CDS (II) 2022, Elementary Mathematics question paper

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

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1 $x^2 - 20 = \sqrt{20 + \sqrt{20 + \sqrt{20 + \sqrt{20 + \ldots}}}}$
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2 $\frac{a + b}{b + c} = \frac{c + d}{d + a}$
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3 $(a^3 + b^3)$
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4 $\frac{\sqrt{x + 20} + \sqrt{x - 1}}{\sqrt{x + 20} - \sqrt{x - 1}} = \frac{7}{3}$
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5 $(x^8 - y^8)$
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6 $x^2 - 14x + k = 0$
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7 $(x + k)$
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8 $If three times the greater of two numbers is divided by the smaller number, the quotient will be 6 and the remainder will be 6. If five times the smaller number is divided by the greater number, the quotient will be 2 and the remainder will be 3. What is the difference between the numbers ?$
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9 $(ab + bc + ca)$
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10 $\frac{1}{x(x - y)(x - z)} + \frac{1}{y(y - z)(y - x)} + \frac{1}{z(z - x)(z - y)}$
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11 $x^4(y - z) + y^4(z - x) + z^4(x - y)$
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12 $\frac{x^6 - 24x^4 + 144x^2}{(x^2 + 4\sqrt{3}x + 12)(x - 2\sqrt{3})^2}$
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13 $X and Y can do a piece of work in 45 days and 40 days respectively. They begin to work together, but X leaves after n days and then Y completes the remaining work in 23 days. What is n equal to ?$
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14 $480 persons working 10 hours per day complete one-fourth of a work in 10 days. How many additional persons are to be employed in order to complete the remaining work in 20 days, working 8 hours per day ?$
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15 $12 women and 16 men can do a piece of work in 5 days. 13 women and 24 men can do it in 4 days. How long will 25 women and 50 men take to do it ?$
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16 $There are two stations X and Y, 1320 km apart. A train starts from station X at 6 a.m. and moves at an average speed of 60 km/hr. At 2 p.m. another train starts from Y towards X and moves at an average speed of 80 km/hr. When do they meet ?$
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17 $Two glasses of equal volume are filled with a mixture of alcohol and water in the ratio 3 : 2 and 4 : 1, respectively. These glasses are emptied into a third glass. What is the ratio of alcohol and water in the third glass ?$
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18 $\frac{3}{5}, \frac{6}{25}, \frac{9}{20}, \frac{27}{50}$
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19 $x > y$
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20 $The value of a 2-digit number is 5 times the sum of the digits. What is the product of the digits ?$
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21 $N = 12^6 \times 3^8 \times 5^3$
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22 $\frac{\log_{10} a}{b - c} = \frac{\log_{10} b}{c - a} = \frac{\log_{10} c}{a - b}$
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23 $What is the angle between the hour hand and the minute hand of a clock when the clock shows 4 hours 40 minutes ?$
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24 $\log_{10}\left[995 + \sqrt{x^2 - 12x + 60}\right] = 3$
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25 $qx^2 - 2px + q = 0$
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26 $p(x) = x^4 + x^2 + 1$
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27 $Consider the question and two statements given below : A 2-digit number is added to the number formed by reversing the digits of the 2-digit number. Question : What is the 2-digit number ? Statement-1 : The sum is divisible by 9. Statement-2 : The sum is divisible by 2. Which one of the following is correct in respect of the question and the statements ?$
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28 $x > y$
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29 $(x^n + y^n)$
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30 $xy > 0$
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31 $a = 3b$
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32 $(k - 3)x^2 - kx - 1$
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33 $bc + cd = 2bd$
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34 $In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 100 km/hr and time of flight increased by 30 minutes. What was the original average speed of the aircraft ?$
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35 $x = b + c$
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36 $A person X starts from a place A and another person Y starts simultaneously from another place B which is d km away from A. They walk in the same direction. X walks at an average speed of u km/hr and Y walks at an average speed of v km/hr. How far will X have walked before he overtakes Y ?$
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37 $p^4 + 4p^2$
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38 $x = 2 + 2^{1/2}$
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39 $X takes 3 hours longer than Y to walk 30 km. If X doubles his speed, he takes 2 hours less than Y. What is the speed of Y ?$
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40 $If C is the compound interest on Rs. 10,000 for one year at 4% per annum when compounded quarterly, then which one of the following is correct ?$
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Consider the following for the next two (02) items that follow : Consider the equation $6x^2 - 25x + \dfrac{6}{x^2} + \dfrac{25}{x} + 12 = 0$ .
41 $x - \dfrac{1}{x}$
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42 $x^2 + \dfrac{1}{x^2}$
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Consider the following for the next three (03) items that follow : A triangle ABC with sides AB = 15 cm, BC = 9 cm, CA = 12 cm is inscribed in a circle.
43 $\cos^2 A + \cos^2 B + \cos^2 C$
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44 $\sin^2 A + \sin^2 B + \sin^2 C$
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45 $What is the radius of the circle ?$
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Consider the following for the next two (02) items that follow : The angle of elevation of a cloud at C from a point (P), H metres above the surface of a lake is $30^\circ$ . The height of the cloud from the surface of the lake is 2H metres. Let $\theta$ be the angle of depression of the reflection of the cloud in the lake from the point P.
46 $\theta$
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47 $What is PC equal to ?$
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Consider the following for the next three (03) items that follow : A frequency distribution table is given below :
x 0 1 2 3 4 5
f 46 p q 25 10 5
Total frequency is 200 and mean of the distribution is 1.46.
48 $What is the value of p ?$
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49 $What is the value of q ?$
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50 $What is the median of the distribution ?$
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51 $\tan\theta = \dfrac{\sqrt{q^2 - p^2}}{p}$
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52 $11\sin\theta + 60\cos\theta = 61$
