CDS (II) 2016, Elementary Mathematics question paper

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

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1 $\log_{10}\left[1 - \left\{1 - \left(1 - x^2\right)^{-1}\right\}^{-1}\right]^{-\frac{1}{2}} = 1$
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2 $\lambda$
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3 $\frac{6^2 + 7^2 + 8^2 + 9^2 + 10^2}{\sqrt{7 + 4\sqrt{3}} - \sqrt{4 + 2\sqrt{3}}}$
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4 $x^2 = y + z$
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5 $Q$
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6 $A$
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7 $In an examination, a student was asked to divide a certain number by 8. By mistake he multiplied it by 8 and got the answer 2016 more than the correct answer. What was the number ?$
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8 $x^2 + ax + b = 0$
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9 $The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller number, then what is the difference of the larger and smaller numbers ?$
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10 $A = \{7, 8, 9, 10, 11, 12\}$
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11 $A boy saves Rs. 4.65 daily. What is the least number of days in which he will be able to save an exact number of rupees ?$
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12 $A$
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13 $Which one of the following rational numbers has non-terminating and repeating decimal expansion ?$
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14 $\alpha$
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15 $2^{100}$
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16 $In an office, one-third of the workers are women, half of the women are married and one-third of the married women have children. If three-fourth of the men are married and one-third of the married men have children, then what is the ratio of married women to married men ?$
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17 $A$
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18 $When the speed of a train is increased by 20%, it takes 20 minutes less to cover the same distance. What is the time taken to cover the same distance with the original speed ?$
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19 $x\%$
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20 $p$
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21 $The cost of 2.5 kg rice is Rs. 125. The cost of 9 kg rice is equal to that of 4 kg pulses. The cost of 14 kg pulses is equal to that of 1.5 kg tea. The cost of 2 kg tea is equal to that of 5 kg nuts. What is the cost of 11 kg nuts ?$
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22 $Which one of the following is correct in respect of the number 1729 ?$
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23 $A shopkeeper increases the cost price of an item by 20% and offers a discount of 10% on this marked price. What is his percentage gain ?$
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24 $p$
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25 $\sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \ldots + \sqrt{1 + \frac{1}{2007^2} + \frac{1}{2008^2}}$
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26 $A person can row downstream 20 km in 2 hours and upstream 4 km in 2 hours. What is the speed of the current ?$
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27 $x$
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28 $2^{x+2} \cdot 27^{\frac{x}{x-1}} = 9$
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29 $A, B, C, D, E, F, G, H, I, J, K$
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30 $2^{122} + 4^{62} + 8^{42} + 4^{64} + 2^{130}$
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31 $2p + 3q = 12$
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32 $3^{30}$
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33 $7^{139}$
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34 $4x + 3a = 0$
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35 $Incomes of Mahesh and Kamal are in the ratio 1 : 2 and their expenses are in the ratio 1 : 3. Which one of the following statements is correct ?$
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36 $X$
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37 $X$
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38 $A train is travelling at 48 km/hour completely crosses another train having half its length and travelling in opposite direction at 42 km/hour in 12 s. It also passes a railway platform in 45 s. What is the length of the platform ?$
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39 $The speeds of three cars are in the ratio 2 : 3 : 4. What is the ratio between the times taken by these cars to travel the same distance ?$
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40 $If the mean age of combined group of boys and girls is 18 years and the mean of age of boys is 20 and that of girls is 16, then what is the percentage of boys in the group ?$
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41 $120\ \text{m}^2$
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42 $x$
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43 $616\ \text{cm}^2$
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44 $A cubic metre of copper weighing 9000 kg is rolled into a square bar 9 m long. An exact cube is cut off from the bar. How much does the cube weigh ?$
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45 $1.96\ \text{m}^3$
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46 $ABC$
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47 $130\pi\ \text{cm}^2$
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48 $AB$
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49 $\frac{5}{11}$
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50 $67\frac{1}{21}\ \text{m}^3$
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51 $1617\ \text{cm}^3$
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52 $The cost of painting a spherical vessel of diameter 14 cm is Rs. 8008. What is the cost of painting per square centimetre ?$
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53 $If a quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals$
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54 $A circle and a square have the same perimeter. Which one of the following is correct ?$
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55 $\pi x\ \text{cm}^3$
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56 $In an equilateral triangle another equilateral triangle is drawn inside joining the mid-points of the sides of given equilateral triangle and the process is continued up to 7 times. What is the ratio of area of fourth triangle to that of seventh triangle ?$
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57 $22\ \text{m} \times 20\ \text{m}$
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58 $\frac{1}{64}$
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59 $60^\circ$
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60 $What is the volume of a sphere of radius 3 cm ?$
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61 $\sqrt{a^2 + b^2}$
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62 $r$
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63 $2x$
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For the next three (03) items that follow : A tent of a circus is made of canvas and is in the form of right circular cylinder and right circular cone above it. The height and diameter of the cylindrical part of the tent are 5 m and 126 m respectively. The total height of the tent is 21 m.
