NDA 2022 (II), Paper I, Mathematics question paper

120 questions · 300 marks · 150 minutes · +2.5 per correct answer, -0.83 per wrong one

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1 $How many four-digit natural numbers are there such that all of the digits are odd?$
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2 $\sum_{r=0}^{n} 2^r C(n,r)$
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3 $If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) are there in the list before the first word that starts with C?$
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Consider the following statements: 1. If f is the subset of Z × Z defined by f = {(xy, x−y); x, y ∈ Z}, then f is a function from Z to Z. 2. If f is the subset of N × N defined by f = {(xy, x+y); x, y ∈ N}, then f is a function from N to N.
4 $Which of the statements given above is/are correct?$
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5 $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$
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Consider the following for the next three (03) items that follow: If $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta \\ 0 & \sin\theta & -\cos\theta \end{pmatrix}$ , which of the following are correct? 1. $A + adjA$ is a null matrix 2. $A^{-1} + adjA$ is a null matrix 3. $A - A^{-1}$ is a null matrix
6 $Select the correct answer using the code given below:$
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7 $3 \times 3$
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8 $For how many quadratic equations, the sum of roots is equal to the product of roots?$
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Consider the following statements: 1. The set of all irrational numbers between $\sqrt{2}$ and $\sqrt{5}$ is an infinite set. 2. The set of all odd integers less than 100 is a finite set.
9 $Which of the statements given above is/are correct?$
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Consider the following statements: 1. 2 + 4 + 6 + … + 2n = n² + n 2. The expression n² + n + 41 always gives a prime number for every natural number n
10 $Which of the above statements is/are correct?$
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11 $p, q$
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12 $What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm?$
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13 $A = \{7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}$
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Consider the following for the next three (03) items that follow: Let R be a relation from N to N defined by $R = \{(x, y): x, y \in N \text{ and } x^2 = y^3\}$ . Which of the following are NOT correct? 1. $(x,x) \in R$ for all $x \in N$ 2. $(x,y) \in R \Rightarrow (y,x) \in R$ 3. $(x,y) \in R$ and $(y,z) \in R \Rightarrow (x,z) \in R$
14 $Select the correct answer using the code given below:$
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Consider the following: 1. A∩B = A∩C ⟹ B = C 2. A∪B = A∪C ⟹ B = C
15 $Which of the above is/are correct?$
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Consider the following for the next three (03) items that follow: Let $z = \frac{1+i\sin\theta}{1-i\sin\theta}$ where $i = \sqrt{-1}$
16 $z$
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17 $\theta$
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18 $\theta$
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Consider the following for the next three (03) items that follow: Let P be the sum of first n positive terms of an increasing arithmetic progression A. Let Q be the sum of first n positive terms of another increasing arithmetic progression B. Let P : Q = (5n+4) : (9n+6)
19 $What is the ratio of the first term of A to that of B?$
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20 $What is the ratio of their 10th terms?$
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21 $If d is the common difference of A, and D is the common difference of B, then which one of the following is always correct?$
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Consider the following for the next three (03) items that follow: Consider the binomial expansion of $(p + qx)^9$ :
22 $x^3$
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23 $What is the ratio of the coefficients of middle terms in the expansion (when expanded in ascending powers of x)?$
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24 $x^2$
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Consider the following for the next three (03) items that follow: Consider the word 'QUESTION':
25 $How many 4-letter words each of two vowels and two consonants with or without meaning, can be formed?$
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26 $How many 8-letter words with or without meaning, can be formed such that consonants and vowels occupy alternate positions?$
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27 $How many 8-letter words with or without meaning, can be formed so that all consonants are together?$
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Consider the following for the next three (03) items that follow: Let $\Delta$ be the determinant of a matrix A, where $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$ and $C_{11}, C_{12}, C_{13}$ be the cofactors of $a_{11}, a_{12}, a_{13}$ respectively.
28 $a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$
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29 $a_{21}C_{11} + a_{22}C_{12} + a_{23}C_{13}$
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30 $\begin{vmatrix} a_{21} & a_{31} & a_{11} \\ a_{23} & a_{33} & a_{13} \\ a_{22} & a_{32} & a_{12} \end{vmatrix}$
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Consider the following for the next three (03) items that follow: Let $f(x)$ be a function satisfying $f(x+y) = f(x)f(y)$ for all $x, y \in N$ such that $f(1) = 2$ :
31 $\sum_{x=2}^{n} f(x) = 2044$
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32 $\sum_{x=1}^{5} f(2x-1)$
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33 $\sum_{x=1}^{6} 2^x f(x)$
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Consider the following for the next three (03) items that follow: A university awarded medals in basketball, football and volleyball. Only x students (x < 6) got medal in all the three sports and the medals went to a total of 15x students. It awarded 5x medals in basketball, (4x + 15) medals in football and (x + 25) medals in volleyball.
