NDA 1 2025 Maths Question Paper with Solutions

The sum of the first $k$ terms of a series $S$ is $3k^{2} + 5k$. Which one of the following is correct?

The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If $r \neq 1$ is the common ratio, then what is the number of possible real values of $r$?

If one root of the equation $x^{2} - kx + k = 0$ exceeds the other by $2\sqrt{3}$, then which one of the following is a value of $k$?

If $x + \frac{5}{y} = 4$ and $y + \frac{5}{x} = - 4$, then what is $(x + y)$ equal to?

If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

If $p,1,q$ are in AP and $p,2,q$ are in GP, then which of the following statements is/are correct?

I. $p,4,q$ are in HP.

II. $(1/p),1/4,(1/q)$ are in AP.

Select the answer using the code given below.

If $x = (1111)_{2}$, $y = (1001)_{2}$ and $z = (110)_{2}$, then what is $x^{3} - y^{3} - z^{3} - 3xyz$ equal to?

If $\Delta = \left| \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right|$ and $A,B,C,D,G$ are the cofactors of the elements $a,b,c,d,g$ respectively, then what is $bB + cC - dD - gG$ equal to?

Consider the following statements in respect of the determinant $\Delta = \left| \begin{matrix} k(k + 2) & 2k + 1 & 1 \\ 2k + 1 & k + 2 & 1 \\ 3 & 3 & 1 \end{matrix} \right|$.

I. $\Delta$ is positive if $k > 0$.

II. $\Delta$ is negative if $k < 0$.

III. $\Delta$ is zero if $k = 0$.

How many of the statements given above are correct?

If $\left| \begin{matrix} 2 & 3 + i & - 1 \\ 3 - i & 0 & i - 1 \\ - 1 & - 1 - i & 1 \end{matrix} \right| = A + iB$, where $i = \sqrt{- 1}$, then what is $A + B$ equal to?

If $A^{2} + B^{2} + C^{2} = 0$, then what is the value of the following? $\left| \begin{matrix} 1 & \cos C & \cos B \\ \cos C & 1 & \cos A \\ \cos B & \cos A & 1 \end{matrix} \right|$

If $\omega$ is a non-real cube root of unity, then what is a root of the following equation? $\left| \begin{matrix} x + 1 & \omega & \omega^{2} \\ \omega & x + \omega^{2} & 1 \\ \omega^{2} & 1 & x + \omega \end{matrix} \right| = 0$

What is $\left( \frac{\sqrt{3} + i}{\sqrt{3} - i} \right)^{3}$ equal to?

If $x^{2} - x + 1 = 0$, then what is $\left( x - \frac{1}{x} \right)^{2} + \left( x - \frac{1}{x} \right)^{4} + \left( x - \frac{1}{x} \right)^{8}$ equal to?

How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all consonants come together in each word?

If $z \neq 0$ is a complex number, then what is $amp(z) + amp\left( \overline{z} \right)$ equal to?

How many sides are there in a polygon which has 20 diagonals?

In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?

What is the number of positive integer solutions of $x + y + z = 5$?

What is the number of rational terms in the expansion of $\left( 3^{1/2} + 5^{1/4} \right)^{12}$?

If the sum of binomial coefficients in the expansion of $(x + y)^{n}$ is 256, then the greatest binomial coefficient occurs in which one of the following terms?

If $k < \left( \sqrt{2} + 1 \right)^{3} < k + 2$, where $k$ is a natural number, then what is the value of $k$?

If $\begin{bmatrix} x & 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}\begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} = \begin{bmatrix} 45 \end{bmatrix}$, then which one of the following is a value of $x$?

If $A = \begin{bmatrix} y & z & x \\ z & x & y \\ x & y & z \end{bmatrix}$, where $x,y,z$ are integers, is an orthogonal matrix, then what is the value of $x^{2} + y^{2} + z^{2}$?

