NDA Mathematics 2024 Question Paper

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 $|2B(\text{adj}(3A))|$?

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?

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

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?

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?

If
$$Z=\frac{1}{3}\left|\begin{array}{ccc}i& 2i& 1\\ 2i& 3i& 2\\ 3& 1& 3\end{array}\right|=x+iy;i=\sqrt{-1}$$
then what is modulus of $Z$ equal to?

If $26!=n{8}^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?