CDS (I) 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 $2^{35}-1$
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2 $29 \times 28 \times 27 \times \ldots 2 \times 1$
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3 $65^{99}$
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4 $x^2 - bx + c = 5$
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5 $In a party of 150 persons, 75 persons take tea, 60 persons take coffee and 50 persons take milk. 15 of them take both tea and coffee, but no one taking milk takes tea. If each person in the party takes at least one drink, then what is the number of persons taking milk only?$
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6 $A, B, C, D and E enter into a business. They invest money in the ratio 2 : 3 : 4 : 5 : 6. However, the time invested by them is in the ratio 6 : 5 : 4 : 3 : 2. If the profit distributed is directly proportional to time and money invested, then who receives the highest amount of profit?$
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7 $Consider the following numbers : 1. 437 2. 797 3. 1073 How many of the above numbers are prime?$
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8 $A can do a certain work at twice the speed of B. Further, B can do the same work at 1.5 times the speed of C. All of them together can finish the work in 12 days. In how many days can C alone finish the work?$
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9 $The sum of digits of a 2-digit number is 12. When the digits are reversed, the number becomes greater by eighteen. What is the difference between the digits in the number?$
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10 $The time taken by a train to cross a man travelling in another train is 10 seconds, when the other train is travelling in the opposite direction. However, it takes 20 seconds, if both the trains are travelling in the same direction. The length of the first train is 200 m and that of the second train is 150 m. What is the speed of the first train?$
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11 $2a = 3b = 6c = 9d = 12e = 18f$
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12 $a + b + c = 0$
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13 $Twelve percent of bananas bought by a fruit vendor got lost during transportation. On selling the remaining bananas, the vendor's overall profit turned out to be 4%. If the vendor had not lost any bananas and had sold them at the price of the remaining bananas, what would have been his profit percentage?$
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14 $(5 + 3\sqrt{2})(5 - 3\sqrt{2})$
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15 $When every even power of every odd integer (greater than 1) is divided by 8, what is the remainder?$
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16 $\frac{n(n^2 + 2)}{3}$
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17 $n - 1$
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18 $What is the largest 5-digit number, which leaves remainder 7, when divided by 18 as well as by 11?$
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19 $In a business dealing, A owes B Rs. 20,000 payable after 5 years, whereas B owes A Rs. 12,000 payable after 4 years. They want to settle it now at the rate of 5% simple interest. Who gives how much money in this settlement?$
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20 $Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74?$
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21 $\alpha$
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22 $Consider the following statements in respect of all factors of 360 : 1. The number of factors is 24. 2. The sum of all factors is 1170. Which of the above statements is/are correct?$
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23 $Consider a 6-digit number of the form XYXYXY. The number is divisible by :$
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24 $3^{29} - 9$
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25 $x = \sqrt{4\sqrt{4\sqrt{4\sqrt{4 \ldots}}}}$
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26 $(m + n)$
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27 $The product of two numbers is 2160 and their HCF is 12. If the sum of the squares of the two numbers is 4896, then what is the mean of the two numbers?$
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28 $The age of Q exceeds the age of P by 3 years. The age of R is twice the age of P and the age of Q is twice the age of S. Further, the age difference of R and S is 30 years. What is the sum of the ages of P and Q?$
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29 $\sqrt{a} + \sqrt{b} - \sqrt{c}$
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30 $There are four bells which ring at an interval of 15 minutes, 25 minutes, 35 minutes and 45 minutes respectively. If all of them ring at 9 A.M., how many more times will they ring together in the next 72 hours?$
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31 $a + b + c + d = 200$
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32 $97^{30} - 14^{30}$
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33 $\log_{10} 50$
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34 $If 17 women and 24 men can do a piece of work in 5 days and 12 women and 23 men can do it in 6 days, then which one of the following is correct ?$
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35 $Three taps A, B and C together can fill a tank in 6 hours. Tap C alone can fill the tank in 12 hours. To fill the tank, when it is empty, all the three taps are started together. After working t hours, tap C is closed and the tank is filled in 8 more hours. What is t equal to ?$
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36 $x$
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37 $n^3 - n$
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38 $S = 9^{27} + 27^9$
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39 $x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$
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40 $The speed of a boat in still water is 15 km/hr. If it can travel 42 km downstream and 28 km upstream in the same time, then what is the speed of the stream ?$
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41 $What is the difference between simple interest and compound interest on Rs. 10,000 for two years at 20% per annum compounded half-yearly ?$
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42 $a(b - c)(x - b)(x - c) + b(c - a)(x - c)(x - a) + c(a - b)(x - a)(x - b)$
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43 $1 - x - x^n + x^{n+1}$
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44 $m$
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45 $x = a + b + \frac{(a - b)^2}{4a + 4b}$
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46 $\cos^4\theta - \sin^4\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$
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47 $\sin\theta = \frac{12}{13}$
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48 $\tan^8\theta + \cot^8\theta = m$
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49 $6 - 4\sin\theta$
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50 $x$
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51 $\frac{\cos^2 32^\circ + \cos^2 58^\circ}{\sec^2 50^\circ - \cot^2 40^\circ} + 4\tan 13^\circ \tan 37^\circ \tan 53^\circ \tan 77^\circ$
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52 $(1 + \cot^2\theta)(1 + \cos\theta)(1 - \cos\theta) - (1 + \tan^2\theta)(1 + \sin\theta)(1 - \sin\theta)$
