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CDS (II) 2024, 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
    A real number xx is such that the sum of the number and four times its square is the least. What is that number ?

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  2. 2
    The difference of the square of two natural numbers mm and nn (m>n)(m > n) is 72. How many pairs of natural numbers will satisfy ?

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  3. 3
    Let NN be a 5-digit number. When NN is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of NN ?

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  4. 4
    What is the remainder when 111222+222333+333444111^{222} + 222^{333} + 333^{444} is divided by 5 ?

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  5. 5
    What are the last three digits in the multiplication of 4321012345×987667894321012345 \times 98766789 ?

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  6. 6
    pp varies directly as (x2+y2+z2)(x^2 + y^2 + z^2). When x=1x = 1, y=2y = 2, z=3z = 3, then p=70p = 70. What is the value of pp when x=−1x = -1, y=1y = 1, z=5z = 5 ?

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  7. 7
    Let NN be the least positive multiple of 11 that leaves a remainder of 5 when divided by 6, 12, 15, 18. Which one of the following is correct ?

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  8. 8
    What is 110+9+111+10+112+11+…+1196+195\frac{1}{\sqrt{10} + \sqrt{9}} + \frac{1}{\sqrt{11} + \sqrt{10}} + \frac{1}{\sqrt{12} + \sqrt{11}} + \ldots + \frac{1}{\sqrt{196} + \sqrt{195}} equal to ?

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  9. 9
    Train XX crosses a man standing on the platform in 24 seconds and train YY crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of XX to speed of YY ?

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  10. 10
    Let p,qp, q be the roots of the equation x2+mx−n=0x^2 + mx - n = 0 and m,nm, n be the roots of the equation x2+px−q=0x^2 + px - q = 0 (m,n,p,q(m, n, p, q are non-zero numbers). Which of the following statements is/are correct ? I. m(m+n)=−1m(m + n) = -1 II. p+q=1p + q = 1 Select the answer using the code given below :

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  11. 11
    What is the maximum value of 8sin⁡θ−4sin⁡2θ8\sin\theta - 4\sin^2\theta ?

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  12. 12
    What is (1+tan⁡α tan⁡β)2+(tan⁡α−tan⁡β)2(1 + \tan\alpha\,\tan\beta)^2 + (\tan\alpha - \tan\beta)^2 equal to ?

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  13. 13
    Consider the following statements : I. tan⁡50∘−cot⁡50∘\tan 50^\circ - \cot 50^\circ is positive II. cot⁡25∘−tan⁡25∘\cot 25^\circ - \tan 25^\circ is negative Which of the statements is/are correct ?

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  14. 14
    If 0≤(α−β)≤(α+β)≤π20 \le (\alpha - \beta) \le (\alpha + \beta) \le \frac{\pi}{2}, tan⁡(α+β)=3\tan(\alpha + \beta) = \sqrt{3} and tan⁡(α−β)=13\tan(\alpha - \beta) = \frac{1}{\sqrt{3}}, then what is tan⁡α⋅cot⁡2β\tan\alpha \cdot \cot 2\beta equal to ?

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  15. 15
    What is the value of sin⁡2θcos⁡2θ(sec⁡2θ+cosec2θ)\sin^2\theta \cos^2\theta (\sec^2\theta + \mathrm{cosec}^2\theta) equal to ?

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  16. 16
    If 64sin⁡2θ+64cos⁡2θ=1664^{\sin^2\theta} + 64^{\cos^2\theta} = 16 where 0≤θ≤π20 \le \theta \le \frac{\pi}{2}, then what is the value of tan⁡θ+cot⁡θ\tan\theta + \cot\theta ?

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  17. 17
    If cosec θ−cot⁡θ=m\mathrm{cosec}\,\theta - \cot\theta = m and sec⁡θ−tan⁡θ=n\sec\theta - \tan\theta = n, then what is cosec θ+sec⁡θ\mathrm{cosec}\,\theta + \sec\theta equal to ?

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  18. 18
    From a point XX on a bridge across a river, the angles of depression of two points PP and QQ on the banks on opposite side of the river are α\alpha and β\beta respectively. If the point XX is at a height hh above the surface of the river, what is the width of the river if α\alpha and β\beta are complementary ?

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  19. 19
    In a triangle ABCABC, ∠ABC=60∘\angle ABC = 60^\circ and ADAD is the altitude. If AB=6AB = 6 cm and BC=8BC = 8 cm, then what is the area of the triangle ?

