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# (Paper) CDS: Elementary Mathematics Solved Exam Paper (II): 2005

**Combined Defence Services**

**Elementary Mathematics Solved Paper: 2005**

**1. What is the value of (1 + x ^{2a }– b)^{–1} + **

(1 + xb – a)

^{–1?}

(a) – 1 (b) 1

(c) a – b (d) 0

**2. The sum of the digits in a two-digit number is 10. If 18 is subtracted from the number, the result is the number with the digits reserved. What is the number? **

(a) 46

(b) 64

(c) 73

(d) 37

**3. If the roots of the equation ax2 + bx + c = 0 are equal in magnitude but opposite in sign, then which one of the following is correct?**

(a) a = 0

(b) b = 0

(c) c = 0

(d) b = 0, c ¹ 0, a ¹ 0

**4. If 5 = 1/log**`x`^{3} what is the value of x?

(a) 1

(b) 3

(c) 5

(d) 15

**5. If the roots of the equation x**^{2}** + x + 1 = 0 are in the ratio of m : n, then which one of the following relations holds?**

(a) m + n + 1 = 0

(b) m/n n/m 1 0

(c) √m √n 1 0

(d)√m/n √n/m 1 0

**7. If b/y z/c 1 and c/z x/z 1, the which one of the following corresponds to the value of the expression a/x y/b?**

(a) 0

(b) 1

(c) – 1

(d) 2

**8. If x ^{2} + xy = 40 and y^{2} + xy = 60, what are the values of x + y?**

(a) ± 15

(b) ± 5

(c) ± 10

(d) √10

**9. If √16 6√7 a b√7, what can be the value of b?**

(a) – 2

(b) 3/2

(c) – 1

(d)1/2

**10. If √a √b √c, what is the simplified value of a b c ^{2}/a b c?**

(a) 1

(b) b/c

(c) c/a

(d) a/b

**12. If x ^{3} + px + q and x^{3} + qx + p have a common factor, which one of the following is correct?**

(a) p + q = 0

(b) p + q – 1 = 0

(c) p + q + 1 = 0

(d) p – q + 1 = 0

**13. For what value of m will the expression 3x ^{2} + mx^{2} + 4x – 4x – 4m be divisible (x + 2)?**

(a) – 4

(b) 0

(c) For any value of m

(d) No such value of m exits

**14. If a + b + c = 0, then what is the value of a ^{4} + b^{4} + c^{4} – 2a^{2}b^{2} – 2a^{2}b^{2} – 2b^{2}c^{2} – 2c^{2}a^{2}?**

(a) 0

(b) 1

(c) – 0

(d) 1/2

**15. For what values of p and q, were p, q are real numbers and p ¹ 0, does the equation 3x – 5 + q = px + 1 have solution?**

(a) For all p and q

(b) p = 3, q ¹ 6

(c) p ¹ 3 for any q

(d) q ¹ 6 for any p

**16. If √3x ^{2} 4x 34 √3x^{2} 4x 11 9 what is the value of expression √3x^{2} 4x 34 √3x^{2} 4x 11?**

(a) 0

(b) 3

(c) 5

(d) 9

**17. If the angle inlcuded between the sides of lengths 3 cm and 4 cm of a triangle is 60°, then what is the area of the triangle? **

(a) 3√3cm^{2}

(b) 4√3cm^{2}

(c) 3√3/2cm^{2}

(d) 2√3cm^{2}

**18. If O is the circumference of a triangle ABC such that ** Ð** BAC = 58°, what is ** Ð**OBC? **

(a) 29°

(b) 32°

(c) 48°

(d) 58°

**19. ABC is a cyclic qudrilateral whose diagonals intersect at O. If **Ð**CAB = 25° and ** Ð**BCD = 85°, what is ** Ð**CBD?**

(a) 60°

(b) 70°

(c) 80°

(d) 85°

**20. Which one of the following statements is correct? If the diagonals of a quadrilateral bisect each other at right angles, then the quadrilateral is**

