NDA 2019 (I), 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 $n^{th}$
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2 $X = \{1, 2, 3, 4\}$
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3 $xRy \Rightarrow x^2 - 4xy + 3y^2 = 0$
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4 $A = \{x \in \mathbb{Z} : x^3 - 1 = 0\}$
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5 $(A \cap B) \cup (A \cap \bar{B}) \cup (\bar{A} \cap B) = A \cup B$
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6 $A \subset C \Rightarrow (A \cap B) \subset (C \cap B),\ (A \cup B) \subset (C \cup B)$
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7 $B = \begin{pmatrix} 3 & 2 & 0 \\ 2 & 4 & 0 \\ 1 & 1 & 0 \end{pmatrix}$
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8 $|x^2 - x - 6| = x + 2$
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9 $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
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10 $\begin{vmatrix} x & -3i & 1 \\ y & 1 & i \\ 0 & 2i & -i \end{vmatrix} = 6 + 11i$
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11 $z^3 + 2z^2 + 2z + 1 = 0$
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12 $C(20, n+2) = C(20, n-2)$
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13 $There are 10 points in a plane. No three of these points are in a straight line. What is the total number of straight lines which can be formed by joining the points?$
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14 $px^2 + qx + r = 0$
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15 $A = \{\lambda, \{\lambda, \mu\}\}$
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Consider the following for the next 02 (two) items that follow: In a school, all the students play at least one of three indoor games – chess, carrom and table tennis. 60 play chess, 50 play table tennis, 48 play carrom, 12 play chess and carrom, 15 play carrom and table tennis, 20 play table tennis and chess.
16 $What can be the minimum number of students in the school?$
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17 $What can be the maximum number of students in the school?$
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18 $(A^{-1})$
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19 $A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose?$
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20 $From 6 programmers and 4 typists, an office wants to recruit 5 people. What is the number of ways this can be done so as to recruit at least one typist?$
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21 $\left[(2x-3y)^2(2x+3y)^2\right]^2$
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22 $(1+ax)^n$
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23 $The numbers 1, 5 and 25 can be three terms (not necessarily consecutive) of$
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24 $(p+q)^{th}$
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25 $n > 1$
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26 $25\csc^2 x + 36\sec^2 x$
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Consider the following for the next 02 (two) items: Let A and B be (3×3) matrices with det A = 4 and det B = 3.
27 $\det(2AB)$
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28 $\det(3AB^{-1})$
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Consider the following for the next 02 (two) items: A complex number is given by $z = \dfrac{1+2i}{1-(1-i)^2}$ .
29 $What is the modulus of z?$
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30 $What is the principal argument of z?$
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31 $\dfrac{\sin 34° \cos 236° - \sin 56° \sin 124°}{\cos 28° \cos 88° + \cos 178° \sin 208°}$
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32 $\tan 54°$
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Consider the following for the next 03 (three) items: If $p = X\cos\theta - Y\sin\theta$ , $q = X\sin\theta + Y\cos\theta$ and $p^2 + 4pq + q^2 = AX^2 + BY^2$ , $0 \leq \theta \leq \dfrac{\pi}{2}$ .
33 $\theta$
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34 $What is the value of A?$
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35 $What is the value of B?$
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Consider the following for the next 02 (two) items: It is given that $\cos(\theta-\alpha)=a$ , $\cos(\theta-\beta)=b$ .
