NDA Mathematics 21 April 2024
Paper was held on Sun, Apr 21, 2024 4:30 AM
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## Mathematics

Let $A$ and $B$ be matrices of order $3 \times 3$. If $|A| = \frac{1}{2 \sqrt{2}}$ and $|B| = \frac{1}{729}$, then what
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If $z$ is any complex number and $iz^2 + z - z + i = 0$, where $i = \sqrt{-1}$, then what is the value of $(|z| + 1)^2$?
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What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?
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If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?
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A man has 7 relatives (4 women and 3 men). His wife also has 7 relatives (3 women and 4 men). In how many ways can they
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A triangle $PQR$ is such that 3 points lie on the side $PQ$, 4 points on $QR$ and 5 points on $RP$ respectively. Triangl
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If $\log{a} = p$, $\log{c} = 2p$ and $\log{e} = 3p$, then what is $(ace)^\frac{1}{p}$ equal to?
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If $\sqrt{2}$ and $\sqrt{3}$ are roots of the equation $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + x^4 = 0$ where $a_0, a_1, a_2, View Question Let$z_1$and$z_2$be two complex numbers such that$\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is$\oper
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If $26! = n8^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?
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Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$: 1.$adj(AB) View Question Consider the following statements in respect of a non-singular matrix$A$of order$n$:1.$A(\text{adj}A^T) = A(\text{ad
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How many four-digit natural numbers are there such that all of the digits are even?
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If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?
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What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?
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Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$:1. All diagonal elements are ze
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Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisibl
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What is the remainder when $2^{120}$ is divided by 7?
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For what value of $n$ is the determinant $$\left|\begin{array}{ccc} C(9,4) & C(9,3) & C(10, n-2) \\ C(11,6) & C(11,5) & View Question If ABC is a triangle, then what is the value of the determinant$$ \left|\begin{array}{ccc} \cos C & \sin B & 0 \\ \tan
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What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allow
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Let $A = \{x \in \mathbb{R} : -1 &lt;x &lt;1\}$. Which of the following is/are bijective functions from A to itself?1. $View Question Let$R$be a relation on the open interval$(-1, 1)$and is given by$R = \{(x, y) : |x + y|
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For any three non-empty sets $A, B, C$, what is $$(A \cup B - \{(A - B) \cup (B - A) \cup (A \cap B)\})$$ equal to?
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If $a, b, c$ are the sides of a triangle $ABC$, then what is $$\begin{vmatrix} a^2 & b \sin A & c \sin A \\ b \sin A & View Question If a, b, c are in AP; b, c, d are in GP; c, d, e are in HP, then which of the following is/are correct?1. a, c, View Question What is the number of solutions of \log_4(x - 1) = \log_2(x - 3)? View Question For x \geq y > 1, let \log_x\left( \frac{x}{y} \right) + \log_y\left(\frac{y}{x}\right) = k, then the value of k c View Question If A=\left[\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\ View Question What is the coefficient of x^{10} in the expansion of (1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}? View Question If the 4th term in the expansion of \left(mx + \frac{1}{x}\right)^n is \frac{5}{2}, then what is the value of mn? View Question If a, b, and c (a > 0, c > 0) are in GP, then consider the following in respect of the equation ax^2 + bx + c View Question If x^2 + mx + n is an integer for all integral values of x, then which of the following is/are correct?1. m must b View Question In a binomial expansion of (x+y)^{2 n+1}(x-y)^{2 n+1}, the sum of middle terms is zero. What is the value of \left(\f View Question Let A = \{1, 2, 3, 4, 5\} and B = \{6, 7\}. What is the number of onto functions from A to B? View Question$$ \text { What is } \frac{\sqrt{3} \cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ} \cos 25^{\circ}} \text { equal to ?
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What is $\sin 9^\circ - \cos 9^\circ$ equal to?
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If in a triangle $ABC$, $\sin^3A + \sin^3B + \sin^3C = 3\sin A \sin B \sin C$, then what is the value of the determinant
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If $\cos^{-1} x = \sin^{-1} x$, then which one of the following is correct?
