NDA 2021 (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 $n$
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2 $x$
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3 $\Delta$
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4 $C_0, C_1, C_2, \ldots, C_n$
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5 $a+b+c=4$
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6 $k$
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7 $a_1, a_2, a_3, \ldots, a_9$
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8 $x^2 + 2x + k = 0$
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9 $n = 100!$
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10 $Z = 1+i$
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11 $A$
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12 $How many matrices of different orders are possible with elements comprising all prime numbers less than 30?$
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13 $A = \begin{vmatrix}p & q\\r & s\end{vmatrix}$
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14 $A$
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15 $\cot 2x \cot 4x - \cot 4x \cot 6x - \cot 6x \cot 2x$
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16 $\tan x = -\frac{3}{4}$
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17 $\csc\left(\frac{7\pi}{6}\right)\sec\left(\frac{5\pi}{3}\right)$
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18 $\begin{vmatrix}x & 1 & 3\\0 & 0 & 1\\1 & x & 4\end{vmatrix} = 0$
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19 $\tan 31° \tan 33° \tan 35° \cdots \tan 57° \tan 59°$
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20 $f(x) = \begin{vmatrix}1 & x & x+1\\2x & x(x-1) & x(x+1)\\3x(x-1) & 2(x-1)(x-2) & x(x+1)(x-1)\end{vmatrix}$
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21 $\sin^{-1} x - \cos^{-1} x = \frac{\pi}{6}$
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22 $(\sin 24° + \cos 66°)(\sin 24° - \cos 66°)$
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23 $A chord subtends an angle 120° at the centre of a unit circle. What is the length of the chord?$
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24 $(1+\cot\theta - \csc\theta)(1+\tan\theta+\sec\theta)$
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25 $\frac{1+\tan^2\theta}{1+\cot^2\theta} - \left(\frac{1-\tan\theta}{1-\cot\theta}\right)^2$
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26 $What is the interior angle of a regular octagon of side length 2 cm?$
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27 $7\sin\theta + 24\cos\theta = 25$
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28 $A ladder 6 m long reaches a point 6 m below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is 75°. What is the height of the flagstaff?$
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29 $x$
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30 $3\cos\theta = 4\sin\theta$
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31 $\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$
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32 $\tan A = \frac{1}{7}$
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33 $m$
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34 $ABC$
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35 $A = \{1,3,5\}$
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36 $Consider the following statements: 1. The null set is a subset of every set. 2. Every set is a subset of itself. 3. If a set has 10 elements, then its power set will have 1024 elements. Which of the above statements are correct?$
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37 $R$
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38 $f:\mathbb{Z}\to\mathbb{Z}$
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39 $Z$
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40 $Z$
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41 $i^{2n+1}(-i)^{2n-1}$
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42 $\alpha$
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43 $5x^2+26x+k=0$
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44 $In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player?$
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45 $(1+4x+4x^2)^5$
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46 $C(n,1)+C(n,2)+\cdots+C(n,n)$
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47 $(1+x)^{2n}$
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48 $If the first term of an AP is 2 and the sum of the first five terms is equal to one-fourth of the sum of the next five terms, then what is the sum of the first ten terms?$
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49 $Consider the following statements: 1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP. 2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP. Which of the above statements is/are correct?$
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50 $How many 5-digit prime numbers can be formed using the digits 1, 2, 3, 4, 5 if the repetition of digits is not allowed?$
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51 $f(x+1) = x^2 - 3x + 2$
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52 $x^2, x, -8$
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53 $The third term of a GP is 3. What is the product of its first five terms?$
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54 $i$
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55 $With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?$
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56 $4x^2+4y^2-20x+12y-15=0$
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57 $(-3, 4)$
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58 $y + px = 1$
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59 $A$
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60 $(-4, 2)$
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61 $(p, p-3)$
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62 $x - 2 = 0$
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63 $ABCD$
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64 $(3\tan\theta, 2\sec\theta)$
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65 $(e)$
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66 $\langle 6, 3, 6\rangle$
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67 $l$
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68 $A(1,7,-5)$
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69 $k$
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70 $x+y+z=3$
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71 $\vec{r} = a\hat{i} + b\hat{j}$
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72 $\vec{c}=\vec{a}+\vec{b}$
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73 $\vec{a}$
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74 $\vec{a}$
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75 $\textbf{\text{not}}$
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76 $\lim_{x\to a}\frac{a^x-x^a}{x^a-a^a}=-1$
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77 $\frac{dx}{dt}=x+1$
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78 $\int_0^a\frac{f(a-x)}{f(x)+f(a-x)}\,dx$
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79 $\lim_{x\to -1}\frac{x^3+x^2}{x^2+3x+2}$
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80 $\int_0^a[f(x)+f(-x)]\,dx=\int_{-a}^a g(x)\,dx$
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81 $y=\sqrt{16-x^2}$
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82 $y=-x^3+3x^2+2x-27$
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83 $A 24 cm long wire is bent to form a triangle with one of the angles as 60°. What is the altitude of the triangle having the greatest possible area?$
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84 $f(x)=e^{|x|}$
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85 $\int\frac{dx}{\sec x+\tan x}$
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86 $\int\frac{dx}{\sec^2(\tan^{-1}x)}$
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87 $x+y=20$
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88 $\sin(\ln x)+\cos(\ln x)$
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89 $x=e^t\cos t$
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90 $\sin2x\cdot\cos2x$
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91 $e^x$
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92 $f(x)$
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93 $f(x)=\begin{cases}a+bx, & x<1\\5, & x=1\\b-ax, & x>1\end{cases}$
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94 $f(x)=\sin x$
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95 $f(x)=3^x$
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96 $y^2+2cy-cx+c^2=0$
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97 $x=\sqrt{1+\frac{d^2y}{dx^2}}$
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98 $y=ae^x+be^{-x}$
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99 $\ln\left(\frac{dy}{dx}\right)+y=x$
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100 $\int e^{(2\ln x+\ln x^2)}\,dx$
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101 $N$
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102 $The numbers of Science, Arts and Commerce graduates working in a company are 30, 70 and 50 respectively. If these figures are represented by a pie chart, then what is the angle corresponding to Science graduates?$
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103 $For a histogram based on a frequency distribution with unequal class intervals, the frequency of a class should be proportional to$
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104 $The coefficient of correlation is independent of$
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105 $The following table gives the frequency distribution of number of peas per pea pod of 198 pods: Number of peas 1234567 Frequency 43376502681 What is the median of this distribution?$
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106 $M$
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107 $What is the sum of deviations of the variate values 73, 85, 92, 105, 120 from their mean?$
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108 $x$
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109 $If the mean of a frequency distribution is 100 and the coefficient of variation is 45%, then what is the value of the variance?$
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110 $A$
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111 $For which of the following sets of numbers do the mean, median and mode have the same value?$
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112 $The mean of 12 observations is 75. If two observations are discarded, then the mean of the remaining observations is 65. What is the mean of the discarded observations?$
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113 $k$
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114 $x_i$
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115 $P(A\cup B)=\frac{5}{6}$
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116 $The average of a set of 15 observations is recorded, but later it is found that for one observation, the digit in the tens place was wrongly recorded as 8 instead of 3. After correcting the observation, the average is$
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117 $E$
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118 $\frac{2}{3}$
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119 $x$
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120 $A$
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Number of peas	1	2	3	4	5	6	7
Frequency	4	33	76	50	26	8	1

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