NDA Mathematics 2025 Question Paper

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 $\frac{z}{y}=120$, 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 $1+3+5+7+\dots$ amount to a sum equal to $12345678987654321$?

How many terms are identical in the two APs $19,21,23,\dots$ up to $110$ terms and $19,22,25,28,\dots$ 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 $\left[\begin{array}{cc}-4& -5\\ 2& 2\end{array}\right]$?