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What is the number of solutions of $(\sin \theta - \cos \theta)^2 = 2$ where $-\pi View Question$ABC$is a triangle such that angle$C = 60^{\circ}$, then what is$\frac{\cos A + \cos B}{\cos \left(\frac{A - B}{2}\ri
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What is $\sqrt{15 + \cot^2 \left( \frac \pi 4 - 2 \cot^{-1} 3 \right)}$ equal to?
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What is the value of $\sin10^{\circ} \cdot \sin50^{\circ} + \sin50^{\circ} \cdot \sin250^{\circ} + \sin250^{\circ} \cdot View Question What is$\tan^{-1} \left( \frac{a}{b} \right) - \tan^{-1} \left( \frac{a - b}{a + b} \right)$equal to? View Question Under which one of the following conditions does the equation$\left(\cos \beta-1\right)x^2+(\cos \beta)x+\sin \beta=0$View Question In a triangle$ABC$,$AB=16 \text{ cm}, BC=63 \text{ cm}$and$AC=65 \text{ cm}$. What is the value of$\cos 2A+\cos 2B+
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If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha) f(\beta)$
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If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2-6x+8=0$, then what is the value of $\cos(2 \alpha+2 View Question What is the value of$\tan 65^\circ +2 \tan 45^\circ-2 \tan 40^\circ-\tan 25^\circ$? View Question Consider the following statements:1. In a triangle$ABC$, if$\cot A \cdot \cot B \cdot \cot C>0$, then the triangle is View Question If (a, b) is the centre and c is the radius of the circle$x^2 + y^2 + 2x + 6y + 1 = 0$, then what is the value of$a^2
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If (1, −1, 2) and (2, 1, −1) are the end points of a diameter of a sphere $x^2 + y^2 + z^2 + 2ux + 2vy + 2wz − 1 = 0$, t
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The number of points represented by the equation $x = 5$ on the $xy$-plane is
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If $\langle l, m, n \rangle$ are the direction cosines of a normal to the plane $2x − 3y + 6z + 4 = 0$, then what is the
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A line through $(1, −1, 2)$ with direction ratios $\langle 3, 2, 2 \rangle$ meets the plane $x + 2y + 3z = 18$. What is
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If $p$ is the perpendicular distance from origin to the plane passing through $(1, 0, 0)$, $(0, 1, 0)$ and $(0, 0, 1)$,
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If the direction cosines &lt;l, m, n&gt; of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then wh
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If a variable line passes through the point of intersection of the lines $x + 2y - 1 = 0$ and $2x - y - 1 = 0$ and meets
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What is the equation to the straight line passing through the point $(-sin\theta, cos\theta)$ and perpendicular to the l
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Two points $P$ and $Q$ lie on line $y = 2x + 3$. These two points $P$ and $Q$ are at a distance 2 units from another poi
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If two sides of a square lie on the lines $2x + y - 3 = 0$ and $4x + 2y + 5 = 0$, then what is the area of the square in
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ABC is a triangle with A(3, 5). The mid-points of sides AB, AC are at (-1, 2), (6, 4) respectively. What are the coordin
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ABC is an acute angled isosceles triangle. Two equal sides AB and AC lie on the lines 7x - y - 3 = 0 and x + y - 5 = 0.
