NDA 1 2024 Maths Question Paper with Solutions

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 is the value of $|2B(\text{adj}(3A))|$ ?

If $z$ is any complex number and $iz^{3}+z^{2}-z+i=0$, where $i=\sqrt{-1}$, then what is the value of $(|z|+1)^{2}$ ?

What is the sum of all four digit numbers formed by using all digits $0,1,4,5$ without repetition of digits ?

If $x,y$ and $z$ are the cube roots of unity, then what is the value of $xy+yz+zx$ ?

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 invite $3$ women and $3$ men so that $3$ of them are man's relatives and $3$ of them are his wife's relatives ?

A triangle $PQR$ is such that $3$ points lie on the side $PQ$, $4$ points on $QR$ and $5$ points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed ?

If $\log _{b}a=p$, $\log _{d}c=2p$ and $\log _{f}e=3p$, then what is $(ace)^{\frac{1}{p}}$ equal to ?

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 $a_{0},a_{1},a_{2},a_{3}$ are integers, then which one of the following is correct ?

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 $Re\left( \frac{z_{1}}{z_{2}} \right)+1$ equal to ?

If $26!=n8^{k}$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$ ?

Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$ :

1. $\text{adj}(AB)=(\text{adj}A)(\text{adj}B)$

2. $\text{adj}(AB)=\text{adj}(BA)$

3. $(AB)\text{adj}(AB)-|AB|I_{n}$ is a null matrix of order $n$

How many of the above statements are correct ?

Consider the following statements in respect of a non-singular matrix $A$ of order $n$ :

1. $A(\text{adj}A^{T})=A(\text{adj}A)^{T}$

2. If $A^{2}=A$, then $A$ is identity matrix of order $n$

3. If $A^{3}=A$, then $A$ is identity matrix of order $n$

Which of the statements given above are correct ?

How many four-digit natural numbers are there such that all of the digits are even ?

If $\omega\ne 1$ is a cube root of unity, then what are the solutions of $(z-100)^{3}+1000=0$ ?

What is $(1+i)^{4}+(1-i)^{4}$ equal to, where $i=\sqrt{-1}$ ?

Consider the following statements in respect of a skew-symmetric matrix $A$ of order $3$ :

1. All diagonal elements are zero.

2. The sum of all the diagonal elements of the matrix is zero.

3. $A$ is orthogonal matrix.

Which of the statements given above are correct ?

Four digit numbers are formed by using the digits $1,2,3,5$ without repetition of digits. How many of them are divisible by $4$ ?

What is the remainder when $2^{120}$ is divided by $7$ ?

For what value of $n$ is the determinant

$\left| \begin{matrix} C(9,4) & C(9,3) & C(10,n-2) \\ C(11,6) & C(11,5) & C(12,n) \\ C(m,7) & C(m,6) & C(m+1,n+1) \end{matrix} \right|=0$

for every $m>n$ ?

If $ABC$ is a triangle, then what is the value of the determinant

$\left| \begin{matrix} \cos C & \sin B & 0 \\ \tan A & 0 & \sin B \\ 0 & \tan (B+C) & \cos C \end{matrix} \right|$ ?

What is the number of different matrices, each having $4$ entries that can be formed using $1,2,3,4$ (repetition is allowed) ?

Let $A=$ {$x\in R:-1<x<1$}. Which of the following is/are bijective functions from $A$ to itself ?

1. $f(x)=x|x|$

2. $g(x)=\cos (\pi x)$

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Let $R$ be a relation on the open interval $(-1,1)$ and is given by $R=$ {$(x,y):|x+y|<2$}. Then which one of the following is correct ?

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 ?

If $a,b,c$ are the sides of triangle $ABC$, then what is

$\left| \begin{matrix} a^{2} & b\sin A & c\sin A \\ b\sin A & 1 & \cos A \\ c\sin A & \cos A & 1 \end{matrix} \right|$

equal to ?

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. $a,c$ and $e$ are in GP

2. $\frac{1}{a},\frac{1}{c},\frac{1}{e}$ are in GP

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What is the number of solutions of $\log _{4}(x-1)=\log _{2}(x-3)$ ?