64 $What is the slant height of the cone ?$
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65 $What is the curved surface area of the cylinder ?$
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66 $How many square metres of canvas are used ?$
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67 $The wheels of a car are of diameter 80 cm each. The car is travelling at a speed of 66 km/hour. What is the number of complete revolutions each wheel makes in 10 minutes ?$
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68 $What is the area of a triangle with sides of length 12 cm, 13 cm and 5 cm ?$
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69 $If the perimeter of a circle is equal to that of a square, then what is the ratio of area of circle to that of square ?$
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70 $ABCDA$
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71 $\sin\theta + \cos\theta = \frac{\sqrt{7}}{2}$
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72 $\sin x + \sin^2 x = 1$
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73 $30^\circ$
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74 $30^\circ$
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75 $\operatorname{cosec}^2 68^\circ + \sec^2 56^\circ - \cot^2 34^\circ - \tan^2 22^\circ$
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76 $2y\cos\theta = x\sin\theta$
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77 $\sin\theta + \cos\theta = \frac{1 + \sqrt{3}}{2}$
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78 $A = \sin^2\theta + \cos^4\theta$
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79 $\frac{\cot A + \operatorname{cosec} A - 1}{\cot A - \operatorname{cosec} A + 1}$
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80 $\sin 1^\circ > \sin 1^c$
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81 $\tan^2 x + \frac{1}{\tan^2 x} = 2$
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82 $If the length of the shadow of a tower is equal to its height, then what is the Sun's altitude at that time ?$
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83 $\frac{\cos 75^\circ}{\sin 15^\circ} + \frac{\sin 12^\circ}{\cos 78^\circ} - \frac{\cos 18^\circ}{\sin 72^\circ} = 1$
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84 $ABC$
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85 $\tan 1^\circ \tan 2^\circ \tan 3^\circ \tan 4^\circ \ldots \tan 89^\circ$
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86 $X$
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87 $x > 0$
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88 $20.5$
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89 $A cricketer has a certain average of 10 innings. In the eleventh inning he scored 108 runs, thereby increasing his average by 6 runs. What is his new average ?$
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90 $The mean of 20 observations is 17. On checking it was found that the two observations were wrongly copied as 3 and 6. If wrong observations are replaced by correct values 8 and 9, then what is the correct mean ?$
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91 $Number of credit cards held by an individual can be treated as$
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92 $In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls ?$
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93 $108^\circ$
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94 $Consider the following statements : 1. The classes of type 15-19, 20-24, 25-29 etc. are exclusive classes. 2. The classes of type 15-20, 20-25, 25-30 etc. are inclusive classes. Which of the above statements is/are correct ?$
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95 $Suppose the class interval 10-15 has frequency 30, then what is the frequency density of this class interval ?$
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96 $S$
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97 $13^5 + 14^5 + 15^5 + 16^5$
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98 $What is the difference between the sum of the cubes and that of squares of first ten natural numbers ?$
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99 $A motorist travels to a place 150 km away at an average speed of 50 km/hour and returns at 30 km/hour. What is the average speed for the whole journey ?$
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100 $x + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{3}}} = 2$
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