CDS (I) 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 $n$
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2 $41^{43} + 43^{43}$
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3 $x = 7 + 4\sqrt{3}$
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4 $4^{61} + 4^{62} + 4^{63} + 4^{64}$
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5 $p$
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6 $\frac{(5.4)^3 - 0.064}{(5.4)^2 + 2.16 + 0.16}$
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7 $A man walks at an average speed of 3 km/hr from his residence and reaches office 40 minutes early. If he walks at an average speed of 2 km/hr, he reaches 40 minutes late. What is the distance between his residence and office?$
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8 $\frac{1}{5\sqrt{4} + 4\sqrt{5}} + \frac{1}{6\sqrt{5} + 5\sqrt{6}} + \frac{1}{7\sqrt{6} + 6\sqrt{7}} + \frac{1}{8\sqrt{7} + 7\sqrt{8}} + \frac{1}{9\sqrt{8} + 8\sqrt{9}}$
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9 $x = 9999$
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10 $(x + \sqrt{1 + x^2})(y + \sqrt{1 + y^2}) = 1$
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11 $\frac{2\sin 68^\circ}{\cos 22^\circ} - \frac{2\cot 15^\circ}{5\tan 75^\circ} - \frac{3\tan 20^\circ \tan 40^\circ \tan 45^\circ \tan 50^\circ \tan 70^\circ}{5}$
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12 $The perpendicular dropped from a vertex of a right-angled triangle upon the hypotenuse divides it into two segments of lengths 9 units and 16 units respectively. What is the length of the perpendicular?$
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13 $43^x \times 47^y = (2021)^2$
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14 $a$
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15 $x^2 = 17x + y$
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16 $n$
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17 $\frac{x - y}{x\sqrt{y} + y\sqrt{x}} = \frac{1}{\sqrt{x}}$
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18 $What is the area of the region enclosed by three identical circles (each of radius 4 cm) touching each other?$
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19 $A$
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20 $What is the smallest natural number from the following which must be subtracted from 9410 to make the remaining number a perfect square?$
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21 $27^5 + 3^{13}$
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22 $p = 2^{2n+2} + m$
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23 $x$
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24 $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{2020} + \sqrt{2021}}$
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25 $x + \frac{1}{x} = \frac{5}{2}$
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26 $a + b = 2,\quad \frac{1}{a} + \frac{1}{b} = 2$
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27 $8 men or 12 women can do a piece of work in 24 days. In how many days can the work be done by 8 men and 12 women?$
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28 $p$
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29 $x$
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30 $p$
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31 $\alpha$
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32 $A shopkeeper marks the price of an article at Rs. 200. After allowing a discount of 10%, he still gains 20% on the cost price. What is the cost price of the article?$
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33 $n$
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34 $> 1$
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35 $What is the number of divisors of 1000 (excluding 1 and 1000)?$
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36 $x^2 - k^2 x + 30kx - 161x - 64 = 0$
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37 $A piece of cloth costs Rs. 10,000. If a 2 m longer piece of the same cloth is purchased for the same amount, it would cost Rs. 250 less per metre. What is the original length of the piece of cloth?$
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38 $ax^2 + bx + c = 0$
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39 $6x^2$
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40 $X$
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41 $3x + 2y$
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42 $ax^2 + bx + c = 0$
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43 $a^2 + b^2 + c^2 - ab - bc - ca$
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44 $\frac{x^6 + y^6}{x^4 + y^4}$
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45 $How many quadratic equations have the sum of their roots equal to the product of their roots?$
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46 $x = \frac{1 - \cos\theta + \sin\theta}{1 + \sin\theta}$
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47 $\cos(x + y) = 0$
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48 $\sin^4\theta + \cos^4\theta - 2\sin^2\theta\cos^2\theta$
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49 $\cos\theta + \sec\theta - 2 = 0$
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50 $y = \cos^2 x + \sec^2 x$
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51 $A$
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52 $A$
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53 $\theta$
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54 $x$
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55 $(\sec^2\alpha + \tan\alpha \cdot \tan\beta - \tan^2\alpha)^2 + (\tan\alpha - \tan\beta)^2 - \sec^2\alpha \cdot \sec^2\beta$
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56 $\tan\theta + \sec\theta = 3$
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57 $\sqrt{\sec^2\theta + \operatorname{cosec}^2\theta} = \tan\theta + \cot\theta$
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58 $2\theta$
