NDA Mathematics 2023 Question Paper

If $\omega$ is a non-real cube root of 1, then what is the value of $\left|\frac{1-\omega }{\omega +{\omega }^2}\right|$ ?

What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once); and divisible by 6 ?

What is the binary number equivalent to decimal number 1011 ?

Let $A$ be a matrix of order $3\times 3$ and $|A|=4$. If $|2\text{ adj}(3A)|=2^{\alpha }{3}^{\beta }$, then what is the value of $(\alpha +\beta )$ ?

If $\alpha$ and $\beta$ are the distinct roots of equation $x^2-x+1=0$, then what is the value of $\left|\frac{{\alpha }^{100}+{\beta }^{100}}{{\alpha }^{100}-{\beta }^{100}}\right|$ ?

Let $A$ and $B$ be symmetric matrices of same order, then which one of the following is correct regarding $(AB-BA)$ ?

1. Its diagonal entries are equal but nonzero
2. The sum of its non-diagonal entries is zero

Select the correct answer using the code given below :

Consider the following statements in respect of square matrices $A,B,C$ each of same order $n$ :

1. $AB=AC\Rightarrow B=C$ if $A$ is non-singular
2. If $BX=CX$ for every column matrix $X$ having $n$ rows then $B=C$

Which of the statements given above is/are correct ?

The system of linear equations $x+2y+z=4$, $2x+4y+2z=8$ and $3x+6y+3z=10$ has

Let $AX=B$ be a system of 3 linear equations with 3-unknowns. Let $X_1$ and $X_2$ be its two distinct solutions. If the combination $a{X}_1+b{X}_2$ is a solution of $AX=B$; where $a,b$ are real numbers, then which one of the following is correct ?

What is the sum of the roots of the equation $\left|\begin{array}{ccc}0& x-a& x-b\\ 0& 0& x-c\\ x+b& x+c& 1\end{array}\right|=0$ ?