(a) a rhobmbus but not a square

(b) a square but not a rhombus

(c) either a rhombus of a square

(d) a rectangle but not a square

**21. In a right-angled triangled ABC, D is the foot of the perpendicular from B on the hypotenuse AC. If AB = 3 cm and BC = 4 cm, what is the area of the triangle ABD?**

(a) 72/25 cm^{2}

(b) 54/25 cm^{2}

(c) 36/25 cm^{2}

(d) 63/25 cm^{2}

**22. AOBD is a quadrilateral such that ** Ð**AOB = ** Ð**ABD = 90° and ** Ð**DAB = 60°. If OA = 3 cm and OB = 4 cm, what is the area of equadrilateral AOBD?**

**23. In a right-angled triangle PQR = 90° and S is the mid-point of the hypotenuse PR. If **Ð**PQS = 40°, what is **Ð**QSR?**

(a) 60°

(b) 80°

(c) 90°

(d) 100°

**24. PR and QS are two diameters of a circle intersecting at O. If PR = 4 and QR = 3, what is the length of PS?**

(a) 2

(b) 3

(c) 4

(d) √5

**25. Three parallel lines l**_{1}**, l _{2}, and l_{3} are such that l2 is between l1 and l3. They are cut by two transversals PRE and QSF such that P, Q lie on l_{1}; R, S lie on l_{2} and E, F lie on l_{3}. If PR = 8, RE = 4 and QS = 6, What is the length of SF?**

(a) 12

(b) 16/3

(c) 3

(d) 4

**26. AOB is the diameter of a circle with centre O.C is any point on the circle such that **Ð**AOC = 70°, what is **Ð**OBC?**

(a) 20°

(b) 35°

(c) 45°

(d) 55°

**27. If the figure given above, DEF is an equilateral triangle. If ** Ð**FLB = 110°, what is **Ð**AKJ? **

(a) 50°

(b) 55°

(c) 60°

(d) 65°

**28. Consider the following statements:**

1. To constract the circumcircel of a triangle ABC, one has to draw perpendicular bisector of any two sides.

2. To construct the incircle of a triangle ABC, one has to draw angle bisectors of any two angles.**Which of the statements given above is/are correct? **

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 or 2

**29. Which one of the following is correct? The areas of two similar triangles are in the ratio of the squares of the corresponding**

(a) medians

(b) altitudes

(c) angle besector segments

(d) All the above

**30. How many common tangents can be drawn if two circles on a plane do not intersect?**

(a) Zero

(b) Two only

(c) Three only

(d) Zero or four

**31. In a right-angled triangle, the square of the hypotenuse is equal to twice the product of the other two sides. What is one of the acute angles of the triangle?**

(a) 15°

(b) 30°

(c) 45°

(d) 60°

**32. A right-angled isosceles triangle is incribed in a circle of radius r. What is the area of remaining portion of the circle?**

(a) r^{2}/2

(b) 1/2 r^{2}

(c) (p – 1) r^{2}

(d) (p – 1) r^{2}

**33. If the sum of the lengths of the diagonals of a rhombus of side 4 cm is 10 cm, what is tis area?**

(a) 8 cm^{2}

(b) 9 cm^{2}

(c) 10 cm^{2 }

(d) 12 cm^{2}

**34. Two metal cubes, of edge 2 cm each, and one metal cube of edge 3 cm are melted. How many cubes of edge 1 cm each, are needed to be melted so that when combined with the previous melt, a cube, having an edge of minimum intergral value, may be moulded?**

(a) 2

(b) 4

(c) 6

(d) 21

**35. There are 5 cones and 5 cylinders each of base radius r and height r. What is the numnber of spheres of radius r that can be moulded out of these (assuming each body to be sold)?**

(a) 2

(b) 3

(c) 4

(d) 5

**36. Three circles of radii r _{1}, r_{2} and r_{3} are drawn cocertric to each other. The radii r_{1}, r_{2} are such that the area of the circle with radius r1 is equal to the area between the circles of raddi r_{2} and r_{1}. The area between circles of raddi r_{3} and r, r_{2} is equal to the area between the circles of raddi r_{2} and r_{1}. What is the value of r_{1} : r_{2} : r_{3}? **