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  20. 20
    If pp and qq are the roots of the equation x2−sin⁡2θ x−cos⁡2θ=0x^2 - \sin^2\theta\,x - \cos^2\theta = 0, then what is the minimum value of p2+q2p^2 + q^2 ?

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  21. 21
    The arithmetic mean of nn numbers is MM. If the sum of first (n−1)(n - 1) terms is kk, then what is the nnth number ?

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  22. 22
    What is the geometric mean of 3, 9, 27, 81, 243, 729, 2187 ?

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  23. 23
    A person purchases one kg of tea powder from each of the four places A,B,C,DA, B, C, D at the rate of Rs. 1000 per 1 kg, 2 kg, 4 kg, 5 kg. If on an average he purchased xx kg of tea powder per Rs. 1000, then what is the approximate value of xx ?

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  24. 24
    What is the sum of the largest and the smallest 4-digit numbers made by using single digit prime numbers (without repetition) ?

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  25. 25
    What is the remainder when 32553^{255} is divided by 28 ?

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  26. 26
    What is the value of x (0≤x≤8)x\,(0 \le x \le 8) if (10097+10054+x+1)(100^{97} + 100^{54} + x + 1) leaves a remainder 0 when divided by 9 ?

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  27. 27
    In a triangle ABCABC, DD is a point on BCBC. If AB⋅DC=AC⋅BDAB \cdot DC = AC \cdot BD, ∠BAD=α\angle BAD = \alpha and ∠CAD=β\angle CAD = \beta then which one of the following is correct ?

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  28. 28
    Let N=12345678ABN = 12345678AB be a 10-digit number, where AA, BB are digits. If NN is divisible by 9, then which of the following statements is/are correct ? I. (A+B)(A + B) is divisible by 9 II. If AA is odd, then BB is odd Select the answer using the code given below :

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  29. 29
    If x3+1x3=658x^3 + \dfrac{1}{x^3} = \dfrac{65}{8} and y3+1y3=73027y^3 + \dfrac{1}{y^3} = \dfrac{730}{27}, then which one of the following is a value of xyxy ?

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  30. 30
    If 11x+5y11x + 5y is a prime number where xx, yy are natural numbers then what is the minimum value of (x+y)(x + y) ?

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  31. 31
    A 4-digit number NN has exactly 15 distinct divisors. What is the total number of distinct divisors of N2N^2 ?

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  32. 32
    If pp, qq and rr are the lengths (in cm) of the sides of a right-angled triangle, then (p−q−r)(q−r−p)(r−p−q)(p - q - r)(q - r - p)(r - p - q) is always

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  33. 33
    What is the minimum value of (a8+a4+1)(b8+b4+1)a4b4\dfrac{(a^8 + a^4 + 1)(b^8 + b^4 + 1)}{a^4 b^4}, where a>0a > 0, b>0b > 0 ?

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  34. 34
    In a class containing 200 students, nn students prefer both tea and coffee; 2n2n students prefer coffee, 3n3n students prefer tea; 4n4n students prefer neither tea nor coffee. What is the value of nn ?

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  35. 35
    Let ABCABC be a triangle with area 36 square cm. If AB=9AB = 9 cm, BC=12BC = 12 cm and ∠ABC=θ\angle ABC = \theta, then what is cos⁡θ\cos\theta equal to ?

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  36. 36
    Let nn be a natural number. The HCF of nn, n+10n + 10 is 10. If the LCM is xx (a 2-digit number), then how many values of xx are possible ?

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  37. 37
    What is HCF of a4+2a3+3a2+2a+1a^4 + 2a^3 + 3a^2 + 2a + 1 and a6−2a3+1a^6 - 2a^3 + 1 ?

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  38. 38
    If the roots of the equation x2−(k−2)x+(k+1)=0x^2 - (k - 2)x + (k + 1) = 0 are equal, then what are the values of kk ?

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  39. 39
    What is (cos⁡θ−sin⁡θ+1cos⁡θ+sin⁡θ−1)(cot⁡θ−cosec θ)\left(\dfrac{\cos\theta - \sin\theta + 1}{\cos\theta + \sin\theta - 1}\right)(\cot\theta - \mathrm{cosec}\,\theta) equal to ?

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  40. 40
    What is sin⁡θ−2sin⁡3θ2cos⁡3θ−cos⁡θ\dfrac{\sin\theta - 2\sin^3\theta}{2\cos^3\theta - \cos\theta} equal to ?