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53 $What is the diameter of a circle inscribed in a regular polygon of 15 sides with side length unity ?$
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54 $x = \frac{1 + \sin\theta}{\cos\theta}$
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55 $\frac{\tan\theta + \sin\theta}{\tan\theta - \sin\theta} = \frac{\sec\theta + 1}{\sec\theta - 1}$
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56 $\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 5$
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57 $\frac{1}{\operatorname{cosec}\theta - \cot\theta} - \frac{1}{\sin\theta} = x$
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58 $2\sin^6\theta + 2\cos^6\theta - 3\sin^4\theta - 3\cos^4\theta$
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59 $3600\sec^2\theta + 121\operatorname{cosec}^2\theta$
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60 $\sqrt{\left(\frac{1 + \sin\theta}{\cos\theta}\right)^2 + \left(\frac{\cos\theta}{1 + \sin\theta}\right)^2 - 2}$
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61 $x$
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62 $What is the median of all possible factors of 120 ?$
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63 $n$
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64 $Consider the following table in respect of students of 4 schools who appeared in a test : School Number of students Average marks in the test I 6060 II 5080 III 5040 IV x 50 If the average marks of the students of all four schools are 58, then how many students appeared from School-IV ?$
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65 $4,\ 7,\ 10,\ 14,\ 2x + 3,\ 2x + 5,\ 22,\ 23,\ 30,\ 50$
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66 $If M is the mean of the first 100 even natural numbers, then what is the mean of the first 100 odd natural numbers ?$
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67 $Mean marks of 50 students were found to be 78.4. But later it was detected that 95 was misread as 59 and 25 was misread as 52. What is the difference between correct mean and incorrect mean ?$
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68 $\angle A = 2\theta$
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69 $\angle EDF$
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70 $\frac{21}{16}$
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Consider the following for the next three (03) items that follow : A conical vessel of radius 12 cm and height 16 cm is filled with water. A sphere is lowered into water and its size is such that it touches the sides of the vessel and it is just immersed.
71 $What is the radius of the sphere ?$
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72 $How much water will remain in the vessel after the overflow ?$
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73 $What is the ratio of lateral surface area of the vessel to the surface area of the sphere ?$
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Consider the following for the next two (02) items that follow : A chord of length $l$ of a circle makes an angle $90^\circ$ at the centre of the circle.
74 $What is the area of the minor segment ?$
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75 $What is the area of the major segment ?$
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76 $Triangle ABC is right-angled at A and AD is perpendicular to BC. If BD = 7.5 cm and DC = 10 cm, then what is AD equal to ?$
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77 $The perpendicular AD on the base BC of a triangle ABC intersects BC at D so that DB = 3 CD. Which one of the following is correct ?$
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78 $AC = \sqrt{3}\,BC$
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79 $\angle A = 60^\circ$
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80 $The diameter of a sphere made of copper is 3 cm. The sphere is melted and recast into a wire. If the length of the wire is half metre, then what is the diameter of the wire ?$
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81 $\frac{1}{27}$
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82 $A rectangular metal sheet is of length 24 cm and breadth 18 cm. From each of its corners a square of side x cm is cut off and an open box is made of the remaining sheet. If the volume of the box is 640 cubic cm, then what is the value of x ?$
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83 $x^{-1} + y^{-1} + z^{-1}$
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84 $Two parallel sides of a trapezium are 29 cm and 21 cm. Non-parallel sides are equal and each is of length 8.5 cm. What is the area of the trapezium ?$
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85 $(\pi = \frac{22}{7})$
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86 $The area of a rhombus is 96 square cm and one of its diagonals is of length 12 cm. What is the perimeter of the rhombus ?$
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87 $A right-angled triangle ABC is inscribed in a circle of radius 10 cm. The altitude drawn to the hypotenuse AC is of length 8 cm. If AB = x cm and BC = y cm, then what is the value of xy ?$
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88 $89\pi$
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89 $A square and a rectangle have same perimeter. They differ in areas by 1 square cm. The length of the rectangle exceeds its breadth by$
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90 $Two rectangles are of same area equal to 480 square cm. They differ in lengths by 6 cm and breadths by 4 cm. What is the difference in their perimeters ?$
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91 $A closed box is built of wood of uniform thickness. Its external dimensions are 12 cm, 10 cm and 8 cm. If the inner surface area is 376 square cm, then what is the thickness of the wood ?$
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92 $X, Y and Z are three equilateral triangles. The sum of the areas of X and Y is equal to the area of Z. If the side lengths of X and Y are 6 cm and 8 cm respectively, then what is the side length of Z ?$
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93 $(\pi = \frac{22}{7})$
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94 $2\sqrt{3}$
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95 $A triangle has side lengths x cm, x + 13 cm and x + 26 cm. If its area is 126 square cm, then what is the value of x ?$
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96 $ABCD is a parallelogram. A circle through A, B and C intersects CD (produced) at E. Which of the following is/are correct ? 1. AE = AD 2. CD = DE Select the correct answer using the code given below :$
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97 $Consider the following statements : 1. In an equilateral triangle, the centroid and centre of circumcircle coincide. 2. Angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral. 3. Every cyclic parallelogram is a rectangle. Which of the statements given above are correct ?$
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98 $\angle A + \angle C = 180^\circ$
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99 $\triangle EBC$
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100 $\frac{BD}{AB}$
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x	0	1	2	3	4	5
f	46	p	q	25	10	5

School	Number of students	Average marks in the test
I	60	60
II	50	80
III	50	40
IV	x	50

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