For $x\ge y>1$, let $\log _{x}\left( \frac{x}{y} \right)+\log _{y}\left( \frac{y}{x} \right)=k$, then the value of $k$ can never be equal to

If $A=\begin{bmatrix} \sin 2\theta & 2\sin ^{2}\theta-1 & 0 \\ \cos 2\theta & 2\sin \theta\cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then which of the following statements is/are correct ?

1. $A^{-1}=\text{adj}A$

2. $A$ is skew-symmetric matrix

3. $A^{-1}=A^{T}$

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What is the coefficient of $x^{10}$ in the expansion of $(1-x^{2})^{20}(2-x^{2}-\frac{1}{x^{2}})^{-5}$ ?

If the 4th term in the expansion of $(mx+\frac{1}{x})^{n}$ is $\frac{5}{2}$, then what is the value of $mn$ ?

If $a,b$ and $c$ $(a>0,c>0)$ are in GP, then consider the following in respect of the equation $ax^{2}+bx+c=0$ :

1. The equation has imaginary roots.

2. The ratio of the roots of the equation is $1:\omega$ where $\omega$ is a cube root of unity.

3. The product of roots of the equation is $\left( \frac{b^{2}}{a^{2}} \right)$.

Which of the statements given above are correct ?

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 be an integer

2. $n$ must be an integer

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In a binomial expansion of $(x+y)^{2n+1}(x-y)^{2n+1}$, the sum of middle terms is zero. What is the value of $\left( \frac{x^{2}}{y^{2}} \right)$ ?

Let $A=$ {$1,2,3,4,5$} and $B=$ {$6,7$}. What is the number of onto functions from $A$ to $B$ ?

What is $\frac{\sqrt{3}\cos 10^{\circ}-\sin 10^{\circ}}{\sin 25^{\circ}\cos 25^{\circ}}$ equal to ?

What is $(\sin 9^{\circ}-\cos 9^{\circ})$ equal to ?

If in a triangle $ABC$, $\sin ^{3}A+\sin ^{3}B+\sin ^{3}C=3\sin A\sin B\sin C$, then what is the value of the determinant

$\left| \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right|$;

where $a,b,c$ are sides of the triangle ?

If $\cos ^{-1}x=\sin ^{-1}x$, then which one of the following is correct ?

What is the number of solutions of $(\sin \theta-\cos \theta)^{2}=2$ where $-\pi<\theta<\pi$ ?

$ABC$ is a triangle such that angle $C=60^{\circ}$, then what is $\frac{\cos A+\cos B}{\cos \left( \frac{A-B}{2} \right)}$ equal to ?

What is $\sqrt{15+\cot ^{2}\left( \frac{\pi}{4}-2\cot ^{-1}3 \right)}$ equal to ?

What is the value of $\sin 10^{\circ}\cdot \sin 50^{\circ}+\sin 50^{\circ}\cdot \sin 250^{\circ}+\sin 250^{\circ}\cdot \sin 10^{\circ}$ equal to ?

What is $\tan ^{-1}\left( \frac{a}{b} \right)-\tan ^{-1}\left( \frac{a-b}{a+b} \right)$ equal to ?

Under which one of the following conditions does the equation $(\cos \beta-1)x^{2}+(\cos \beta)x+\sin \beta=0$ in $x$ have a real root for $\beta\in [0,\pi]$ ?

In a triangle $ABC$, $AB=16\text{ cm}$, $BC=63\text{ cm}$ and $AC=65\text{ cm}$. What is the value of $\cos 2A+\cos 2B+\cos 2C$ ?

If $f(\theta)=\frac{1}{1+\tan \theta}$ and $\alpha+\beta=\frac{5\pi}{4}$, then what is the value of $f(\alpha)f(\beta)$ ?

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^{2}-6x+8=0$, then what is the value of $\cos (2\alpha+2\beta)$ ?

What is the value of $\tan 65^{\circ}+2\tan 45^{\circ}-2\tan 40^{\circ}-\tan 25^{\circ}$ ?