Consider the following in respect of a non-singular matrix $M$:

I. $\left| M^{2} \right| = |M|^{2}$

II. $|M| = \left| M^{- 1} \right|$

III. $|M| = \left| M^{T} \right|$

How many of the above are correct?

If $f(\theta) = \begin{bmatrix} \cos\theta & \sin\theta \\ - \sin\theta & \cos\theta \end{bmatrix}$, then what is $\{ f(\pi)\}^{2}$ equal to?

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then what is $A^{2} - 4A$ equal to? where $I_{3}$ is the identity matrix of order 3.

If the number of selections of $r$ as well as $(n + r)$ things from $5n$ different things are equal, then what is the value of $r$?

What is the number of selections of at most 3 things from 6 different things?

If $A = \begin{bmatrix} x & y & z \\ y & z & x \\ z & x & y \end{bmatrix}$, where $x,y,z$ are integers, is an orthogonal matrix, then what is $A^{2}$ equal to?

What is $(p + q + r)$ equal to?

What is $(pq + qr + rp)$ equal to?

What is $\left( p^{2} + q^{2} + r^{2} \right)$ equal to?

What is the minimum value of $p$?

What is the maximum value of $p$?

Consider the following statements:

I. The triangle is obtuse-angled triangle.

II. The sum of acute angles of the triangle is also acute.

Which of the statements given above is/are correct?

What is $\angle B$ equal to?

What is the area of the triangle?

What is $PN$ equal to?

What is $MN$ equal to?

What is $(p/q)$ equal to?

What is $(p + q)$ equal to?

What is $\tan^{2}\alpha$ equal to?

What is $\tan\alpha$ equal to?

What is $2\sin 2\alpha + \cos 2\alpha$ equal to?

One of the angles of the triangle is

Consider the following statements: I. The triangle is right-angled. II. One of the sides of the triangle is 3 times the other. III. The angles $A$, $C$ and $B$ of the triangle are in AP. Which of the statements given above is/are correct?

A man at $M$, standing 100 m away from the base $(P)$ of a chimney of height 50 m, observes the angle of elevation of the highest point $(Q)$ of the smoke to be $45^{\circ}$. The highest point of the chimney is at $R$. Further $P$, $R$ and $Q$ are in a straight line and the straight line is perpendicular to $PM$. What is the angle $RMQ$ equal to?

If $k$ is a root of $x^{2} - 4x + 1 = 0$, then what is $\tan^{- 1}k + \tan^{- 1}(1/k)$ equal to?

If $\tan^{- 1}k + \tan^{- 1}(1/2) = \pi/4$, then what is the value of $k$?

Under what condition will the lines $m^{2}x + ny - 1 = 0$ and $n^{2}x - my + 2 = 0$ be perpendicular?

If $p$ and $q$ are real numbers between 0 and 1 such that the points $(p,1)$, $(1,q)$ and $(0,0)$ form an equilateral triangle, then what is $(p + q)$ equal to?

The vertices of a triangle are $A(1,1)$, $B(0,0)$ and $C(2,0)$. The angular bisectors of the triangle meet at $P$. What are the coordinates of $P$?

Let $A(3, - 1)$ and $B(1,1)$ be the end points of line segment $AB$. Let $P$ be the middle point of the line segment $AB$. Let $Q$ be the point situated at a distance $\sqrt{2}$ units from $P$ on the perpendicular bisector line of $AB$. What are the possible coordinates of $Q$?

$ABC$ is an equilateral triangle and $AD$ is the altitude on $BC$. If the coordinates of $A$ are $(1,2)$ and that of $D$ are $( - 2,6)$, then what is the equation of $BC$?

What is the equation of the circle whose diameter is 10 cm and the equations of two of its diameters are $x + y = 0$ and $x - y = 0$?

A square is inscribed in a circle $x^{2} + y^{2} + 2x + 2y + 1 = 0$ and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

A tangent to the parabola $y^{2} = 4x$ is inclined at an angle $45^{\circ}$ with the positive direction of $x$-axis. What is the point of contact of the tangent and the parabola?