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59 $\tan^2\theta + 3\sec\theta - 9 = 0$
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60 $\dfrac{\cos\theta}{\operatorname{cosec}\theta + 1} + \dfrac{\cos\theta}{\operatorname{cosec}\theta - 1} = 2$
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61 $AB$
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62 $A circle of radius 25 cm has a chord of length 48 cm. What is the length of the perpendicular drawn from the centre of the circle to the chord?$
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63 $x$
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64 $3 : 7$
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65 $\pi = \dfrac{22}{7}$
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66 $x$
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67 $x$
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68 $n$
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69 $x$
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70 $A tall cylindrical container with circular base of radius 18 cm contains a good quantity of water. Metal balls each of radius 0.9 cm are immersed in it. How many balls are required to raise the water level by 3 cm?$
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71 $9\sqrt{3}$
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72 $A farmland is in the shape of a rhombus. The perimeter of the land is 100 m and the length of one of the diagonals is 40 m. The land is divided into four equal parts. What is the area of each part?$
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73 $ABCD$
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74 $ABCD$
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75 $4(2 + \sqrt{2})$
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76 $12\sqrt{2}$
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77 $Consider the following statements : 1. The sum of any two sides of a triangle is less than twice the median drawn to the third side. 2. The perimeter of a triangle is greater than the sum of the three medians. Which of the above statements is/are correct?$
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78 $D$
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79 $ABC$
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80 $ABC$
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81 $ABC$
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82 $x$
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83 $ABC$
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84 $Water is trickling out of a completely filled cylindrical tank of height 1 m and diameter 2 m. Every second a spherical droplet of 1 cm radius trickles down from the bottom of the tank. The tank will be emptied in approximately$
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85 $27 : 8 : 1$
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86 $ABC$
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87 $ABC$
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88 $What is the radius of the circle inscribed in a triangle whose sides are 4 cm, 7.5 cm and 8.5 cm?$
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89 $In a shower, 5 cm of rain falls. What is the volume of water that falls on 2 hectare area of land?$
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90 $n$
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Direction : Consider the following data for the two (02) items that follow : The table below gives the age-wise population percentage of a city :
Age group Percentage
Below 30 years 14.00
30–34.99 29.75
35–39.99 26.25
40–44.99 0
45–49.99 18.50
50 years and above 11.50
The number of persons below the age of 40 years is 10.5 lakhs.
91 $What is the total population of the city (in lakhs)?$
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92 $1 : 2$
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Direction : Consider the following data for the two (02) items that follow : The expenditure (in lakhs of rupees) of a company for the years 2011 to 2017 is as under :
Year Expenditure
2011 13.8
2012 15.4
2013 10.4
2014 13.1
2015 15.8
2016 17.2
2017 19.4
93 $How many times the increase in expenditure in a year exceeded by more than 15% as compared to previous year?$
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94 $In which year, the percentage increase in expenditure is maximum as compared to its previous year?$
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Direction : Consider the following for the two (02) items that follow : The budget allocations represented in a pie diagram under five different heads $A$ , $B$ , $C$ , $D$ and $E$ are respectively 40%, 18%, 9%, 25% and 8%. The total budget allocation is Rs. 300.4 lakhs.
95 $A$
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96 $A$
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Direction : Consider the following for the four (04) items that follow : 500 candidates appeared in an examination comprising tests in English, Hindi and Mathematics. 30 candidates failed in English only; 75 failed in Hindi only; 50 failed in Mathematics only; 15 failed in both English and Hindi; 17 failed in both Hindi and Mathematics; 17 failed in both Mathematics and English; 5 failed in all three tests.
97 $What is the percentage of candidates who failed in at least two subjects?$
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98 $What is the percentage of candidates who failed in only one subject?$
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99 $What is the percentage of candidates who failed in at least one subject?$
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100 $How many candidates passed in two or more subjects?$
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Age group	Percentage
Below 30 years	14.00
30–34.99	29.75
35–39.99	26.25
40–44.99	0
45–49.99	18.50
50 years and above	11.50

Year	Expenditure
2011	13.8
2012	15.4
2013	10.4
2014	13.1
2015	15.8
2016	17.2
2017	19.4

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