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CDS (II) 2023, Elementary Mathematics question paper

100 questions · 100 marks · 120 minutes · +1 per correct answer, -0.33 per wrong one

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  1. 1
    If a2−bc=αa^2 - bc = \alpha, b2−ac=βb^2 - ac = \beta, c2−ab=γc^2 - ab = \gamma, then what is aα+bβ+cγ(a+b+c)(α+β+γ)\dfrac{a\alpha + b\beta + c\gamma}{(a + b + c)(\alpha + \beta + \gamma)} equal to ?

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  2. 2
    If (x−1)3(x - 1)^3 is a factor of x4+αx3+βx2+γx−1x^4 + \alpha x^3 + \beta x^2 + \gamma x - 1, then the other factor will be :

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  3. 3
    A 2-digit number is such that the sum of the number and the number obtained by reversing the order of the digits of the number is 55. Further, the difference of the given number and the number obtained by reversing the order of the digits of the number is 45. What is the product of the digits ?

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  4. 4
    If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, C and A can finish the same work in 15 days; then in how many days can A, B and C together finish half of the work ?

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  5. 5
    A person borrowed Rs. 10,000 at 12% rate of interest per annum compounded quarterly for a period of 9 months. What is the interest paid by him to settle his account after 9 months ?

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  6. 6
    For what relation between a and b is the equation sin⁡θ=a+b2ab\sin \theta = \dfrac{a + b}{2\sqrt{ab}} possible ?

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  7. 7
    Three persons A, B and C together can do a piece of work in 36 days. A and B together can do five times as much work as C alone; B and C together can do as much work as A alone. If A and C together can do n times as much work as B alone, then what is the value of n ?

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  8. 8
    If 2a3=4b5=3c4\dfrac{2a}{3} = \dfrac{4b}{5} = \dfrac{3c}{4}, then what is the value of 18aa2+c2−b2\dfrac{18}{a}\sqrt{a^2 + c^2 - b^2} ?

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  9. 9
    The sum of deviations of n numbers from 10 and 20 are a, b respectively. If ba=−4\dfrac{b}{a} = -4, then what is the mean of these n numbers ?

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  10. 10
    If the median of observations 12, 1, 8, 54, 61, 28, 45, 35, 21, 17 is M, then what is the value of 2M+52M + 5 ?

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  11. 11
    How many real roots does the equation x+9=x−3\sqrt{x + 9} = x - 3 have ?

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  12. 12
    If x=97+563x = 97 + 56\sqrt{3}, then what is the value of x4+1x4\sqrt[4]{x} + \dfrac{1}{\sqrt[4]{x}} ?

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  13. 13
    Let L be the LCM and H be the HCF of two given numbers. L and H are in the ratio 3 : 2. If the sum of the two numbers is 45, then what is the product of the numbers ?

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  14. 14
    A man walks at an average speed of 3 km/hr from his home and reaches office 40 minutes early. If he walks at an average speed of 2 km/hr, he would reach office 40 minutes late. What is the distance between his home and office ?

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  15. 15
    If 3x−1+33−x=63^{x - 1} + 3^{3 - x} = 6, then what is 2x−1+23−x2^{x - 1} + 2^{3 - x} equal to ?

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  16. 16
    If x(a−b+aba−b)=y(a+b−aba+b)x\left(a - b + \dfrac{ab}{a - b}\right) = y\left(a + b - \dfrac{ab}{a + b}\right) and x+y=2a3x + y = 2a^3, then what is x−yx - y equal to ?

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  17. 17
    Which one of the following is a factor of 33x3+22y3−18xy+663\sqrt{3}x^3 + 2\sqrt{2}y^3 - 18xy + 6\sqrt{6} ?

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  18. 18
    What is number of digits in the expansion of 125100125^{100} ? (Given log⁡102=0.301\log_{10} 2 = 0.301)

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  19. 19
    What is the HCF of acx3+bcx2+adx2+acdx+bdx+bcdacx^3 + bcx^2 + adx^2 + acdx + bdx + bcd and adx3+acx2+bdx2+bcx+acdx+bcdadx^3 + acx^2 + bdx^2 + bcx + acdx + bcd if HCF (c,d)=1\text{HCF}\,(c, d) = 1, c≠dc \ne d ?