(a) 1:√2: √3

(b) 2:√2: 2√3

(c) 1 : 2 : 3

(d) 1:2√2: 3√3

**37. If V is volume of cuboid of dimensions q, b, c and S is its surface area, then what is S/V equal to?**

**38. A closed right circular cone contains water up to a height h/2 above the base, where h is height of the cone. To what height does water rise if the cone is inverted?**

(a) h/2

(b) 3h/4

(c) 7/8^{1/2} h

(d) 7/8^{1/3} h

**39. A line through the vertex C of a quadrialateral ABCD parallel to its diagonal BD meets AB produced in E, while a line through D parallel to AB meets a line through A in F. If X, Y, Z are respectively the areas of the quadrialateral ABCD, D AED and DAEF, then which one of the following is correct?**

(a) X ¹Y ¹ Z

(b) X = Y ¹ Z

(c) X ¹ Y = Z

(d) X = Y = Z

**40. What is the last digit in the expansion of 3100?**

(a) 1

(b) 3

(c) 7

(d) 9

**41. What least number must be added to 6591 to make the sum a perfect square?**

(a) 33

(b) 78

(c) 133

(d) 278

**42. What is the compound interest on Rs. 24,000 at the rate of 10% per annum for 1y years according to six-montly system?**

(a) Rs. 3,873

(b) Rs. 3,783

(c) Rs. 3,600

(d) Rs. 3,200

**43. A vessel can be filled by a pipe in 25 minutes and it can be emptied by a waste pipe in 20 minutes. If both the pipes are opened when the vessel is full, how much is the time taken to empty the vessel?**

(a) 5 minutes

(b) 45 minutes

(c) 50 minutes

(d) 100 minutes

**44. If 1.525252... is converted to a fraction, then what is the sume of its numerator and denominator?**

(a) 152

(b) 249

(c) 250

(d) 252

**45. Match List-I with List-II and select the correct answer using the code given below the Lists:
List-I List-II**

(a) 27

^{log}

_{3}2 1. 1

(b) 49

^{log}

_{3}4 2. 2

(c) 25

^{log}

_{3}2 3. 4

4. 8

5. 16

**Code:**

A B C

(a) 4 1 3

(b) 2 1 4

(c) 2 5 4

(d) 4 5 3

**46. Match List-I with List-II and select the correct answer using the code given below the Lists:
List-I List-II**

(a) (A – A) – B 1. f

(b) A – ( A – B) 2. A

(c) B – (A – B) 3. B

4. A Ç B

5. A È B

**Code:**

A B C

(a) 1 2 3

(b) 3 4 5

(c) 1 4 3

(d) 3 2 5

**47. While dividing a number, the divisor is 8 times the quotient and 4 times the remainder. If the remainder is 12, what is the dividend ?**

(a) 288

(b) 300

(c) 360

(d) 396

**48. A train travels of distance of 300 km at a constant speed. If the speed of the train is increased by 5 km/hr, the journey would have taken 2 hours less. What was the initial speed of the train in km/hr?**

(a) 20

(b) 25

(c) 30

(d) 35

**49. A can do a piece of work in 4 days. B can do the same work in 6 days and C in 12 days. A started the work but had no leave after 1 day. In how many days can both B and C complete the remaining work? **

(a) 2 days

(b) 3 days

(c) 4 days

(d) 6 days

**50. To cover a distance of 1 km down the stream a rower takes 10 min, while up the stream the rower takes 30 minutes for the same distance. What is the speed of the stream in km/hr?**

(a) 1

(b) 2

(c) 3

(d) 4

**Combined Defence Services**

**Elementary Mathematics Solved Paper: 2005**

**51. The simple interest on a sum during a period is 9/25 of the sum. It the rate of interest is R% and the time period is R years, then what is the value of R?**

(a) 2

(b) 4

(c) 6

(d) 8

**52. A train takes 18 seconds to pass completely through a station of 162 m long and 15 seconds to pass completely through another station 120 meters long. What is the length of the train?**

(a) 90 m

(b) 100 m

(c) 110 m

(d) 120 m

**53. Equilateral triangles BCP, CAQ and ABR are drawn extremely on the slides BC, CA and AB respectively of a scalene triangle ABC.
Consider the following statements:**

