NDA 2 2024 Maths Question Paper with Solutions

Let $X$ be a matrix of order $3\times 3$, $Y$ be a matrix of order $2\times 3$ and $Z$ be a matrix of order $3\times 2$. Which of the following statements are correct?

I. $(ZY)X$ is defined and is a square matrix of order $3$.

II. $Y(XZ)$ is defined and is a square matrix of order $2$.

III. $X(YZ)$ is not defined.

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Consider the following statements :

I. The set of all irrational numbers between $\sqrt{12}$ and $\sqrt{15}$ is an infinite set.

II. The set of all odd integers less than $1000$ is a finite set.

Which of the statements given above is/are correct?

How many 4-digit numbers are there having all digits as odd?

If $\omega\ne 1$ is a cube root of unity, then what is $(1+\omega-\omega^{2})^{100}+(1-\omega+\omega^{2})^{100}$ equal to?

Let $A$ and $B$ be two square matrices of same order. If $AB$ is a null matrix, then which one of the following is correct?

In the expansion of $(1+x)^{p}(1+x)^{q}$, if the coefficient of $x^{3}$ is $35$, then what is the value of $(p+q)$?

If $p$ times the $pth$ term of an AP is equal to $q$ times the $qth$ term $(p\ne q)$, then what is the $(p+q)th$ term equal to?

Let $p=\ln (x)$, $q=\ln (x^{3})$ and $r=\ln (x^{5})$, where $x>1$. Which of the following statements is/are correct?

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

II. $p,q$ and $r$ can never be in GP.

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If

$Z=\frac{1}{3}\left| \begin{matrix} i & 2i & 1 \\ 2i & 3i & 2 \\ 3 & 1 & 3 \end{matrix} \right|=x+iy;\ i=\sqrt{-1}$

then what is modulus of $Z$ equal to?

What is the value of the sum

$\sum _{n=1}^{20}(i^{n-1}+i^{n}+i^{n+1})$

where $i=\sqrt{-1}$?

Let $x>1,y>1,z>1$ be in GP. Then

$\frac{1}{1+\ln x},\ \frac{1}{1+\ln y},\ \frac{1}{1+\ln z}$

are

If

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

then what is

$\left| \begin{matrix} 1+\omega & 1+\omega^{2} & \omega+\omega^{2} \\ 1 & \omega & \omega^{2} \\ \frac{1}{\omega} & \frac{1}{\omega^{2}} & 1 \end{matrix} \right|$

equal to?

If the sum of the first $n$ terms of a series is $n(2n+1)$, then what is the $nth$ term?

In how many ways can the letters of the word INDIA be permutated such that in each combination, vowels should occupy odd positions?

The letters of the word EQUATION are arranged in such a way that all vowels as well as consonants are together. How many such arrangements are there?

If $n$ is a root of the equation $x^{2}+px+m=0$ and $m$ is a root of the equation $x^{2}+px+n=0$, where $m\ne n$, then what is the value of $p+m+n$?

In how many ways can a student choose $(n-2)$ courses out of $n$ courses if $2$ courses are compulsory $(n>4)$?

If

$D_{n}=\left| \begin{matrix} n & 20 & 30 \\ n^{2} & 40 & 50 \\ n^{3} & 60 & 70 \end{matrix} \right|$

then what is the value of $\sum _{n=1}^{4}D_{n}$?

Consider the following in respect of the matrices

$P=\begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}$ and $Q=\begin{bmatrix} a^{2} & ab & ac \\ ab & b^{2} & bc \\ ac & bc & c^{2} \end{bmatrix}$

I. $PQ$ is a null matrix.

II. $QP$ is an identity matrix of order $3$.

III. $PQ=QP$

Which of the above is/are correct?

If $P$ is a skew-symmetric matrix of order $3$, then what is $\text{det}(P)$ equal to?

If $4\sin ^{-1}x+\cos ^{-1}x=\pi$, then what is $\sin ^{-1}x+4\cos ^{-1}x$ equal to?

What is $\cot ^{2}(\sec ^{-1}2)+\tan ^{2}(\operatorname{cosec} ^{-1}3)$ equal to?

In a triangle $ABC$

$\frac{a}{\cos A}=\frac{b}{\cos B}=\frac{c}{\cos C}$

What is the area of the triangle if $a=6\text{ cm}$?

The roots of the equation $7x^{2}-6x+1=0$ are $\tan \alpha$ and $\tan \beta$, where $2\alpha$ and $2\beta$ are the angles of a triangle. Which one of the following is correct?

In a triangle $ABC$, $\angle A=75^{\circ}$ and $\angle B=45^{\circ}$. What is $2a-b$ equal to?

What is the number of solutions of the equation $\cot 2x\cdot \cot 3x=1$ for $0<x<\pi$?

What is the general solution of $\cos ^{100}x-\sin ^{100}x=1$?

where $n$ is an integer.

In a triangle $ABC$

$\tan A+\tan B+\tan C=k$

What is the value of $\cot A\cot B\cot C$?

What is $\sin 12^{\circ}\sin 48^{\circ}$ equal to?

What is $\frac{\cos 17^{\circ}-\sin 17^{\circ}}{\cos 17^{\circ}+\sin 17^{\circ}}$ equal to?

Consider the following numbers :

I. $\tan 22.5^{\circ}$

II. $\cot 22.5^{\circ}$

III. $\tan 22.5^{\circ}-\cot 22.5^{\circ}$

How many of the above are irrational numbers?