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  20. 20
    If xn−pyn+qznx^n - py^n + qz^n is divisible by x2+abyz−bzx−axyx^2 + abyz - bzx - axy, then what is pan−qbn\dfrac{p}{a^n} - \dfrac{q}{b^n} equal to ?

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  21. 21
    Consider the following statements : 1. If (a+b)(a + b) is directly proportional to (a−b)(a - b), then (a2+b2)(a^2 + b^2) is directly proportional to ab. 2. If a is directly proportional to b, then (a2−b2)(a^2 - b^2) is directly proportional to ab. Which of the statements given above is/are correct ?

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  22. 22
    If (3a+6b+c+2d)×(3a−6b−c+2d)=(3a−6b+c−2d)×(3a+6b−c−2d)(3a + 6b + c + 2d) \times (3a - 6b - c + 2d) = (3a - 6b + c - 2d) \times (3a + 6b - c - 2d), then which one of the following is correct ?

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  23. 23
    If 3sin⁡θ+5cos⁡θ=53\sin\theta + 5\cos\theta = 5, then what is the value of 5sin⁡θ−3cos⁡θ5\sin\theta - 3\cos\theta equal to ?

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  24. 24
    The combined age of a man and his wife is 6 times the combined age of their children. Two years ago their combined age was 10 times the combined age of their children; and six years later their combined age will be 3 times the combined age of their children. How many children do they have if each child is at least 2 years old ?

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  25. 25
    What is 3(sin⁡x−cos⁡x)4+6(sin⁡x+cos⁡x)2+4(sin⁡x)6+4(cos⁡x)63(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin x)^6 + 4(\cos x)^6 equal to ?

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  26. 26
    What is the value of sin⁡θ+cos⁡θ\sin\theta + \cos\theta, if θ\theta satisfies the equation cot⁡2θ−(3+1)cot⁡θ+3=0\cot^2\theta - (\sqrt{3} + 1)\cot\theta + \sqrt{3} = 0; 0<θ<π40 < \theta < \dfrac{\pi}{4} ?

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  27. 27
    Which one of the following is a value of θ\theta, if θ\theta satisfies the equation tan⁡2θtan⁡4θ−1=0\tan 2\theta \tan 4\theta - 1 = 0; 0<θ<π20 < \theta < \dfrac{\pi}{2} ?

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  28. 28
    If tan⁡x=sin⁡θ+cos⁡θsin⁡θ−cos⁡θ\tan x = \dfrac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta}, π4<θ<π2\dfrac{\pi}{4} < \theta < \dfrac{\pi}{2}, then what is 2 sin⁡x\sqrt{2}\,\sin x equal to ?

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  29. 29
    How many values of θ\theta will satisfy the equation (sin⁡2θ−4sin⁡θ+3)(4−cos⁡2θ+4sin⁡θ)=0(\sin^2\theta - 4\sin\theta + 3)(4 - \cos^2\theta + 4\sin\theta) = 0, where 0<θ<π20 < \theta < \dfrac{\pi}{2} ?

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  30. 30
    If xsin⁡3θ+ycos⁡3θ=sin⁡θcos⁡θx\sin^3\theta + y\cos^3\theta = \sin\theta\cos\theta and xsin⁡θ−ycos⁡θ=0x\sin\theta - y\cos\theta = 0, for every θ∈(0,π2)\theta \in \left(0, \dfrac{\pi}{2}\right), then what is x2+y2x^2 + y^2 equal to ?

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  31. Consider the following for the next two (02) items that follow : A flagstaff stands on the top of a vertical tower. The angle of elevation of the top of the flagstaff from a certain place on the same horizontal level with the base of the tower is found to be α\alpha. Advancing a distance d towards the tower in the same horizontal plane, the angle of elevation of the top of the flagstaff is observed to be β\beta and that of the top of the tower is observed to be γ\gamma. Let H be the height of the top of the flagstaff from the base of the tower and h be the height of the tower.
    31
    Which one of the following is correct ?