1. DDACP and DDBCQ are similar.

2. DDBPR and DDAQR are similar.

**Which of the following statements given above is/are correct?**

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

**54. Consider the following:**

1. cos^{6} A + sin^{6} A = 1 – 3 sin^{2} A cos^{2} A

2. cos^{4} A + sin^{4} A = 1 – 2 sin^{2} A cos^{2} A

3. cos^{4}A – sin^{4}A = 2cos^{2} A – 1**Which of the statements given above are correct?**

(a) 1 and 2

(b) 2 and 3

(c) 1 and 3

(d) 1, 2 and 3

**56. If tan (x ^{2} – 8x + 60)° = cot (6x – 5)°, what is one of the values of x?**

(a) 7

(b) 8

(c) 9

(d) 10

**57. What is the value of**

(a) 0

(b) sin x

(c) cos x

(d) 1

**what is the vlaue of p + q +r?**

(a) 0

(b) 1

(c) 2

(d) 3

**59. A child of height 4 fit. wants that his shadow should be negligibly small (practically zero). What should be the angle of elevation of sum for sum thing to happen?**

(a) Angle of elevation of sun should be very small tending to 0°.

(b) Angle of elevation of sun should be nearer to 90°.

(c) Angle of elevation of sun should be 45°.

(d) Angle of elevation of sun should be between 30° and 60°.

**60. If sec q = √2 √2 √2.....,
what is the value of cos q (1 + 2 cos q)?**

(a) 0

(b) – 1

(c) 1

(d) 2

**61. A person standing on the bank of a river observes that that the angle subtended by a tree located on the opposite bank is 60°; when he retires 40 m from the bank, he finds the angle to be 30°. What is the breadth of the river?**

(a) 40 m

(b) 20 m

(c) 10 m

(d) 20√3m

**62. What is the value of (cosec A – sin A)
(sec A – cos A) (tan A + cot A)?**

(a) 0

(b) 1

(c) 2

(d) 3

**63. If x = cos ^{2}q + sec^{2}q, which one of the following is correct?**

(a) 0 < x < 1

(b) 1 < x < 2

(c) x = 2

(d) x ³ 2

**64. If 0 < q < j and x = sin q – cos q,
which one of the following is correct?**

(a) x = 0

(b) x is always positive

(c) x is always negative

(d) x may be positive or negative

**65. If sin A + cos B = x and cos A + sin B = y what is the value of sin A cos B + cos A sin B?**

**67. If sin q + cos q = what is the value of sin ^{6}q + cos^{6}q? **

(a) 1/4

(b) 1/2

(c) 1

(d) 2

**68. If sin A + sin B + sin C + sin D = 4, what is the value of cos A + cos B + cos C + cos D?**

(a) 0

(b) 1

(c) 2

(d) 4

**69. What is the angle of elevation of sum if a flag-staff of 3 m high placed on top of a tower throws a shadow of √3?**

(a) 30°

(b) 45°

(c) 60°

(d) Cannot be determined

**70. What value must be given to* so that the number 8* 76246 is divisible by 11?**

(a) 0

(b) 1

(c) 2

(d) 3

**71. Consider the following statements:**

1. The produt of an integrer and a rational number can never be a natural number.

2. The quotient of division of an integer by rational number can never be an integer.**Which of the statements given above is/are correct?**

(a) 1 only

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

**72. A tradesman marks two prices on two similar items, one for the cash payment and the other at the credit of 6 months. What will be the ratio between the two prices if the rate of interest is 10% per annum?**

(a) 10 : 11

(b) 20 : 21

(c) 40 : 41

(d) 50 : 51

**73. An equilateral triangle is inscribed in a circle and another circle is inscribed in this equilateral triangle. What is the ratio of area of the bigger circle to that of smaller circle? **

(a) 2 :1

(b) 3:2

(c) 3:1

(d) 4:1

**74. A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. What is the total amount of metal wasted?**

(a) 1/2 times the area of the original square piece.

(b) 1/4 times the area of the original square piece

(c) 1/2 times the area of the circular piece.

(d) 1/4 times the area of the circular piece.

