Class 12 Mathematics 2023 Board Exam

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

If $z\bar{z}=|z+\bar{z}|$, where $z=x+iy$, $i=\sqrt{-1}$, then the locus of $z$ is a pair of :

If $1!+3!+5!+7!+\dots +199!$ is divided by $24$, what is the remainder ?

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 value of $\sqrt{12+5i}+\sqrt{12-5i}$, where $i=\sqrt{-1}$ ?

What is the binary number equivalent to decimal number $1011$?

If $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$, then what is the value of $\det (I+A{A}^{\prime })$, where $I$ is the $3\times 3$ identity matrix ?

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 $A$, $B$ and $C$ are square matrices of order $3$ and $\det (BC)=2\det (A)$, then what is the value of $\det (2A^{-1}BC)$ ?

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