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  32. 32
    Which one of the following is correct ?

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  33. AB is a straight road leading to the foot P of a tower of height h. Q is at distance x from P and R is at a distance y from Q (R is farther from P than Q; R, Q are on the same side). The angle of elevation of the top of the tower at Q is twice of that at R. (Use the formula tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta})
    33
    Which one of the following is correct ?

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  34. 34
    Which one of the following is correct ?

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  35. Two parallel chords AB and CD of a circle are of lengths 60 cm and 80 cm respectively. They are on the same side of the centre O and 10 cm apart.
    35
    What is the diameter of the circle ?

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  36. 36
    If the chord AB subtends an angle α\alpha and chord CD subtends an angle β\beta at the centre O, then what is the value of tan⁡(β2)−tan⁡(α2)\tan\left(\frac{\beta}{2}\right) - \tan\left(\frac{\alpha}{2}\right) ?

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  37. Let p=x4−y2z2p = x^4 - y^2z^2, q=y4−z2x2q = y^4 - z^2x^2, r=z4−x2y2r = z^4 - x^2y^2.
    37
    What is px2+qy2+rz2px^2 + qy^2 + rz^2 equal to ?

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  38. 38
    What is x2(px2+qy2+rz2)+qr−p2x^2(px^2 + qy^2 + rz^2) + qr - p^2 equal to ?

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  39. A right conical cap just covers two spheres placed one above the other on a table such that it touches both the spheres. Let r be the radius of the smaller sphere and R be the radius of the bigger sphere. Let 2θ2\theta be the vertical angle of the cone.
    39
    What is the height of the cone ?

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  40. 40
    What is the radius of the base of the cone ?

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  41. A line segment AB is bisected at C and semi-circles S1S_1, S2S_2 and S3S_3 are drawn respectively on AB, AC and CB as diameters such that they all lie on same side of AB. A circle S is drawn touching internally S1S_1 and externally S2S_2 and S3S_3.
    41
    If r is the radius of S and R is the radius of S2S_2, then which one of the following is correct ?

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  42. 42
    If m is the area of the circle S and n is the area of semi-circle S1S_1, then which one of the following is correct ?

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  43. Let (x−a)(x−b)(x−ma)(x−mb)=(x+a)(x+b)(x+ma)(x+mb)\frac{(x-a)(x-b)}{(x-ma)(x-mb)} = \frac{(x+a)(x+b)}{(x+ma)(x+mb)}; m,a,b>0m, a, b > 0.
    43
    What is x2+abx2+m2ab\frac{x^2 + ab}{x^2 + m^2ab} equal to ?

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  44. 44
    What is x equal to ?

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  45. The total monthly electricity bill for a house consists of the sum of two parts, one part is proportional to number of rooms and the other part is proportional to number of units consumed. Rs. 400 is the monthly electricity bill for a house with 8 rooms and consuming 240 units and Rs. 320 is the monthly electricity bill for a house with 6 rooms and consuming 200 units.
    45
    What is the monthly electricity bill for a house with m rooms and consuming n units ?

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  46. 46
    What is the monthly electricity bill for a house with 7 rooms consuming 300 units ?

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  47. A grouped frequency distribution is given below :
    Weekly wages in Rupees (Rs.)Numbers of workers
    2050 - 25505
    2550 - 305010
    3050 - 3550k
    3550 - 40508
    4050 - 45502
    4550 - 505010
    47
    If average weekly wages earned by a worker is Rs. 3,520, then what is the value of k ?

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  48. 48
    What is the median (approximate value) of the distribution ?

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  49. A quadratic equation is given by (a+b+c)x2−(2a+2b)x+(a+b−c)=0(a + b + c)x^2 - (2a + 2b)x + (a + b - c) = 0; where a, b and c are real and distinct.
    49
    What are the roots of the equation ?

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  50. 50
    Consider the following statements : 1. One of the roots of the equation is always less than 1 if a, b and c are all positive. 2. One of the roots of the equation is always negative if a, b and c are all negative. Which of the statements given above is/are correct ?