34 $How many received medals in exactly two of the three sports?$
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35 $How many received medals in at least two of three sports?$
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36 $How many received medals in exactly one of three sports?$
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Consider the following for the next three (03) items that follow: Let $A = \begin{pmatrix} 0 & \sin^2\theta & \cos^2\theta \\ \cos^2\theta & 0 & \sin^2\theta \\ \sin^2\theta & \cos^2\theta & 0 \end{pmatrix}$ and $A = P + Q$ where P is symmetric matrix and Q is skew-symmetric matrix.
37 $What is P equal to?$
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38 $What is Q equal to?$
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39 $What is the minimum value of determinant of A?$
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Consider the following for the next three (03) items that follow: ABC is a triangular plot with AB = 16 m, BC = 10 m and CA = 10 m. A lamp post is situated at the middle point of the side AB. The lamp post subtends an angle 45° at the vertex B.
40 $What is the height of the lamp post?$
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41 $\frac{AB}{\sin C}$
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42 $\cos A + \cos B + \cos C$
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Consider the following for the next three (03) items that follow: There are two points P and Q due south of a leaning tower, which leans towards north. P is at a distance x and Q is at a distance y from the foot of the tower (x > y). The angles of elevation of the top of the tower from P and Q are 15° and 75° respectively.
43 $At what height is the top of the tower above the ground level?$
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44 $\theta$
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45 $What is the length of the tower?$
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46 $\csc\!\left(-\frac{73\pi}{3}\right)$
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47 $\cos\!\left(\frac{5\pi}{17}\right)+\cos\!\left(\frac{7\pi}{17}\right)+2\cos\!\left(\frac{11\pi}{17}\right)\cos\!\left(\frac{\pi}{17}\right)$
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48 $\tan\!\left(\frac{3\pi}{8}\right)$
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49 $\tan^{-1}\cot(\csc^{-1}2)$
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50 $a = 4, b = 3, c = 2$
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51 $\cos36° - \cos72°$
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52 $\sec x = \frac{25}{24}$
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53 $\tan^2 165° + \cot^2 165°$
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54 $\sin\!\left(2n\pi+\frac{5\pi}{6}\right)\sin\!\left(2n\pi-\frac{5\pi}{6}\right)$
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55 $1+2(\sin x+\cos x)(\sin x - \cos x) = 0$
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56 $\left(1, \frac{1}{2-\sqrt{3}}\right)$
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57 $If P(3, 4) is the mid-point of a line segment between the axes, then what is the equation of the line?$
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58 $The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the origin and B lies above the origin. What is the equation of line passing through (8, 0) and parallel to the side AC?$
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59 $The centre of the circle passing through origin and making positive intercepts 4 and 6 on the coordinate axes, lies on the line$
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60 $The centre of an ellipse is at (0, 0), major axis is on the y-axis. If the ellipse passes through (3, 2) and (1, 6), then what is its eccentricity?$
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61 $x^2 = \sqrt{3}y$
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62 $Consider the points A(2, 4, 6), B(-2, -4, -2), C(4, 6, 4) and D(8, 14, 12). Which of the following statements is/are correct? 1. The points are the vertices of a rectangle ABCD. 2. The mid-point of AC is the same as that of BD. Select the correct answer using the code given below:$
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63 $x^2 + y^2 + z^2 - 4x - 6y - 8z - 16 = 0$
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64 $A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?$
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65 $Consider the following statements: 1. The direction ratios of y-axis can be <0, 4, 0> 2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0> Which of the statements given above is/are correct?$
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66 $\overrightarrow{PR} = \vec{a}$
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67 $\vec{a}$
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68 $\vec{a}, \vec{b}$
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69 $2x^2\hat{i}+3x\hat{j}+\hat{k}$
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70 $\hat{j}+\hat{k}$
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71 $z = [y]$
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72 $f(x) = 4x+1$
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73 $f(x) = \log_{10}(x^2+2x+11)$
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74 $\int (x^x)^2(1+\ln x)\,dx$
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75 $\int e^x\{1+\ln x + x\ln x\}\,dx$
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76 $\int \frac{(\cos x)^{1.5}-(\sin x)^{1.5}}{\sqrt{\sin x\cdot\cos x}}\,dx$
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77 $y = \frac{x\sqrt{x^2-16}}{2} - 8\ln\left|x+\sqrt{x^2-16}\right|$
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78 $y = (x^x)^x$
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79 $3(\sin x - \cos x) + 4(\cos^3 x - \sin^3 x)$