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In the parabola $y^2 = 8x$, the focal distance of a point P lying on it is 8 units. Which of the following statements is
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What is the eccentricity of the ellipse if the angle between the straight lines joining the foci to an extremity of the
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Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \t View Question If a vector of magnitude 2 units makes an angle$\frac{\pi}{3}$with$2\hat{i}$,$\frac{\pi}{4}$with$3\hat{j}$and an View Question Consider the following in respect of moment of a force:1. The moment of force about a point is independent of point of a View Question For any vector$\vec{r}$, what is$\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdo
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Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle b
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Let $y_1(x)$ and $y_2(x)$ be two solutions of the differential equation $\frac{dy}{dx} = x$. If $y_1(0) = 0$ and $y_2(0) View Question The differential equation, representing the curve$y = e^{x}(a\cos{x} + b\sin{x})$where$a$and$b$are arbitrary const View Question If$f(x)=ax-b$and$g(x)=cx+d$are such that$f(g(x))=g(f(x))$, then which one of the following holds? View Question What is$\int^{1}_{-1}(3\sin x-\sin 3x)\cos^2 xdx$equal to? View Question What are the order and degree respectively of the differential equation $$\left\{2-\left(\frac{d y}{d x}\right)^2\right View Question If \frac{dy}{dx} = 2e^xy^3, y(0)= \frac{1}{2} then what is 4y^2(2-e^x) equal to? View Question Let p=\int_a^b f(x) d x and q=\int_a^b|f(x)| d x. If f(x)=e^{-x}, then which one of the following is correct ? View Question What is \int^{\pi/2}_0 \frac{a+\sin x}{2a+\sin x+\cos x} dx equal to? View Question The non-negative values of b for which the function \frac{16x^3}{3} - 4bx^2 + x has neither maximum nor minimum in t View Question Which one of the following is correct in respect of f(x) = \frac{1}{\sqrt{|x| - x}} and g(x) = \frac{1}{\sqrt{x - |x| View Question What is the value of \alpha ? View Question What is the value of \beta ? View Question Consider the following statements :1. f(x) is increasing in the interval (e, \infty)2. f(x) is decreasing in the i View Question Consider the following statements :1. f''(e) = \frac{1}{e}2. f(x) attains local minimum value at x = e3. A local m View Question What is g[f(x) - 3x] equal to? View Question What is f''(x) equal to? View Question Consider the following statements:1. f(x) is differentiable for all x 2. g(x) is continuous at x = 0.00013. The d View Question What is \lim\limits_{x \to 0-} h(x) + \lim\limits_{x \to 0+} h(x) equal to? View Question What is \varphi(a) equal to? View Question What is \varphi'(a) equal to? View Question Which of the following is/are correct?1. f'(0) = 02. f''(0) Select the correct answer using the code given below: View Question The function y has a relative maxima at x = 0 for View Question What is \int_{-1}^{0} h(x) dx equal to? View Question What is \int_{0}^{2} h(x) dx equal to? View Question What is the value of \alpha? View Question What is the value of \beta? View Question What is the value of A_1? View Question What is the value of \frac{2(A_1 + A_2)}{ A_1 - 3A_2}? View Question What is f(x) equal to? View Question What is 8\int_1^2 f(x)dx equal to? View Question A bag contains 5 black and 4 white balls. A man selects two balls at random. What is the probability that both of these View Question If a random variable (x) follows binomial distribution with mean 5 and variance 4 , and 5^{23} P(X=3)=\lambda 4^\lamb View Question From data (-4, 1), (-1, 2), (2, 7) and (3, 1), the regression line of y on x is obtained as y = a + bx, then what is t View Question Let x+2 y+1=0 and 2 x+3 y+4=0 are two lines of regression computed from some bivariate data. If \theta is the acut View Question If a, b, c are in HP, then what is \frac{1}{b-a} + \frac{1}{b-c} equal to? 1. \frac{2}{b}2. \frac{1}{a} + \frac{1 View Question If two random variables X and Y are connected by relation \frac{2 X-3 Y}{5 X+4 Y}=4 and X follows Binomial distr View Question An edible oil is sold at the rates 150, 200, 250, 300 rupees per litre in four consecutive years. Assuming that an equal View Question If the letters of the word 'TIRUPATI' are written down at random, then what is the probability that both Ts are always c View Question Let m = 77^n. The index n is given a positive integral value at random. What is the probability that the value of m View Question Three different numbers are selected at random from the first 15 natural numbers. What is the probability that the produ View Question What is the minimum value of P(A) + P(B)? View Question What is the maximum value of P(A) + P(B)? View Question What is the minimum value of P(B \cap C)? View Question What is the maximum value of P(B \cap C)? View Question What is the value of n? View Question What is the value of p + q? View Question$$ \text { What is } \sum_i^n x_i f_i \text { equal to? }$\$
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What is the mean of the distribution?
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What is the mean deviation of the largest five observations?
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What is the variance of the largest five observations?
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