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  51. 51
    What is the radius of the sphere passing through the corners of the cuboid with edges 8 cm, 12 cm and 24 cm ?

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  52. 52
    A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end ?

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  53. 53
    A sum of money at 20% rate of compound interest per annum becomes more than 100 times in n years. What is the least value of n ? (Use log⁡102=0.301\log_{10} 2 = 0.301, log⁡103=0.477\log_{10} 3 = 0.477)

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  54. 54
    The corners of an equilateral triangular plate were cut in such a manner that it forms a regular hexagonal plate. What is the ratio of the area of the triangular plate to the area of the hexagonal plate ?

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  55. 55
    Two equal arcs of different circles C1C_1 and C2C_2 subtend angles of 60∘60^\circ and 75∘75^\circ respectively, at the centres. What is the ratio of the radius of C1C_1 to the radius of C2C_2 ?

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  56. 56
    ABC is a triangle with sides AB = 41 cm, BC = 28 cm and CA = 15 cm. If D, E and F are the mid-points of AB, BC and CA respectively, then what is the area of the triangle DEF ?

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  57. 57
    A triangle has sides 13 cm, 14 cm and 15 cm long. What is the length of the smallest altitude of the triangle ?

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  58. 58
    The circumference of a circle exceeds the diameter by 16.8 cm. What is the diameter of the circle ? (Take π=227\pi = \frac{22}{7})

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  59. 59
    The hypotenuse AC of a right-angled ABC is 353\sqrt{5} cm. If AB is doubled and BC is tripled such that ABC remains a right-angled triangle, the hypotenuse becomes 15 cm. What is AB + BC equal to ?

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  60. 60
    What is the area of the region between two concentric circles if the chord of the outer circle of length 14 cm is a tangent of the inner circle ? (Take π=227\pi = \frac{22}{7})

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  61. 61
    A pendulum swings through an angle of 9∘9^\circ and its end describes an arc of length 14.3 cm. What is the length of the pendulum ? (Take π=227\pi = \frac{22}{7})

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  62. 62
    The arch of a bridge is in the form of an arc of a circle. If the span of the bridge is 40 m and height in the middle is 8 m, then what is the radius of curvature of the bridge ?

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  63. 63
    If a, b and c are the sides of a right-angled triangle, where a > b > c, then what is the value of the expression (a+b+c)(a+b−c)(a−b+c)(a−b−c)(a + b + c) (a + b - c) (a - b + c) (a - b - c) ?

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  64. 64
    The cube root of x varies inversely as the square root of y. x = 8 when y = 3. What is the value of x when y=33y = \sqrt[3]{3} ?

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  65. 65
    Three solid lead spheres of diameters 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres ?

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  66. 66
    A solid sphere of radius 3 cm is melted to form a hollow cylinder of height 4 cm and external diameter 10 cm. What is the thickness of the cylinder ?

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  67. 67
    Three glasses P, Q and R have capacities in the ratio 1 : 2 : 3. All these glasses are completely filled with mixtures of milk and water. The ratio of milk to water in P is 1 : 2, in Q it is 2 : 3 and in R it is 3 : 1. If the content of all three glasses are put into a bigger container, what will be the ratio of milk to water in the container ?

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  68. 68
    What is the LCM of x4+x2y2+y4x^4 + x^2 y^2 + y^4, x3+y3x^3 + y^3, x3−y3x^3 - y^3 ?

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  69. 69
    What is x2−y2−z2−2yzx2+y2−z2+2xy+x2−y2−z2−2yzx2−y2+z2−2xz\frac{x^2 - y^2 - z^2 - 2yz}{x^2 + y^2 - z^2 + 2xy} + \frac{x^2 - y^2 - z^2 - 2yz}{x^2 - y^2 + z^2 - 2xz} equal to ?

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  70. 70
    If tan⁡A+cot⁡A=2\tan A + \cot A = 2, where 0<A<90∘0 < A < 90^\circ, then what is the value of tan⁡2A+tan⁡3A+tan⁡4A+....+tan⁡nA\tan^2 A + \tan^3 A + \tan^4 A + .... + \tan^n A ?