**75. A hemi-spherical bowl has its external diameter as 10 cm. Its thickness is 1 cm. What is the whole surface area of the bowl?**

(a) 258 cm^{2}

(b) 280 cm^{2}

(c) 284 cm^{2}

(d) 286 cm^{2}

**76. A right circular cone is separated into three solids of curved surface areas S _{1}, S_{2}, S_{3} by two planes which are curved surface areas S_{1}, S_{2}, S_{3} by two planes which are parallel to the base and trisect the altitude. What is the vlaue of S_{1} : S_{2} : S_{3}?**

(a) 1:1:1

(b) 1:2:3

(c) 1:3:5

(d) 1:3:6

**77. A hollow right circular cylinder with height 8 cm and base radius 7 cm is opened out into a rectangle. What are the length, breadth of the rectangle respectively.**

(a) 22 cm, 16 cm

(b) 44 cm, 8 cm

(c) 22 cm, 8 cm

(d) 44 cm, 16 cm

**78. On increasing the radius of a cylinder by 6 units, the volume increases by x cubic units. On increasing the altitude of the cylinder by 6 units, the volume also increases by x cubic units. If the original altitude is 2 units, what is the original radius?**

(a) 2 units

(b) 4 units

(c) 6 units

(d) 8 units

**79. A cube having each side of unit length is cut into two parts by a plane through two diagonals of two opposite faces. What is the total surface of each of these parts?**

(a) 3 √2 sq.units

(b) 2 √3 sq.units

(c) √2 sq.units

(d) 3 sq. units

**80. The minute hand of a clock is 14 cm long. If it moves between 8.00 a.m., what is the area covered by it on the face of the clock?**

(a) 512 cm^{2}

(b) 462 cm^{2}

(c) 264 cm^{2}

(d) 196 cm^{2}

**81. If A **Ç **B = ** **F****, then which one of the following holds for A **Ç** B?**

(a) B´

(b) A´

(c) A

(d) B

**what is the value of (x + y + z)?**

(a) a + b + c

(b) a^{2} + b^{2} + c^{2}

(c) 0

(d) 1

**83. When a + b + c + 3a ^{1/3} b^{2/3} + 3a^{2/3} b^{1/3} is divided by a1/3 + b1/3 + c1/3, what is the remainder**

(a) 3a

(b) 3b

(c) 0

(d) c

^{2/3}

**84. What is the HCF of the polynomials n ^{2} (n + 1)^{2} – 1, n^{4} – 3n^{2} + 1, n^{4} – (n – 1)^{2} ?**

(a) n

^{2}– n – 1

(b) n

^{2}– n + 1

(c) n

^{2}+ n +1

(d) n

^{2}+ n – 1

**85. Consider the following all real numbers
a, b and c:**

1. If a

^{2}+ b

^{2}+ c

^{2}= ab + bc + ca, then a = b = c

2. If a + b + c = 0, then a

^{3}+ b

^{3}+ c

^{3}= 3abc

3. If a + b = c, then a

^{2}+ b

^{3}+ c

^{3}3abc = 0

**Which of the above is/are correct?**

(a) 1 only

(b) 2 only

(c) 1 and 2

(d) 1, 2 and 3

**86. The value of a machine depreciates every year by 4%. Its value at present is Rs. 6,250. What will be its value after two years.**

(a) Rs. 5,000

(b) Rs. 5,550

(c) Rs. 5,760

(d) Rs. 6,000

**87. If a, b are natural numbers such that 7a + 5b is divisible by 19, which one of the following is divisible by 19?**

(a) –7a – 5b

(b) 14a + 2b

(c) 9a + 20b

(d) 13a + 6b

**88. (2x – 3y) ^{3} + (3y – 4z)^{3} + (4z – 2x)^{3} can be factorised into which one of the following?**

(a) (2x + 3y + 4z) (2x – 3y – 4z)

(b) (2x + 3y – 4z) (2x – 3y – 4z)

(c) (2x – 3y) (3y – 4z) (4z – 2x)

(d) 6 (2x – 3y) (3y – 4z) (2x – x)