What is the distance between the two foci of the hyperbola $25x^{2} - 75y^{2} = 225$?

If any point on an ellipse is $\left( 3\sin\alpha,5\cos\alpha \right)$, then what is the eccentricity of the ellipse?

If a line in 3 dimensions makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes, then what is $\cos(\alpha + \beta)\cos(\alpha - \beta)$ equal to?

$A(1,2, - 1)$, $B(2,5, - 2)$ and $C(4,4, - 3)$ are three vertices of a rectangle. What is the area of the rectangle?

$ABC$ is a triangle right-angled at $B$. If $A(k,1, - 1)$, $B(2k,0,2)$ and $C(2 + 2k,k,1)$ are the vertices of the triangle, then what is the value of $k$?

If a line $\frac{x + 1}{p} = \frac{y - 1}{q} = \frac{z - 2}{r}$, where $p = 2q = 3r$, makes an angle $\theta$ with the positive direction of $y$-axis, then what is $\cos 2\theta$ equal to?

What is the equation of the plane passing through the point $(1,1,1)$ and perpendicular to the line whose direction ratios are $(3,2,1)$?

A line makes angles $\alpha$, $\beta$ and $\gamma$ with the positive directions of the coordinate axes. If $\overrightarrow{a} = \left( \sin^{2}\alpha \right)\widehat{i} + \left( \sin^{2}\beta \right)\widehat{j} + \left( \sin^{2}\gamma \right)\widehat{k}$ and $\overrightarrow{b} = \widehat{i} + \widehat{j} + \widehat{k}$, then what is $\overrightarrow{a} \cdot \overrightarrow{b}$ equal to?

Consider the following statements in respect of a vector $\overrightarrow{d} = \left( \overrightarrow{a} \times \overrightarrow{b} \right) \times \overrightarrow{c}$: I. $\overrightarrow{d}$ is coplanar with $\overrightarrow{a}$ and $\overrightarrow{b}$. II. $\overrightarrow{d}$ is perpendicular to $\overrightarrow{c}$. Which of the statements given above is/are correct?

The position vectors of three points $A$, $B$ and $C$ are $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ respectively such that $3\overrightarrow{a} - 4\overrightarrow{b} + \overrightarrow{c} = \overrightarrow{0}$. What is $AB:BC$ equal to?

The position vectors of three points $A$, $B$ and $C$ are $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ respectively, where

$\overrightarrow{c} = \left( \cos^{2}\theta \right)\overrightarrow{a} + \left( \sin^{2}\theta \right)\overrightarrow{b}$. What is $\left( \overrightarrow{a} \times \overrightarrow{b} \right) + \left( \overrightarrow{b} \times \overrightarrow{c} \right) + \left( \overrightarrow{c} \times \overrightarrow{a} \right)$ equal to?

Let $\overrightarrow{a}$, $\overrightarrow{b}$, $\left( \overrightarrow{a} \times \overrightarrow{b} \right)$ be unit vectors. What is $\left( \overrightarrow{a} \cdot \overrightarrow{b} \right)$ equal to?

What is $\left( \frac{dy}{dx} \right)^{2}$ equal to?

What is $\left( \frac{x^{2} + 4}{y^{2} + 4} \right)\frac{dy}{dx}\left\lbrack \left( x^{2} + 4 \right)\frac{d^{2}y}{dx^{2}} - 16y \right\rbrack$ equal to?

What is $\angle A$ equal to if the area of the triangle is maximum?

What is the maximum area of the triangle?

The derivative of $y$ with respect to $x$

If $p + q = 10$, then what is $\frac{dy}{dx}$ equal to?

What is the nature of the curve?

What is the area bounded by the curve, the $x$-axis and the line $x = 4$?

What is $\lim_{x \rightarrow 0}{f\prime(x)}$ equal to?

Consider the following statements: I. The function is continuous at $x = - 1$. II. The function is differentiable at $x = 1$. Which of the statements given above is/are correct?

What is the range of the function?