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80 $y = \sqrt{1-x^2}$
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81 $x - |y| = 0$
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82 $f(\alpha) = \sqrt{\sec^2\alpha - 1}$
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83 $f(x) = \ln(x+\sqrt{1+x^2})$
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84 $\lim_{x\to 0}\frac{x}{\sqrt{1-\cos4x}}$
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85 $\lim_{x\to\pi/2}\frac{4x-2\pi}{\cos x}$
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86 $f(x) = \frac{x^2+x+|x|}{x}$
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87 $\lim_{h\to 0}\frac{\sin^2(x+h)-\sin^2x}{h}$
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88 $f(x)$
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89 $y = \ln^2\!\left(\frac{x^2-x+1}{x^2+x+1}\right)$
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90 $\frac{d}{dx}\!\left(\frac{1+x^4+x^8}{1-x^2+x^4}\right) = ax+bx^3$
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91 $f(x) = (p\sec x)^2+(q\csc x)^2$
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92 $f(x) = \sum_{j=1}^{7}(x-j)^2$
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93 $f(x) = \begin{cases}|x|+1, & 0<|x|\leq3\\1, & x=0\end{cases}$
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94 $\int_0^1 \ln\!\left(\frac{1}{x}-1\right)dx$
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95 $\int_0^{\pi/2}(\sin^4x+\cos^4x)\,dx = k$
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96 $\int_{-\pi/2}^{\pi/2}(e^{\cos x}\sin x+e^{\sin x}\cos x)\,dx$
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97 $x^2+y^2=\pi^2$
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98 $\frac{dy}{dx}+\cos\!\left(\frac{dy}{dx}\right)=0$
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99 $What is the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis?$
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100 $(dy-dx)+\cos x(dy+dx)=0$
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101 $\frac{x}{y} = \frac{55}{42}$
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102 $What is the probability of getting a composite number in the list of natural numbers from 1 to 50?$
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103 $n > 7$
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104 $Two numbers x and y are chosen at random from a set of first 10 natural numbers. What is the probability that (x + y) is divisible by 4?$
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105 $x + \frac{1}{x} > 2$
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106 $Three fair dice are tossed once. What is the probability that they show different numbers that are in AP?$
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107 $P(A) = 0.5$
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108 $Five coins are tossed once. What is the probability of getting at most four tails?$
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109 $Three fair dice are thrown. What is the probability of getting a total greater than or equal to 15?$
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110 $The probability that a person hits a target is 0.5. What is the probability of at least one hit in 4 shots?$
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111 $A box contains 2 white balls, 3 black balls and 4 red balls. What is the number of ways of drawing 3 balls from the box with at least one black ball?$
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112 $During war one ship out of 5 was sunk on an average in making a certain voyage. What is the probability that exactly 3 out of 5 ships would arrive safely?$
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113 $A card is drawn from a pack of 52 cards. A gambler bets that it is either a spade or an ace. The odds against his winning are$
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114 $The coefficient of correlation between ages of husband and wife at the time of marriage for a given set of 100 couples was noted to be 0.7. Assume that all these couples survive to celebrate the silver jubilee of their marriage. The coefficient of correlation at that point of time will be$
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115 $The completion of a construction job may be delayed due to strike. The probability of strike is 0.6. The probability that the construction job gets completed on time if there is no strike is 0.85 and the probability that the construction job gets completed on time if there is a strike is 0.35. What is the probability that the construction job will not be completed on time?$
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Consider the following for the next two (02) items that follow: The mean and standard deviation (SD) of marks obtained by 50 students of a class in 4 subjects are given below:
Subject Mathematics Physics Chemistry Biology
Mean Marks 40 28 38 36
SD 15 12 14 16
116 $Which one of the following subjects shows highest variability of marks?$
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117 $What is the coefficient of variation of marks in Mathematics?$
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Consider the following for the next three (03) items that follow: Consider the following grouped frequency distribution:
Class 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 1 2 4 6 4 3
118 $What is the median of the distribution?$
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119 $What is mean deviation about the median?$
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120 $What is the mean deviation about the mean?$
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Subject	Mathematics	Physics	Chemistry	Biology
Mean Marks	40	28	38	36
SD	15	12	14	16

Class	0-10	10-20	20-30	30-40	40-50	50-60
Frequency	1	2	4	6	4	3

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