**89. **

(a) – 1, – 4

(b) – 1, 4

(c) 3, – 4

(d) 3, 4

**90. What is the digit in the unit place of the number represented by (7 ^{29} + 9^{26})?**

(a) 5

(b) 6

(c) 7

(d) 8

**91. If y + z exceeds x by 11 and z + x exceeds x by 7, how much does by exceed x?**

(a) 1

(b) 2

(c) 4

(d) Cannot be determined

**92. Which one of the following is correct?**

(a)

(b)

(c)

(d)

**Directions: The following five (5) items consist of two statements: one labelled as the ‘Assertion (A)’ and the other as ‘Reason (R)’. You are to examine these two statements carefully and select the answers to these items using the code given below:
Code:**

(a) Both A and R are individually true, and R is the correct explanation of A

(b) Both A and R are individually true but R is not the correct explanation of A

(c) A is true but R is false

(d) A is false but R is true

**93. Assertion (A): 4x – 5y = 1 and 8/5x 2y 2/5 have infinite number of solutions.
Reason (R): a _{1} x + b_{1} y = c_{1} and a_{2} x + b_{2}y = c_{2} have infinite number of solutions, if **

**a _{1}/a_{2} b_{1}/b_{2} c_{1}/c_{2}**

**94. Assertion (A): The more than and less than type frequency curves cut each other at median.
Reason (R): Point of intersection divides the distribution into two euqal parts. **

**95. Assertion (A): Mean is the best mesure of central tendency.
Reason (R): It is very much affected by the extreme values.**

**96. Assertion (A): 3 and 8 are relatively prime;
8 and 13 are relatively prime; 3 and 13 are relatively prime.
Reason (R): If a and b are relatively prime; b and c are relatively primes then, a and c must be relatively prime.**

**97. Assertion (A): The rectangle whose perim-eter is 20 units will have an area less than or equal to 25 square units.
Reason (R): In comparison to all rectangles of given perimeter, a square of the same perimeter has the maximum area.**

**98. For a given data of 50 sampled observations, there are 30 observations that are less than or equal to the value 5.45. What does the value 5.45 indicate?**

(a) Mean

(b) 30^{th} %

(c) 60th %

(d) 54.5^{th} %

**99. What is the median of the data 1, 2, 2, 3, 4, 4, 4, 4, .... up to 36 terms?**

(a) 4.5

(b) 6.0

(c) 6.5

(d) 18

**100. Consider the following statements in respect of graphical representation of data:**

1. It makes reading more interesting.

2. It is less time-consuming.

3. It is easily understandable.**Which of the statements given above are correct?**

(a) 1 and 2

(b) 1 and 3

(c) 2 and 3

(d) 1, 2 and 3

## Answers

**1. (b) 2. (b) 3. (d) 4. (d) 5. (b) 6. (b) 7. (b) 8. (c) 9. (c) 10. (b) 11. (a) 12. (d) 13. (c) 14. (a) 15. (c) 16. (c) 17. (a) 18. (b) 19. (b) 20. (c) 21. (b) 22. (b) 23. (b) 24. (b) 25. (c) 26. (b) 27. (a) 28. (c) 29. (d) 30. (d) 31. (c) 32. (c) 33. (*) 34. (d) 35. (d) 36. (a) 37. (c) 38. (d) 39. (c) 40. (a) 41. (c) 42. (b) 43. (d) 44. (c) 45. (d) 46. (c) 47. (b) 48. (b) 49. (b) 50. (a) 51. (c) 52. (a) 53. (a) 54. (d) 55. (c) 56. (a) 57. (d) 58. (d) 59. (b) 60. (c) 61. (b) 62. (b) 63. (d) 64. (c) 65. (c) 66. (c) 67. (a) 68. (a) 69. (d) 70. (c) 71. (d) 72. (b) 73. (d) 74. (a) 75. (d) 76. (b) 77. (b) 78. (c) 79. (a) 80. (b) 81. (d) 82. (c) 83. (c) 84. (d) 85. (c) 86. (c) 87. (c) 88. (d) 89. (a) 90. (d) 91. (b) 92. (d) 93. (a) 94. (a) 95. (c) 96. (c) 97. (b) 98. (c) 99. (b) 100. (d)**