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  41. 41
    The mean weight of 150 students in a class is 60 kg. The mean weight of boys in the class is 70 kg and that of girls is 55 kg. What is the ratio of number of boys to number of girls ?

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  42. 42
    Two towers AA and BB of height 23 m and 11 m respectively, stand 9 m apart. A straight rod is joined to the two tops of the towers. A monkey sitting on the top of AA, climbs the rod to reach the top of BB. If the monkey takes 5 minutes to reach the other end, what is the average speed of the monkey ?

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  43. 43
    A spherical wooden ball of radius rr is to be divided into eight identical parts by cutting by planes passing through the same diameter. What is the surface area of each final piece ?

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  44. 44
    A trolley with two wheels one metre apart is moved clockwise on the circular track around a ground with radius 50 m (described by right wheel). If the size of each wheel is of 1 foot radius and the right wheel turns 1000 times, how many times will the other wheel turn ?

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  45. 45
    What is the remainder when 70×71×72×73×74×75×76×77×78×7970 \times 71 \times 72 \times 73 \times 74 \times 75 \times 76 \times 77 \times 78 \times 79 is divided by 1000 ?

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  46. 46
    A vertical pole of length 80 m is situated on the horizontal plane. The base of the pole is at PP. There are two points AA and BB such that PP, AA, BB are on the same straight line. Let the angles of elevation of top of the pole from AA and BB be α\alpha and β\beta (α>β\alpha > \beta) respectively. If PA=64PA = 64 m and AB=36AB = 36 m, then what is (α+β)(\alpha + \beta) equal to ?

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  47. 47
    Let kk be a positive integer. What is the quotient when x8k+3+x8k+6+x8k+9+x8k+12x^{8k+3} + x^{8k+6} + x^{8k+9} + x^{8k+12} is divided by (1+x3)(1+x6)(1 + x^3)(1 + x^6) ?

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  48. 48
    A square is drawn inside a square of side 14 cm in such a way that the corners of the inner square coincide with the mid points of the sides of the outer square. What is the area lying between the two squares ?

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  49. 49
    The base of a right-angled triangle is 43\dfrac{4}{3} times the height of triangle. If the area of the triangle is 54 square cm, then what is the perimeter of the triangle ?

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  50. 50
    What is the area of a triangle having sides 4, 4 and 6 units ?

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  51. Consider the following for the next three (03) items that follow : Let ABCABC be a triangle right-angled at BB. Let PP be the point on BCBC such that BP=PCBP = PC. If AB=10AB = 10 cm, ∠BAP=45∘\angle BAP = 45^\circ and ∠CAP=θ\angle CAP = \theta (use tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha + \beta) = \dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha \tan\beta})
    51
    What is tan⁡θ\tan\theta equal to ?

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  52. 52
    If ∠ACP=γ\angle ACP = \gamma, then what is tan⁡γ\tan\gamma equal to ?

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  53. 53
    Consider the following statements : I. The line segment APAP divides the area of the triangle ABCABC into two equal parts II. The perimeter of the triangle APCAPC is more than 46 cm III. The area of the triangle APCAPC is 50 square cm Which of the statements given above are correct ?

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  54. A frequency distribution is as follows :
    Marks18-2627-3536-4445-5354-6263-7172-80
    Number of students571015832
    54
    What is the median of the distribution ?

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  55. 55
    What is the mode of the distribution ?

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  56. ABCABC is a triangle right-angled at BB. Given that AC−AB=2AC - AB = 2 cm and BC=16BC = 16 cm
    56
    If ∠BAC=θ\angle BAC = \theta then what is sin⁡θ+cos⁡θ\sin\theta + \cos\theta equal to ?

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  57. 57
    If BDBD is the perpendicular on the side ACAC, then what is the length of BDBD ?

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  58. Let MNMN be a chord of length 16 cm of a circle with centre at OO and radius 10 cm. The tangents at MM and NN intersect at a point PP. Further, OPOP intersects MNMN perpendicularly at QQ.
    58
    What is OQOQ equal to ?

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  59. 59
    What is PMPM equal to ?

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  60. 60
    What is the area of triangle OMNOMN ?

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  61. 61
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : What is the integral value of kk for which the expression 4x2−kx+14x^2 - kx + 1 is positive ? Statement-I : k<−2k < -2 Statement-II : k>−4k > -4 Which one of the following is correct in respect of the above Question and the Statements ?

