NDA 2 2025 Maths Question Paper with Solutions

If $p^{x}=q^{y}=r^{z}$, where $x,y$ and $z$ are in GP, then consider the following statements :

I. $p,q$ and $r$ are in AP.

II. $\ln p,\ln q$ and $\ln r$ are in GP.

Which of the statements given above is/are correct?

If $A$ and $B$ are non-empty subsets of a set, and $A^{c}$ and $B^{c}$ represent their complements, then which of the following is/are correct?

I. $A-B=B^{c}-A^{c}$

II. $A-B^{c}=A^{c}-B$

Select the answer using the code given below.

Let $y=x!$ and $z=(2x)!$. If $(z/y)=120$, then what is the value of $(3x)!$?

Let $n$ be a natural number. The number of consecutive zeros at the end of the expansion of $n!$ is exactly $2$. How many values of $n$ are possible?

If $(10+\log _{10}x)$, $(10+\log _{10}y)$ and $(10+\log _{10}z)$ are in AP, then consider the following statements :

I. The GM of $x$ and $z$ is $y^{2}$.

II. The AM of $\log _{10}x$ and $\log _{10}z$ is $\log _{10}y$.

Which of the statements given above is/are correct?

How many terms of the series $1+3+5+7+\ldots$ amount to a sum equal to $12345678987654321$?

How many terms are identical in the two APs $19,21,23,\ldots$ up to $110$ terms and $19,22,25,28,\ldots$ up to $75$ terms?

If

$\alpha=\frac{-1+\sqrt{-3}}{2}$

then what is the value of $(1+\alpha^{19}-\alpha^{35})^{100}-(1-3\alpha^{25}+\alpha^{38})^{50}$?

What is the remainder when $5^{99}$ is divided by $13$?

What is the value of the determinant of the inverse of the matrix

$\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?

In a class of $45$ students, $34$ like to play cricket and $26$ like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

The system of equations

$2x-3y-5=0, 15y-10x+50=0$

If

$(\frac{1-i}{1+i})^{2m}(\frac{1+i}{1-i})^{2n}=1$

where $i=\sqrt{-1}$, then what is the smallest positive value of $(m-n)$?

In obtaining the solution of the system of equations $x+y+z=7$, $x+2y+3z=16$ and $x+3y+4z=22$ by Cramer’s rule, the value of $y$ is obtained by dividing $D$ by $D_{2}$, where

$D=\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{matrix} \right|$

What is the value of the determinant $D_{2}$?

Consider the following in respect of non-singular matrices $A$ and $B$ :

I. $(AB)^{-1}=A^{-1}B^{-1}$

II. $(BA)(AB)^{-1}=I$, where $I$ is the identity matrix

III. $(AB)^{T}=A^{T}B^{T}$

How many of the above are correct?

The value of the determinant

$\left| \begin{matrix} a & b & c \\ l & m & n \\ p & q & r \end{matrix} \right|$

is equal to

Let $1,\omega,\omega^{2}$ be three cube roots of unity. If $x=a+b$, $y=a\omega+b\omega^{2}$, $z=a\omega^{2}+b\omega$, then what is $x^{2}+y^{2}+z^{2}$ equal to?

How many 4-digit numbers that are divisible by $4$ can be formed using the digits $1,2,3$ and $4$ (repetition of digits is not allowed)?

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

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

equal to?

Which one of the following is the greatest coefficient in the expansion of $(1+x)^{100}$?

What is $\beta$ equal to?

What is the relation between $\alpha$ and $\beta$?

If $p+q=66$, then which one of the following is correct?

If $p+q=15$, then what is $q-p$ equal to?

What is $\frac{p}{q}$ equal to?

What is $\frac{p^{2}-q^{2}}{p^{2}+q^{2}}$ equal to?

What is $\left( \frac{\sqrt{3}p}{4}-\frac{q}{4} \right)$ equal to?

What is $\frac{p^{2}+q^{2}}{p^{2}q^{2}}$ equal to?

How many values does $(x+y)$ have?

How many values does $(y-x)$ have?

If the roots of the equation are equal, then which one of the following is correct?

If the roots of the equation are equal, then $a,b,c$ are in

What is the relation between $m$ and $n$?

How many values of $m$ are possible?

How many triangles can be formed by joining these points?

How many quadrilaterals can be formed by joining these points?

What is the other root of $f(x)=0$?

What is $(a+b+c)$ equal to?

What is the value of the determinant of the matrix $A^{4}$?

What is $[\text{adj}A]^{-1}$ equal to?

What is the sum of the binary numbers $(101101101)_{2}$ and $(100011)_{2}$?

Set $X$ contains $3n$ elements and set $Y$ contains $2n$ elements, and they have $n$ elements in common. How many elements does $(X-Y)\times (Y-X)$ have?