If

$\frac{x}{\cos \theta}=\frac{y}{\cos \left( \frac{2\pi}{3}-\theta \right)}=\frac{z}{\cos \left( \frac{2\pi}{3}+\theta \right)}$

then what is $x+y+z$ equal to?

If $p\tan (\theta-30^{\circ})=q\tan (\theta+120^{\circ})$, then what is $(p+q)/(p-q)$ equal to?

Let $P$ and $Q$ be two non-void relations on a set $A$. Which of the following statements are correct?

I. $P$ and $Q$ are reflexive $⇒P\cap Q$ is reflexive.

II. $P$ and $Q$ are symmetric $⇒P\cup Q$ is symmetric.

III. $P$ and $Q$ are transitive $⇒P\cap Q$ is transitive.

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If $A$ and $B$ are two non-empty sets having $10$ elements in common, then how many elements do $A\times B$ and $B\times A$ have in common?

What is the remainder when $7^{n}-6n$ is divided by $36$ for $n=100$?

What is the maximum number of possible points of intersection of four straight lines and a circle (intersection is between lines as well as circle and lines)?

In an AP, the ratio of the sum of the first $p$ terms to the sum of the first $q$ terms is $p^{2}:q^{2}$. Which one of the following is correct?

What is the number of real roots of the equation $(x-1)^{2}+(x-3)^{2}+(x-5)^{2}=0$?

In a class of $240$ students, $180$ passed in English, $130$ passed in Hindi and $150$ passed in Sanskrit. Further, $60$ passed in only one subject, $110$ passed in only two subjects and $10$ passed in none of the subjects. How many passed in all three subjects?

What is the value of $\left| \frac{Z_{1}}{Z_{2}} \right|$?

What is the value of $\frac{1}{2}+Re\left( \frac{Z_{1}}{Z_{2}} \right)$?

What is the common difference?

What is the sum of all five terms?

What is $V+W$ equal to?

What is the value of $(U+V)W$?

Which one of the following statements is correct?

Which one of the following is a root of the equation?

What is $A(\text{adj}A)$ equal to?

What is $A^{-1}$ equal to?

What is $3\alpha+2\beta$ equal to if $(2\hat{i}+6\hat{j}+27\hat{k})\times (\hat{i}+\alpha\hat{j}+\beta\hat{k})$ is a null vector?

For what value of the angle between the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is the quantity $|\overrightarrow{a}\times \overrightarrow{b}|+\sqrt{3}|\overrightarrow{a}\cdot \overrightarrow{b}|$ maximum?

Let $\theta$ be the angle between two unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{a}+2\overrightarrow{b}$ is perpendicular to $5\overrightarrow{a}-4\overrightarrow{b}$, then what is $\cos \theta+\cos 2\theta$ equal to?

Let $ABCDEF$ be a regular hexagon. If $\overrightarrow{AD}=m\overrightarrow{BC}$ and $\overrightarrow{CF}=n\overrightarrow{AB}$, then what is $mn$ equal to?

The vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are of the same length. If taken pairwise, they form equal angles. If $\overrightarrow{a}=\hat{i}+\hat{j}$ and $\overrightarrow{b}=\hat{j}+\hat{k}$, then what can $\overrightarrow{c}$ be equal to?

I. $\hat{i}+\hat{k}$

II. $\frac{-\hat{i}+4\hat{j}-\hat{k}}{3}$

Select the correct answer using the code given below.

The diagonals of a quadrilateral $ABCD$ are along the lines $x-2y=1$ and $4x+2y=3$. The quadrilateral $ABCD$ may be a

The foci of the ellipse $4x^{2}+9y^{2}=1$ are at $Q$ and $R$. If $P(x,y)$ is any point on the ellipse, then what is $PQ+PR$ equal to?

If $P(2,4)$, $Q(8,12)$, $R(10,14)$ and $S(x,y)$ are vertices of a parallelogram, then what is $(x+y)$ equal to?

The equation of a circle is

$(x^{2}-4x+3)+(y^{2}-6y+8)=0$

Which of the following statements are correct?

I. The end points of a diameter of the circle are at $(1,2)$ and $(3,4)$.

II. The end points of a diameter of the circle are at $(1,4)$ and $(3,2)$.

III. The end points of a diameter of the circle are at $(2,4)$ and $(4,2)$.

Select the answer using the code given below.

Consider the points $P(4k,4k)$ and $Q(4k,-4k)$ lying on the parabola $y^{2}=4kx$. If the vertex is $A$, then what is $\angle PAQ$ equal to?

What is $\angle BAC$ equal to?

What are the coordinates of $A$?

What are the coordinates of vertex $D$?

What is the point of intersection of the diagonals of the trapezium?

What is the diameter of the sphere?

The centre of the sphere lies on the plane

Which of the following are the direction ratios of $S$?

If $⟨l,m,n⟩$ are direction cosines of $S$, then what is the value of $43(l^{2}-m^{2}-n^{2})$?

What are the direction ratios of the line?

What is the point of intersection of $L$ and $P$?

Let $z=[y]$ and $y=[x]-x$, where $[\cdot ]$ is the greatest integer function. If $x$ is not an integer but positive, then what is the value of $z$?

If $f(x)=4x+1$ and $g(x)=kx+2$ such that $f\circ g(x)=g\circ f(x)$, then what is the value of $k$?

What is the minimum value of the function $f(x)=\log _{10}(x^{2}+2x+11)$?

Which one of the following is correct regarding $\underset{x\to 3}{\lim }\frac{|x-3|}{x-3}$?