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  62. 62
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : In how many days can AA, BB and CC together finish the work ? Statement-I : AA and BB together can finish the work in 24 days Statement-II : BB and CC together can finish the work in 36 days Which one of the following is correct in respect of the above Question and the Statements ?

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  63. 63
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Can we have a common solution which is prime ? Statement-I : x2−26x+133=0x^2 - 26x + 133 = 0 Statement-II : x2−44x+475=0x^2 - 44x + 475 = 0 Which one of the following is correct in respect of the above Question and the Statements ?

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  64. 64
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Is 327n+173n327^n + 173^n divisible by 500 ? Statement-I : nn is odd natural number Statement-II : nn is a positive integer Which one of the following is correct in respect of the above Question and the Statements ?

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  65. 65
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : If the price of petrol goes up by 20%, by what percentage should the consumption be reduced so that the expenditure remains the same ? Statement-I : Price of petrol per litre was Rs. 90 Statement-II : Consumption was 24 litre before price hike Which one of the following is correct in respect of the above Question and the Statements ?

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  66. 66
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : The ratio of PP's salary to QQ's salary is 6:56 : 5. How much is PP's expenditure ? Statement-I : The ratio of PP's saving to QQ's saving is 3:23 : 2 Statement-II : The ratio of PP's expenditure to QQ's expenditure is 1:11 : 1 Which one of the following is correct in respect of the above Question and the Statements ?

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  67. 67
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : The largest of five different integers is 8 and least is 2. What is the average of these integers ? Statement-I : The sum of all the 5 integers is a multiple of 5 Statement-II : The number of odd integers is odd Which one of the following is correct in respect of the above Question and the Statements ?

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  68. 68
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : There are three different weights. All the weights are integers and their sum is a prime number. What are the weights ? Statement-I : One of the weights is twice the another weight Statement-II : One of the weights is thrice the another weight Which one of the following is correct in respect of the above Question and the Statements ?

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  69. 69
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : What is the amount at the end of 10 years ? Statement-I : The principal amount is Rs. 1,00,000 Statement-II : Rate of interest is 10% per annum Which one of the following is correct in respect of the above Question and the Statements ?

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  70. 70
    A question is given followed by two statements I and II. Consider the Question and the Statements and mark the correct option. Question : Is p2+pq+q2p^2 + pq + q^2 odd where pp, qq are integers ? Statement-I : p+qp + q is even Statement-II : pqpq is odd Which one of the following is correct in respect of the above Question and the Statements ?

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  71. 71
    The total population of an area is 10,000 out of which males and females are equal in number. Out of the total population 30% are Newspaper readers. Out of the total newspaper readers, one-third read English Newspaper. Out of the total English Newspaper readers, 20% are females. What is the number of males who do not read English Newspaper ?

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  72. 72
    What is the maximum area of a rectangle, in square cm, whose perimeter is 400 cm ?

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  73. 73
    What is the remainder if we divide 3103^{10} by 7 ?

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  74. 74
    What is the square root of 64% ?

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  75. 75
    The difference of 1031−510^{31} - 5 and 1030+p10^{30} + p is divisible by 3 where pp is a digit. How many values of pp are possible ?

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  76. 76
    Consider the following statements : I. 61 divides 107100−76100107^{100} - 76^{100} II. 100 divides 675+33567^5 + 33^5 Which of the statements given above is/are correct ?

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  77. 77
    The average of the temperatures recorded at noontime from Monday to Sunday is 31∘31^\circC. If the lowest temperature recorded is 30∘30^\circC, then what is the maximum of temperature that is possible to record at noontime on any one of the days

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  78. 78
    If (x+1yz)−(y+1zx)=(y+1zx)−(z+1xy)\left(x + \frac{1}{yz}\right) - \left(y + \frac{1}{zx}\right) = \left(y + \frac{1}{zx}\right) - \left(z + \frac{1}{xy}\right) and x+z≠2yx + z \ne 2y, then what is xyzxyz equal to ?

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  79. 79
    Consider the following statements in respect of p=n(n+1)(n+2)(n+3)+1p = n(n+1)(n+2)(n+3) + 1, where nn is a natural number : I. pp is always odd II. pp is a perfect square Which of the statements given above is/are correct ?

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  80. 80
    What is the difference between the average of first 50 even natural numbers and the average of first 50 odd natural numbers ?