Let $A=$ {$-3,-2,-1,0,1,2,3$} and $B=$ {$0,1,4,9$}. How many elements does the subset of $A\times B$ corresponding to the relation $R=$ {$(x,y):|x|<y$} have, where $x\in A$ and $y\in B$?

Consider the following statements :

Statement-I : If $X$ is an $n\times n$ matrix, then $\text{det}(mX)=m^{n}\text{det}(X)$, where $m$ is a scalar.

Statement-II : If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\text{det}(Y)=m\text{det}(X)$.

Which one of the following is correct in respect of the above statements?

Consider the following statements about the matrix

$M=\begin{bmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{bmatrix}$ :

Statement-I : The inverse of $M$ does not exist.

Statement-II : $M$ is non-singular.

Which one of the following is correct in respect of the above statements?

What is $\cot ^{-1}9+\operatorname{cosec} ^{-1}\left( \frac{\sqrt{41}}{4} \right)$ equal to?

How many values of $\theta$, where $-\pi<\theta<\pi$, satisfy both the equations $\cot \theta=-\sqrt{3}$ and $\operatorname{cosec} \theta=-2$ simultaneously?

If $x+\frac{1}{x}=2\cos \theta$, then what is $x^{3}+\frac{1}{x^{3}}$ equal to?

If $0\le x\le \frac{\pi}{2}$, then what is the number of values of $x$ satisfying the equation $\tan x+\sec x=2\cos x$?

What is the value of $\tan \left\lbrack \frac{1}{2}\sec ^{-1}(\frac{2}{\sqrt{3}}) \right]$?

Which of the following are the direction ratios of the line of intersection of the given planes?

What is the equation of the plane $P$?

What is the radius of $S$?

On which one of the following planes does the centre of $S$ lie?

What is the fourth vertex $D$?

If angle $BCD$ is $\theta$, then what is $\cos ^{2}\theta$ equal to?

For different values of $m$, the equation $4y=mx-m+2$ represents

The equation of the locus of a point equidistant from the points $(a,b)$ and $(c,d)$ is $(a-c)x+(b-d)y+k=0$. What is the value of $k$?

Consider the following statements in respect of the equation $x^{2}+3y=0$ :

I. The equation represents the equation to parabola that opens upwards.

II. The axis of the parabola is $x=0$.

III. The equation of the latus rectum is $4y-3=0$.

How many of the statements given above are correct?

What is the sum of the intercepts of the line

$\frac{x}{a^{2}}+\frac{y}{b^{2}}=\frac{2}{a^{2}+b^{2}}$

on the coordinate axes?

Which one of the following is the perpendicular form of the straight line $\sqrt{3}x+2y=7$?

If the vertices $B$ and $D$ of a square $ABCD$ are $(2,3)$ and $(4,1)$ respectively, then what is the area of the square?

What is the value of $\sin \theta$ if $\theta$ is the acute angle between the lines whose equations are $px+qy=p+q$ and $p(x-y)+q(x+y)=2q$?

The circle $x^{2}+y^{2}-2kx-2ky+k^{2}=0$ touches the $x$-axis at $P$ and $y$-axis at $Q$. What is $PQ$ equal to?

What is the distance between the foci of the hyperbola $x^{2}-4y^{2}=1$?

Let $\overrightarrow{p}=\overrightarrow{a}-\overrightarrow{b}$, $\overrightarrow{q}=\overrightarrow{a}+\overrightarrow{b}$. If $|\overrightarrow{a}|=|\overrightarrow{b}|=2$ and $\overrightarrow{a}\cdot \overrightarrow{b}=2$, then what is the value of $|\overrightarrow{p}\times \overrightarrow{q}|$?

How many of the following can be a vector perpendicular to both the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+3\hat{k}$?

I. $4\hat{i}+5\hat{j}-3\hat{k}$

II. $-8\hat{i}-10\hat{j}+6\hat{k}$

III. $\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})$

Select the correct answer.

What is the area of the parallelogram whose sides are represented by the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$?

The position vectors of the vertices $A,B,C$ and $D$ of a quadrilateral $ABCD$ are given by $3\hat{i}+4\hat{j}-2\hat{k}$, $4\hat{i}-4\hat{j}-3\hat{k}$, $2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively. What is the angle between the diagonals $AC$ and $BD$ of the quadrilateral?

A force $\overrightarrow{F}=2\hat{i}-\lambda\hat{j}+5\hat{k}$ is applied at the point $A(1,2,5)$. If its moment about the point $B(-1,-2,3)$ is $16\hat{i}-6\hat{j}+2\lambda\hat{k}$, then what is the value of $\lambda$?

What is $\underset{x\to 0-}{\lim }f(x)$ equal to?

What is $\underset{x\to 0+}{\lim }f(x)$ equal to?

What is $\underset{x\to -1}{\lim }f(x)$ equal to?

What is the area bounded by the curve $f(x)$, the $x$-axis and the lines $x=-2$ and $x=1$?