{1} |
29 |
4 |
f_{2} |
45 |
What are the value of f_{1} and f_{2} respectively?
(a) 10 and 17
(b) 17 and 10
(c) 11 and 16
(d) 16 and 11
67. An aeroplane flying at a height of 300 m above the ground passes vertically above another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 60° and 45° respectively. What is the height of the lower plane from the ground?
68. If x = a cos θ + b sin θ and y = a sin θ – b cos θ, then what is x^{2} + y^{2} equal to?
(a) 2ab
(b) a + b
(c) a^{2} + b^{2}
(d) a^{2} – b^{2}
69. From the top of a building 90 m high, the angles of depression of the top and the bottom of a tree are 30° and 45° respectively. What is the height of the tree?
70. Which one of the following triples does not represent the sides of a triangle?
(a) (3, 4, 5)
(b) (4, 7, 10)
(c) (3, 6, 8)
(d) (2, 3, 6)
71. If the perimeter of a rectangle is 10 cm and the area is 4 cm^{2}, then its length is
(a) 6 cm
(b) 5 cm
(c) 4.5 cm
(d) 4 cm
72. The angles of a triangle are in the ratio 2 : 4 : 3. The smallest angle of the triangle is
(a) 20°
(b) 40°
(c) 50°
(d) 60°
73. A ball of radius 1 cm is put into a cylindrical pipe so that it fits inside the pipe. If the length of the pipe is 14 m, what is the surface area of the pipe?
(a) 2200 square cm
(b) 4400 square cm
(c) 8800 square cm
(d) 17600 square cm
74. The areas of two circular fields are in the ratio 16 : 49. If the radius of the bigger field is 14 m, then what is the radius of the smaller field?
(a) 4 m
(b) 8 m
(c) 9 m
(d) 10 m
75. Let ABCD be a rectangle. Let P, Q, R, S be the mid-points of sides AB, BC, CD, DA respectively. Then the quadrilateral PQRS is a
(a) Square
(b) Rectangle, but need not be a square
(c) Rhombus, but need not be a square
(d) Parallelogram, but need not be a rhombus
77. If each of the dimensions of a rectangle is increased by 200%, the area is increased by
(a) 300%
(b) 400%
(c) 600%
(d) 800%
80. ABCDEF is a regular polygon. Two poles at C and D are standing vertically and subtend angles of elevation 30° and 60° at A respectively. What is the ratio of the height of the pole at C to that of the pole at D?
81. Two parallel chords of a circle whose diameter is 13 cm are respectively 5 cm and 12 cm in length. If both the chords are on the same side of the diameter, then the distance between these chords is
(a) 5.5 cm
(b) 5 cm
(c) 3.5 cm
(d) 3 cm
82. If the radius of a right circular cone is increased by p% without increasing its height, then what is the percentage increase in the volume of the cone?
83. A copper wire when bent in the form of a square encloses an area of 121 cm^{2}. If the same wire is bent in the form of a circle, it encloses an area equal to
(a) 121 cm^{2}
(b) 144 cm^{2}
(c) 154 cm^{2}
(d) 168 cm^{2}
(a) 5 cm
(b) 6 cm
(c) 8 cm
(d) 9 cm
85. If the surface area of a sphere is reduced to one-ninth of the area, its radius reduces to
(a) One-forth
(b) One-third
(c) One-fifth
(d) one-ninth
86. In a trapezium ABCD, AB is parallel to CD and the diagonals intersect each other at O. What is the ratio of OA to OC equal to?
(a) Ratio of OB to OD
(b) Ratio of BC to CD
(c) Ratio of AD to AB
(d) Ratio of AC to BD
87. Ice-cream, completely filled in a cylinder of diameter 35 cm and height 32 cm, is to be served by completely filling identical disposable cones of diameter 4 cm and height 7 cm. The maximum number of persons that can be served in this way is
(a) 950
(b) 1000
(c) 1050
(d) 1100
88. The radius of a circle is increased so that its circumference increases by 15%. The area of the circle will increase by
(a) 31.25%
(b) 32.25%
(c) 33.25%
(d) 34.25%
89. ABCD is a rectangle. The diagonals AC and BD intersect at O. If AB = 32 cm and AD = 24 cm, then what is OD equal to?
(a) 22 cm
(b) 20 cm
(c) 18 cm
(d) 16 cm
90. A field is divided into four regions as shown in the given figure. What is the area of the field in square metres?
91. In the figure given below, D is the diameter of each circle. What is the diameter of the shaded circle?
92. In the figure given below, AC is parallel to ED and AB = DE = 5 cm and BC = 7 cm. What is the area ABDE : area BDE : area BCD equal to?
(a) 10 : 5 : 7
(b) 8 : 4 : 7
(c) 2 : 1 : 2
(d) 8 : 4 : 5
93. In the figure given below, PQRS is a parallelogram. PA bisects angle P and SA bisects angle S. What is angle PAS equal to?
(a) 60°
(b) 75°
(c) 90°
(d) 100°
(a) 10 and 130
(b) 10 and 125
(c) 20 and 130
(d) 20 and 125
(a) 55°
(b) 70°
(c) 75°
(d) 80°
Consider the following statements:
1. Triangle ADC is an isosceles triangle.
2. D is the centroid of the triangle ABC.
3. Triangle ABD is congruent to the triangle CBD.
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
(a) SAS rule
(b) SSS rule
(c) ASA rule
(d) AAA rule
Which of the above statements are correct?
(a) 2, 3 and 4 only
(b) 1, 2 and 4 only
(c) 1, 3 and 4 only
(d) 1 and 2 only
100. From an aeroplane vertically over a straight horizontal road, the angles of depression of two consecutive kilometer-stones on the opposite sides of the aeroplane are observed to be α and β. The height of the aeroplane above the road is
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