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  81. 81
    Three amounts xx, yy, zz are such that yy is the compound interest on xx; and zz is the compound interest on yy. The rate of interest per annum and the time period in years are same. Which one of the following is correct ?

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  82. 82
    There are nn concentric squares. The area of the innermost square is 1 unit and the distance between corresponding corners of any two consecutive squares is 1 unit. Consider the following statements : I. The diagonal of the nnth square is 2n+2−22n + \sqrt{2} - 2 II. The area included between nnth square and (n−1)(n - 1)th square is independent of nn Which of the statements given above is/are correct ?

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  83. 83
    In a rectangle ABCDABCD, ACAC is one of the diagonals. If AC+AB=3ADAC + AB = 3AD and AC−AD=4AC - AD = 4 units, then what is the area of the triangle ?

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  84. 84
    The area of the circle circumscribing three identical circles touching each other is π(2+3)23\frac{\pi\left(2 + \sqrt{3}\right)^2}{3} square cm. What is the radius of one of the smaller circles ?

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  85. 85
    In a triangle ABCABC, AB=21AB = 21 cm, BC=20BC = 20 cm and CA=13CA = 13 cm. A perpendicular CDCD is drawn upon the longest side. What is the area of the triangle BCDBCD ?

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  86. 86
    There are two containers AA and BB. In container AA, the ratio of milk and water is 1:31 : 3 and in container BB, the ratio of milk and water is m:nm : n. If the mixture in the containers AA and BB are mixed in the ratio 2:32 : 3 to get 20 litres of a mixture having milk and water in the ratio 3:73 : 7, then what is the value of mn\frac{m}{n} ?

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  87. 87
    A cone, a hemisphere and a cylinder stand on equal base of radius rr and have the same height. If the sum of volumes of cone, the hemisphere and the cylinder is equal to volume of a sphere of radius RR, then what is R3r3\frac{R^3}{r^3} equal to ?

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  88. 88
    If x3+px2+qx+rx^3 + px^2 + qx + r is an integer for all integral values of xx, then consider the following statements : I. pp must be an integer II. qq must be an integer III. rr must be an integer Which of the statements given above is/are correct ?

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  89. 89
    XYZXYZ is a 3-digit number, where XX, YY, ZZ are distinct non-zero digits. The difference between the two 3-digit numbers XYZXYZ and YXZYXZ is 90. How many possible values exist for the sum (X+Y)(X + Y) ?

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  90. 90
    How many times does the minute hand of a clock coincide with the second hand between 2.01 pm and 4.01 pm on the same day ?

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  91. 91
    What is the HCF of 236−12^{36} - 1 and 245−12^{45} - 1 ?

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  92. 92
    The section of a solid right circular cone by a plane containing vertex and perpendicular to base is an equilateral triangle of side 14 cm. What is the volume of the cone ? (π=227)\left(\pi = \frac{22}{7}\right)

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  93. 93
    Three identical cones each with base radius 3 cm are placed on their bases so that each is touching the other two. There will be one and only circle that would pass through each of the vertices of the cones. What is the area of the circle ?

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  94. 94
    A circle is inscribed in a triangle ABCABC right-angled at BB. If AB=5AB = 5 cm and BC=12BC = 12 cm, then what is the radius of the circle ?

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  95. 95
    The ratio of sum of interior angles to sum of exterior angles of a regular polygon of nn sides is 72\frac{7}{2}. What is the measure of an interior angle of polygon ?

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  96. 96
    The number 199 can be written as m2−n2m^2 - n^2, where mm, nn are natural numbers (m>n)(m > n). What is the value of mnmn ?

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  97. 97
    How many numbers of the form 2n−12^n - 1 and less than 2000 are prime ?

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  98. 98
    In a class of 160 students, each of them opt at least one language from among English, Hindi and Sanskrit. It is found that 130 students opt English, 120 students Hindi and 110 Sanskrit. If the students opt either only one language or all three languages, then what is the number of students who study all three languages ?

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  99. 99
    Let S=5a+7b+11c+13dS = 5^a + 7^b + 11^c + 13^d, where aa, bb, cc and dd are natural numbers. What is the number of distinct remainders of SS when it is divided by 10 ?

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  100. 100
    In a right triangle ABCABC, ∠A=90∘\angle A = 90^\circ and ADAD is perpendicular to BCBC. If ∠CAD=60∘\angle CAD = 60^\circ and BC=6BC = 6 cm, then what is ABAB equal to ?

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