Algebra PYQ — Page 37

Let $\alpha$ be a root of the equation $1+{x}^{2}+{x}^{4}=0$. Then the value of ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}$ is equal to:

Let $f(x)=\frac{x-1}{x+1},x\in R-{0,-1,1)$ . If ${f}^{n+1}(x)=f({f}^{n}(x))$ for all $n\in N$, then ${f}^{6}(6)+{f}^{7}(7)$ is equal to

A class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$, then $b+3g$ is equal to

If the system of linear equations $2x+3y-z=-2$ $x+y+z=4$ $x-y+|\lambda |z=4\lambda -4$ where $\lambda \in \mathbb{R}$, has no solution, then

The total number of $3$-digit numbers, whose greatest common divisor with $36$ is $2$, is ______.

Let $A$ and $B$ be two $3\times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

Let $z=a+ib,b\neq 0$ be complex numbers satisfying ${z}^{2}=\bar{z}\cdot {2}^{1-|z|}$. Then the least value of $n\in N$, such that ${z}^{n}={(z+1)}^{n}$, is equal to _____ .

Let $A={x\in R:|x+1|<2}$ and $B={x\in R:|x-1|\geq 2}$. Then which one the following statements is NOT true?

The remainder when ${3}^{2022}$ is divided by $5$ is

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar{z}=i{z}^{2}+{z}^{2}-z$ is equal to

The number of terms in the expansion of (1 + x)²⁰ + (1 - x)²⁰ is:

The value of ¹⁰C₀ + ¹⁰C₁ + ¹⁰C₂ + ... + ¹⁰C₁₀ is:

The total number of $5$-digit numbers, formed by using the digits $1,2,3,5,6,7$ without repetition, which are multiple of $6$, is

Let ${{{a}_{n}}}_{n=0}^{\infty }$ be a sequence such that ${a}_{0}={a}_{1}=0$ and ${a}_{n+2}=3{a}_{n+1}-2{a}_{n}+1,\forall n\geq 0$. Then ${a}_{25}{a}_{23}-2{a}_{25}{a}_{22}-2{a}_{23}{a}_{24}+4{a}_{22}{a}_{24}$ is equal to

The domain of the function ${\mathrm{cos}}^{-1}(\frac{2{\mathrm{sin}}^{-1}(\frac{1}{4{x}^{2}-1})}{\pi })$ is

Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let $A$ be a $3\times 3$ real matrix such that $A(\begin{matrix}1 \\ 1 \\ 0\end{matrix})=(\begin{matrix}1 \\ 1 \\ 0\end{matrix});A(\begin{matrix}1 \\ 0 \\ 1\end{matrix})=(\begin{matrix}-1 \\ 0 \\ 1\end{matrix})$ and $A(\begin{matrix}0 \\ 0 \\ 1\end{matrix})=(\begin{matrix}1 \\ 1 \\ 2\end{matrix})$. If $X={[{x}_{1}{x}_{2}{x}_{3}]}^{T}$ and $I$ is an identity matrix of order $3$, then the system $(A-2I)X=(\begin{matrix}4 \\ 1 \\ 1\end{matrix})$ has

Let $A={1,2,3,4,5,6,7}$ and $B={3,6,7,9}$. Then the number of elements in the set ${C\subseteq A:C\cap B\neq \phi }$ is ______

Let ${{{a}_{n}}}_{n=0}^{\infty }$ be a sequence such that ${a}_{0}={a}_{1}=0$ and ${a}_{n+2}=2{a}_{n+1}-{a}_{n}+1$ for all $n\geq 0$. Then, $\sum _{n=2}^{\infty }\frac{{a}_{n}}{{7}^{n}}$ is equal to

Let $A={1,{a}_{1},{a}_{2}\ldots \ldots {a}_{18},77}$ be a set of integers with $1<{a}_{1}<{a}_{2}<\ldots ..<{a}_{18}<77$. Let the set $A+A={x+y:x,y\in A}$ contain exactly $39$ elements. Then, the value of ${a}_{1}+{a}_{2}+\ldots ..+{a}_{18}$ is equal to ______.

If $\frac{1}{2\cdot {3}^{10}}+\frac{1}{{2}^{2}\cdot {3}^{9}}+\ldots +\frac{1}{{2}^{10}\cdot 3}=\frac{K}{{2}^{10}\cdot {3}^{10}}$, then the remainder when $K$ is divided by $6$ is

The remainder when ${(2021)}^{2023}$ is divided by $7$ is

The sum of the maximum and minimum values of the function $f(x)=|5x-7|+[{x}^{2}+2x]$ in the interval $[\frac{5}{4},2]$, where $[t]$ is the greatest integer $\leq t$, is ______.

Let $A={x\in R:|x+1|<2}$ and $B={x\in R:|x-1|\geq 2}$. Then which one the following statements is NOT true?