NDA Mathematics Medium Questions

A 4-digit number is selected at random formed by using the digits 0, 1, 2, 3 and 4 (where repetition of digits is not allowed). What is the probability that the number selected is divisible by 3 ?

If $p,q,r$ are the cube roots of unity, then what is the value of $\left|\begin{array}{ccc}{p}^2+q^2& r^2& r^2\\ p^2& q^2+r^2& p^2\\ q^2& q^2& r^2+p^2\end{array}\right|$ ?

A 4-digit number is selected at random formed by using the digits 0, 1, 2, 3 and 4 (where repetition of digits is not allowed). What is the probability that the number selected is divisible by 4 ?

A 4-digit number is selected at random formed by using the digits 0, 1, 2, 3 and 4 (where repetition of digits is not allowed). What is the probability that the number selected is divisible by 6 ?

If $M=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right)$, then what is the value of $|M||\mathrm{adj}M|$ ?

If $M$ is a square matrix such that $M^3=M$, then how many values of $|M|$ are possible?

If $M_k=$ $\left[\begin{array}{cc}k& k-1\\ k-1& k\end{array}\right]$ where $k$ is a natural number, then what is $|M_1|+|M_2|+|M_3|+\cdots +|M_{50}|$ equal to?

Let A, B, C and D be mutually exclusive and exhaustive events and $\frac{P(A)}{2}=\frac{P(B)}{3}=\frac{P(C)}{5}=\frac{P(D)}{8}$. What is $P(A)+P(B)+P(C)$ equal to ?

If $A$ is a square matrix such that $|A|=-2$, then $|A{A}^T|$, where $A^T$ is the transpose of $A$, is equal to

A 4-digit number is selected at random formed by using the digits 0, 1, 2, 3 and 4 (where repetition of digits is not allowed). What is the probability that the number selected is divisible by 2 ?

Let A, B, C and D be mutually exclusive and exhaustive events and $\frac{P(A)}{2}=\frac{P(B)}{3}=\frac{P(C)}{5}=\frac{P(D)}{8}$.

What is $\frac{2P(A)+3P(B)}{4P(C)+5P(D)}$ equal to ?

Let $p=(x+y+z)$ and $q=xyz$. If $\left|\begin{array}{ccc}x& 1& 1\\ 1& y& 1\\ 1& 1& z\end{array}\right|$ is positive, then which one of the following is correct?

If $M^T$ is the transpose of a $2\times 2$ matrix $M$, then which of the following is/are correct?

I. $|M+M^T|=|M|+|M^T|$ if $M$ is symmetric.

II. $|M+M^T|=0$ if $M$ is anti-symmetric.

Select the answer using the code given below:

Four dice are rolled. What is the probability that the total of the numbers on the dice is 6 ?

Consider the following statements in respect of the events $A,B,C$ :

I. $(A\cup B\cup C)\cap (\overline{A}\cap \overline{B}\cap \overline{C})$ is an impossible event.

II. $(A\cap B\cap C)\cap (\overline{A}\cup \overline{B}\cup \overline{C})$ is a possible event.

Which of the statements given above is/are correct ?

A 4-digit number is selected at random formed by using the digits $0,1,2,3$ and $4$ (where repetition of digits is not allowed). What is the probability that the number selected does not contain zero at any position?

If $\left|\begin{array}{ccc}a-b& p-q& x-y\\ b-c& q-r& y-z\\ c-a& r-p& z-x\end{array}\right|=k\left|\begin{array}{ccc}a& b& c\\ p& q& r\\ x& y& z\end{array}\right|$, then what is the value of $k$?

Let $\vec{a},\vec{b},\vec{c}$ be unit vectors. Further, $\vec{a}$ is perpendicular to $\vec{b}$; $\vec{c}$ makes an angle $\frac{\pi }{3}$ with both $\vec{a}$ and $\vec{b}$; and $\vec{c}=p\vec{a}+q\vec{b}+r(\vec{a}\times \vec{b})$.

What is the value of $(p+q)$ ?

Let $f(x)=\left|\begin{array}{ccc}3x^2& \cos x& -\sin x\\ 6& -1& 0\\ q& q^2& q^3\end{array}\right|$ where $q$ is any constant. Then what is $\frac{d^2}{d{x}^2}(f(x))$ at $x=0$ equal to?

Let $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{b}\times \vec{c}=\vec{a}$.

Consider the following statements :

I. $\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs.

II. $\vec{a},\vec{b},\vec{c}$ are unit vectors.

Which of the statements given above is/are correct ?