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  71. 71
    Which of the following is/are identity/identities ? 1. sin⁡3θ+cos⁡3θsin⁡θ+cos⁡θ+sin⁡θcos⁡θ=1\frac{\sin^3 \theta + \cos^3 \theta}{\sin \theta + \cos \theta} + \sin \theta \cos \theta = 1; 0<θ<π20 < \theta < \frac{\pi}{2} 2. 1−sin⁡6θ=cos⁡2θ(cos⁡4θ+3sin⁡2θ)1 - \sin^6 \theta = \cos^2 \theta (\cos^4 \theta + 3 \sin^2 \theta) Select the correct answer using the code given below :

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  72. 72
    If 7sin⁡4θ+9cos⁡4θ+42sin⁡2θ=167 \sin^4 \theta + 9 \cos^4 \theta + 42 \sin^2 \theta = 16, 0<θ<π20 < \theta < \frac{\pi}{2}, then what is tan⁡θ\tan \theta equal to ?

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  73. 73
    An isosceles triangle has its base length 2a and its height is h. On each side of the triangle a square is drawn external to the triangle. What is the area of the figure thus formed ?

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  74. 74
    If p=a2(b−a)(c−a)p = \frac{a^2}{(b - a)(c - a)}, q=b2(c−b)(a−b)q = \frac{b^2}{(c - b)(a - b)}, r=c2(a−c)(b−c)r = \frac{c^2}{(a - c)(b - c)}, then what is (p+q+r)2(p + q + r)^2 equal to ?

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  75. 75
    Which one of the following is a factor of a2−b2−c2+2bc+a+b−ca^2 - b^2 - c^2 + 2bc + a + b - c ?

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  76. 76
    Let α\alpha and β\beta be the roots of the equation 1x+a+b=1x+1a+1b\frac{1}{x + a + b} = \frac{1}{x} + \frac{1}{a} + \frac{1}{b}; a≠0a \ne 0, b≠0b \ne 0, x≠0x \ne 0. Which one of the following is a quadratic equation whose roots are α2\alpha^2 and β2\beta^2 ?

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  77. 77
    If x=67−67−67−67−xx = \cfrac{6}{7 - \cfrac{6}{7 - \cfrac{6}{7 - \cfrac{6}{7 - x}}}}; x>1x > 1, then what is the value of x2−3x+2x^2 - 3x + 2 equal to ?

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  78. 78
    A train completely overtakes two persons, walking in the same direction with speeds 3 km/hr and 4 km/hr in 9 seconds and 758\frac{75}{8} seconds respectively. What is the length of the train ?

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  79. 79
    A person bought an article and sold it at a profit of 20%. Had he bought it at 20% less, what would have been the profit percentage if the selling price had been the same ?

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  80. 80
    If 2s=a+b+c2s = a + b + c, then what is s2+(s−a)(s−b)+(s−b)(s−c)+(s−c)(s−a)s^2 + (s - a)(s - b) + (s - b)(s - c) + (s - c)(s - a) equal to ?

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  81. 81
    A sphere of radius 5 cm is dropped in a right circular cylindrical vessel partly filled with water. The radius of the cylindrical vessel is 10 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel ?

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  82. 82
    Consider the following statements : 1. The angle in a sector greater than a semi-circle is less than a right angle. 2. If two sides of a pair of opposite sides of a cyclic quadrilateral are equal, then its diagonals are also equal. Which of the statements given above is/are correct ?

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  83. 83
    If a,b,c,x,y,za, b, c, x, y, z are real numbers such that (a+b+c)2−3(ab+bc+ca)+3(x2+y2+z2)=0(a + b + c)^2 - 3(ab + bc + ca) + 3(x^2 + y^2 + z^2) = 0, then which one of the following is correct ?

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  84. 84
    In a triangle ABC, angle B=90∘B = 90^\circ and p is the length of the perpendicular from B to AC. If BC = 10 cm and AC = 12 cm, then what is the value of p ?