Consider the following statements :

1. In a triangle $ABC$, if $\cot A\cdot \cot B\cdot \cot C>0$, then the triangle is an acute angled triangle.

2. In a triangle $ABC$, if $\tan A\cdot \tan B\cdot \tan C>0$, then the triangle is an obtuse angled triangle.

Which of the statements given above is/are correct ?

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 $a^{2}+b^{2}+c^{2}$ ?

If $(1,-1,2)$ and $(2,1,-1)$ are the end points of a diameter of a sphere $x^{2}+y^{2}+z^{2}+2ux+2vy+2wz-1=0$, then what is $u+v+w$ equal to ?

The number of points represented by the equation $x=5$ on the $xy$-plane is

If $⟨l,m,n⟩$ are the direction cosines of a normal to the plane $2x-3y+6z+4=0$, then what is the value of $49(7l^{2}+m^{2}-n^{2})$ ?

A line through $(1,-1,2)$ with direction ratios $⟨3,2,2⟩$ meets the plane $x+2y+3z=18$. What is the point of intersection of line and plane ?

If $p$ is the perpendicular distance from origin to the plane passing through $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, then what is $3p^{2}$ equal to ?

If the direction cosines $⟨l,m,n⟩$ of a line are connected by relation $l+2m+n=0$, $2l-2m+3n=0$, then what is the value of $l^{2}+m^{2}-n^{2}$ ?

If a variable line passes through the point of intersection of the lines $x+2y-1=0$ and $2x-y-1=0$ and meets the coordinate axes in $A$ and $B$, then what is the locus of the mid-point of $AB$ ?

What is the equation to the straight line passing through the point $(-\sin \theta,\cos \theta)$ and perpendicular to the line $x\cos \theta+y\sin \theta=9$ ?

Two points $P$ and $Q$ lie on line $y=2x+3$. These two points $P$ and $Q$ are at a distance $2\text{ units}$ from another point $R(1,5)$. What are the coordinates of the points $P$ and $Q$ ?

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 square units ?

$ABC$ is a triangle with $A(3,5)$. The mid-points of sides $AB,AC$ are at $(-1,2),(6,4)$ respectively. What are the coordinates of centroid of the triangle $ABC$ ?

$ABC$ is an acute angled isosceles triangle. Two equal sides $AB$ and $AC$ lie on the lines $7x-y-3=0$ and $x+y-5=0$. If $\theta$ is one of the equal angles, then what is $\cot \theta$ equal to ?

In the parabola $y^{2}=8x$, the focal distance of a point $P$ lying on it is $8\text{ units}$. Which of the following statements is/are correct ?

1. The coordinates of $P$ can be $(6,4\sqrt{3})$.

2. The perpendicular distance of $P$ from the directrix of parabola is $8\text{ units}$.

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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 minor axis is $90^{\circ}$ ?

Let $\overrightarrow{a}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$. If $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{a})=\alpha\hat{i}-\beta\hat{j}+\gamma\hat{k}$, then what is the value of $\alpha+\beta+\gamma$ ?

If a vector of magnitude $2\text{ units}$ makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector ?

Consider the following in respect of moment of a force :

1. The moment of force about a point is independent of point of application of force.

2. The moment of a force about a line is a vector quantity.

Which of the statements given above is/are correct ?

For any vector $\overrightarrow{r}$, what is $(\overrightarrow{r}\cdot \hat{i})(\overrightarrow{r}\times \hat{i})+(\overrightarrow{r}\cdot \hat{j})(\overrightarrow{r}\times \hat{j})+(\overrightarrow{r}\cdot \hat{k})(\overrightarrow{r}\times \hat{k})$ equal to ?

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors of magnitude $4$ inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\overrightarrow{a}$ and $\overrightarrow{a}-\overrightarrow{b}$ ?

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 $y_{2}(0)=4$, then what is the number of points of intersection of the curves $y_{1}(x)$ and $y_{2}(x)$ ?

The differential equation, representing the curve $y=e^{x}(a\cos x+b\sin x)$ where $a$ and $b$ are arbitrary constants, is