36 $\cos(\alpha-\beta)$
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37 $\sin^2(\alpha-\beta)+2ab\cos(\alpha-\beta)$
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38 $\sin\alpha + \cos\alpha = p$
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39 $\sin^{-1}\dfrac{4}{5} + \sec^{-1}\dfrac{5}{4} - \dfrac{\pi}{2}$
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40 $\sin^{-1}\dfrac{2p}{1+p^2} - \cos^{-1}\dfrac{1-q^2}{1+q^2} = \tan^{-1}\dfrac{2x}{1-x^2}$
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41 $\tan\theta = \dfrac{1}{2}$
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42 $\cos A = \dfrac{3}{4}$
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43 $\tan 75° + \cot 75°$
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44 $\cos46°\cos47°\cos48°\cos49°\cos50°\ldots\cos135°$
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45 $\sin2\theta = \cos3\theta$
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46 $x^2 + px + q = 0$
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47 $n(n+1)$
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48 $(1+\tan\alpha\tan\beta)^2+(\tan\alpha-\tan\beta)^2-\sec^2\alpha\sec^2\beta$
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49 $p=\csc\theta-\cot\theta$
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50 $1:2:3$
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51 $x\cos\theta+y\sin\theta=p$
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52 $x^2+y^2=a^2$
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53 $The sum of the focal distances of a point on an ellipse is constant and equal to the$
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54 $2x^2-3y^2-6=0$
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55 $y^2=4ax$
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56 $(1,3)$
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57 $3y+4x=1$
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58 $y=x$
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59 $x+y-4=0$
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60 $x^2+y^2+4x-7y+12=0$
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61 $A(2,-3,3)$
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62 $x^2+y^2+z^2-6x+8y-10z+1=0$
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63 $2x+y+2z=9$
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64 $4x-2y+4z+9=0$
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65 $What are the direction cosines of z-axis?$
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66 $\vec{a}=\hat{i}-2\hat{j}+5\hat{k}$
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67 $3\hat{i}-2\hat{j}+\hat{k}$
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68 $AC=p$
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69 $\vec{a}=2\hat{i}-6\hat{j}-3\hat{k}$
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70 $\lambda$
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71 $\sec^2(\tan^{-1}x)$
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72 $f(x)=\log_{10}(1+x)$
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73 $f(x)=\ln(\sqrt{x^2+1}-x)$
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74 $f(x)=\log_x 10$
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75 $\lim_{x\to0}\dfrac{1-\cos^3 4x}{x^2}$
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76 $r>0$
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77 $f(x)=3^{1+x}$
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78 $x^2+9|x|+20=0$
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79 $f(x)=\sin(\cos x)$
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80 $f(x)=\sqrt{(2-x)(x-3)}$
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81 $\dfrac{dy}{dx}=\cos(y-x)+1$
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82 $\displaystyle\int_0^{\pi/2}|\sin x-\cos x|\,dx$
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83 $y=a\cos2x+b\sin2x$
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84 $A given quantity of metal is to be cast into a half cylinder (i.e. with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is$
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85 $\displaystyle\int_0^{\pi/2}e^{\sin x}\cos x\,dx$
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86 $f(x)=\dfrac{x-2}{x+2}$
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87 $\displaystyle\int \ln(x^2)\,dx$
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88 $(4,2)$
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89 $The differential equation of the system of circles touching the y-axis at the origin is$
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90 $\dfrac{d^2y}{dx^2}+2\left(\dfrac{dy}{dx}\right)^2+9y=x$
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91 $\dfrac{dy}{dx}+\dfrac{x}{y}=0$
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92 $k$
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93 $a^2x+b^2y$
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94 $\displaystyle\int e^{x\ln a}\,dx$
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95 $y=c\sin x$
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96 $\sin\theta+\cos\theta=\sqrt{2}\cos\theta$
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97 $In a circle of diameter 44 cm, the length of a chord is 22 cm. What is the length of minor arc of the chord?$
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98 $\sin\theta=-\dfrac{1}{2}$
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99 $\textbf{\text{not}}$
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100 $AB=z$
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101 $From a deck of cards, cards are taken out with replacement. What is the probability that the fourteenth card taken out is an ace?$
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102 $P(A)=0.5$
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103 $\dfrac{1}{2}$
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104 $A pair of fair dice is rolled. What is the probability that the second dice lands on a higher value than does the first?$
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105 $A fair coin is tossed and an unbiased dice is rolled together. What is the probability of getting a 2 or 4 or 6 along with head?$
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106 $If A, B, C are three events, then what is the probability that at least two of these events occur together?$
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107 $If two variables X and Y are independent, then what is the correlation coefficient between them?$
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108 $P(A\cup B)=\dfrac{2}{3}$
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109 $The mean of 100 observations is 50 and the standard deviation is 10. If 5 is subtracted from each observation and then it is divided by 4, then what will be the new mean and the new standard deviation respectively?$
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110 $If two fair dice are rolled then what is the conditional probability that the first dice lands on 6 given that the sum of numbers on the dice is 8?$
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111 $Two symmetric dice flipped with each dice having two sides painted red, two painted black, one painted yellow and the other painted white. What is the probability that both land on the same colour?$
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112 $\dfrac{1}{2}$
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113 $Two cards are chosen at random from a deck of 52 playing cards. What is the probability that both of them have the same value?$
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114 $In eight throws of a die, 5 or 6 is considered a success. The mean and standard deviation of total number of successes is respectively given by$
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115 $\bar{A}$
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116 $Consider the following statements: 1. The algebraic sum of deviations of a set of values from their arithmetic mean is always zero. 2. Arithmetic mean > Median > Mode for a symmetric distribution. Which of the above statements is/are correct?$
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117 $Z=X+5$
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118 $If all the natural numbers between 1 and 20 are multiplied by 3, then what is the variance of the resulting series?$
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119 $What is the probability that an interior point in a circle is closer to the centre than to the circumference?$
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120 $If A and B are two events, then what is the probability of occurrence of either event A or event B?$
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