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  85. 85
    The mean of p, q, r, s and t is 280. If the mean of p, r and t is 240, what is the mean of q and s ?

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  86. Consider the following for the next (05) items that follow : A, B, C, D, E, F and G are cousins. D is thrice as old as A. Further, C is as many years younger to B, as G to E and E to D. The average age of D and G is 16 years; the average age of A and E is 11 years; the average age of B and C is also 11 years. B and C have equal weight. A's weight is 10 kg less than that of B; D is 4 kg heavier than E; E is 4 kg heavier than F; F is 4 kg heavier than G. Further, D has age-weight ratio of 9 : 20, where age is in years and weight in kg; A has age-weight ratio of 2 : 5. Moreover, none of them is more than 40 kg.
    86
    What is D's age (in years) ?

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  87. 87
    What is the average age (in years) of B, C, D, E and G ?

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  88. 88
    What is the difference between the weights (in kg) of G and C ?

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  89. 89
    What is the average weight (in kg) of A, B, C, D, E, F and G ?

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  90. 90
    Consider the following statements : 1. The age of F cannot be determined due to insufficient data. 2. The average weight of D and F is equal to weight of E. 3. The weight difference is maximum for D and A. Which of the statements given above are correct ?

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  91. Consider the following for the next ten (10) items that follow :
    91
    Question : Is m>nm > n if m, n are real numbers ? Statement-I : m=(1−p)(p2+p+1)m = (1 - p)(p^2 + p + 1) and n=(p+1)(p2−p+1)n = (p + 1)(p^2 - p + 1) Statement-II : m=pnm = pn

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  92. 92
    Question : What is the other root of the quadratic equation with real coefficients if one of the roots is −4−102\frac{-4 - \sqrt{10}}{2} ? Statement-I : The product of the roots is −32(3+10)-\frac{3}{2}(3 + \sqrt{10}). Statement-II : The sum of roots of quadratic equation is −1-1.

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  93. 93
    Question : What is the 3-digit number which is divisible by 10 ? Statement-I : If the digits in hundred's place and ten's place of the number are interchanged, the resulting number is diminished by 180. Statement-II : If the digit in hundred's place is halved and digit in ten's place and unit place of the number are interchanged, the resulting number is diminished by 336.

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  94. 94
    Question : Are x, y, z equal, where x, y, z are real numbers ? Statement-I : x2+y2+z2−xy−yz−zx=0x^2 + y^2 + z^2 - xy - yz - zx = 0 Statement-II : x3+y3+z3−3xyz=0x^3 + y^3 + z^3 - 3xyz = 0

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  95. 95
    Question : What is the ratio x:y:zx : y : z equal to if x,y,z≠0x, y, z \neq 0 ? Statement-I : x+zy=zx\frac{x + z}{y} = \frac{z}{x} Statement-II : z−yx=xz\frac{z - y}{x} = \frac{x}{z}

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  96. 96
    Question : What is the sum of two natural numbers ? Statement-I : LCM of the two numbers is 144. Statement-II : One of the numbers is 72.

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  97. 97
    Question : Is average of the largest and the smallest of 4 given numbers greater than the average of the 4 numbers ? Statement-I : The difference between the largest and the second largest numbers is less than the difference between the second smallest and the smallest of the numbers. Statement-II : The difference between the largest and the smallest numbers is greater than the difference between the second largest and the second smallest of the numbers.

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  98. 98
    Question : Is (a−b+c)>(a+b−c)(a - b + c) > (a + b - c), where a, b and c are real numbers ? Statement-I : b is negative. Statement-II : c is negative.

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  99. 99
    Question : What is the cost of 15 pens, 21 pencils and 18 note books ? Statement-I : The cost of 7 pens, 6 pencils and 5 note books is Rs. 200. Statement-II : The cost of 3 pens, 8 pencils and 7 note books is Rs. 210.

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  100. 100
    Question : What is the area of the triangle inscribed in a semi-circle with the diameter as the base ? Statement-I : The diameter of semi-circle is 20 cm. Statement-II : Two shorter sides of the triangle are 12 cm and 16 cm.

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