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53 $2\cos^2\theta + \sin\theta - 2 = 0$
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54 $30^\circ$
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55 $h$
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56 $p$
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57 $32\sqrt{3}$
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58 $The surface area of a cube is increased by 25%. If p is the percentage increase in its length, then which one of the following is correct ?$
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59 $A solid cube is cut into two cuboids of equal volume. What is the ratio of total surface area of the given cube to that of one of the cuboids ?$
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60 $The length of a diagonal of a cuboid is 11 cm. The surface area is 240 square cm. What is the sum of its length, breadth and height ?$
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61 $What is the area of the circle (approximately) inscribed in a triangle with side lengths 12 cm, 16 cm and 20 cm ?$
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62 $Two times the total surface area of a solid right circular cylinder is three times its curved surface area. If h is the height and r is the radius of the base of the cylinder, then which one of the following is correct ?$
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63 $A floor of a big hall has dimensions 30 m 60 cm and 23 m 40 cm. It is to be paved with square tiles of same size. What is the minimum number of tiles required ?$
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64 $How long will a man take to walk around the boundary of a square field of area 25 hectares at the rate of 5 km/hr ?$
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65 $Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct ?$
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66 $66\frac{2}{3}\%$
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67 $What is the maximum area that can be covered by three non-intersecting circles drawn inside a rectangle of sides 8 cm and 12 cm ?$
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68 $\angle APO = \theta$
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69 $A solid iron ball is melted and 64 smaller solid balls of equal size are made using the entire volume of iron. What is the ratio of the surface area of the larger ball to the sum of the surface areas of all the smaller balls ?$
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70 $A triangle ABC has been divided into four smaller triangles P, Q, R, S whose perimeters are 16 cm, 12 cm, 4 cm and 12 cm respectively. P, R and S contain the vertices A, B and C respectively. What is the perimeter of the triangle ABC ?$
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For the next ten (10) items that follow : Each item contains a Question followed by two Statements. Answer each item using the following instructions :
71 $X > 5$
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72 $ax^2 + bx + c = 0$
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73 $m > n$
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74 $Question : Can a circle be drawn through the points A, B and C ? Statement I : AB = 5 cm, BC = 5 cm, CA = 6 cm. Statement II : AB = 3 cm, BC = 4 cm, CA = 7 cm.$
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75 $(x + y)$
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76 $p^2 + q^2 + q$
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77 $30^\circ$
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78 $\triangle$
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79 $\triangle$
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80 $A chord PQ of the circle C divides it into two segments such that 3 times the area of the major segment is 4 times the area of the minor segment. Question : What is the radius of C ? Statement I : Area of the minor segment is 66 square cm. Statement II : Area of the major segment is 88 square cm.$
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Consider the following data for the next two (02) items that follow :
Class 0 - 30 30 - 60 60 - 90 90 - 120
Frequency 4 5 7 4
81 $What is the mode of the distribution ?$
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82 $7(Q - P) = 9R$
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Consider the following data for the next two (02) items that follow :
Class 40 - 50 50 - 60 60 - 70 70 - 80
Frequency 4 3 1 2
83 $What is the mean of the distribution ?$
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84 $If M is the median, then what is the value of 3M ?$
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85 $2 : 1$
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86 $\pi = \frac{22}{7}$
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87 $The volume of a cuboid is 3600 cubic cm. The areas of two adjacent faces are 225 square cm and 144 square cm. What is the area of the other adjacent face ?$
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88 $The perimeter and the area of a right-angled triangle are 36 cm and 54 square cm respectively. What is the length of the hypotenuse ?$
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89 $X = \{x \mid x = 2 + 4k, \text{ where } k = 0,\ 1,\ 2,\ 3, \ldots 24\}$
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90 $The perimeter of a sector of a circle of radius 5.2 cm is 16.4 cm. What is the area of the sector ?$
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Consider the following for the next three (03) items that follow : In the triangle ABC, AB = 6 cm, BC = 8 cm and AC = 10 cm. The perpendicular dropped from B meets the side AC at D. A circle of radius BD (with centre B) cuts AB and BC at P and Q respectively as shown in the figure.
91 $What is the length of QC ?$
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92 $\angle ABD = \theta$
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93 $What is the radius of the circle ?$
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Consider the following for the next three (03) items that follow : In the figure given below, a circle is inscribed in a square PQRS. A rectangle at the corner P that measures 4 cm $\times$ 2 cm and a square at the corner R are drawn.
94 $What is the area of the circle ?$
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95 $What is the area of the smaller square ?$
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96 $What is the area of the shaded region ?$
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Consider the following for the next two (02) items that follow : In the following figure, a rectangle ABCD is inscribed in a circle of radius r. Given $\angle DAE = 30^\circ$ and $\angle ACD = 30^\circ$ .
97 $What is the ratio of the area of the circle to the area of the rectangle ?$
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98 $\triangle AEC$
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Consider the following for the next two (02) items that follow : In the following figure, a triangle ABC is inscribed in a circle with centre at O. Let $\angle POA = x^\circ$ and $\angle OQB = y^\circ$ . Further, OB = BQ.
99 $What is the relation between x and y ?$
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100 $y = 15$
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Class	0 - 30	30 - 60	60 - 90	90 - 120
Frequency	4	5	7	4

Class	40 - 50	50 - 60	60 - 70	70 - 80
Frequency	4	3	1	2

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