Algebra PYQ — Page 52

Let ${S}_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If ${S}_{3n}=3{S}_{2n}$, then the value of $\frac{{S}_{4n}}{{S}_{2n}}$ is :

Let $A=[\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{matrix}].$ Then the number of $3\times 3$ matrices $B$ with entries from the set ${1,2,3,4,5}$ and satisfying $AB=BA$ is ________.

Let ${P}_{1},{P}_{2}\ldots ,{P}_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points ${P}_{i},{P}_{j},{P}_{k}$ such that $i+j+k\neq 15,$ is :

The least positive integer $n$ such that $\frac{(2i{)}^{n}}{(1-i{)}^{n-2}},i=\sqrt{-1},$ is a positive integer, is ______.

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{matrix}].$ Then ${A}^{2025}-{A}^{2020}$ is equal to

Let $A={n\in N:n$ is a $3-$digit number$}$ $B={9k+2:k\in N}$ and $C={9k+l:k\in N}$ for some $l(0<l<9)$. If the sum of all the elements of the set $A\cap (B\cup C)$ is $274\times 400,$ then $l$ is equal to

A function $f(x)$ is given by $f(x)=\frac{{5}^{x}}{{5}^{x}+5}$, then the sum of the series $f(\frac{1}{20})+f(\frac{2}{20})+f(\frac{3}{20})+\ldots +f(\frac{39}{20})$ is equal to:

If $A=[\begin{matrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{matrix}],B=[\begin{matrix}1 & 0 \\ i & 1\end{matrix}],i=\sqrt{-1}$, and $Q={A}^{T}BA$, then the inverse of the matrix $A{Q}^{2021}{A}^{T}$ is equal to:

The number of elements in the set {$A=[\begin{matrix}a & b \\ 0 & d\end{matrix}]:a,b,d\in {-1,0,1}$ and $(I-A{)}^{3}=I-{A}^{3}$}, where $I$ is $2\times 2$ identity matrix, is .

If the matrix $A=[\begin{matrix}0 & 2 \\ K & -1\end{matrix}]$ satisfies $A({A}^{3}+3I)=2I,$ then the value of $K$ is

Which of the following is not correct for relation $R$ on the set of real numbers?

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R={(x,y)\in N\times N:{x}^{3}-3{x}^{2}y-x{y}^{2}+3{y}^{3}=0}$. Then the relation $R$ is

Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true ?

The number of elements in the set ${x\in R:(|x|-3)|x+4|=6}$ is equal to

The number of real roots of the equation ${e}^{4x}-{e}^{3x}-4{e}^{2x}-{e}^{x}+1=0$ is equal to

Let $A={1,2,3,\ldots ,10}$ and $f:A\rightarrow A$ be defined as $f(k)={\begin{matrix}k+1 & \mathrm{if}k\mathrm{is}odd \\ k & \mathrm{if}k\mathrm{is}even\end{matrix}$ Then the number of possible functions $g:A\rightarrow A$ such that $gof=f$ is:

If the matrix $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{matrix}]$ satisfies the equation ${A}^{20}+\alpha {A}^{19}+\beta A=[\begin{matrix}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{matrix}]$ for some real numbers $\alpha$ and $\beta$, then $\beta -\alpha$ is equal to ______.

Let $A=[\begin{matrix}\begin{matrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{matrix}\end{matrix}]$ and $B=7{A}^{20}-20{A}^{7}+2I$, where $I$ is an identity matrix of order $3\times 3.$ If $B=[{b}_{ij}]$, then ${b}_{13}$ is equal to

If ${\mathrm{log}}_{3}2,{\mathrm{log}}_{3}({2}^{x}-5),{\mathrm{log}}_{3}({2}^{x}-\frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to _____.

If the remainder when $x$ is divided by $4$ is $3$, then the remainder when ${(2020+x)}^{2022}$ is divided by $8$ is ___ .

If $x,y,z$ are in arithmetic progression with common difference $d,x\neq 3d,$ and the determinant of the matrix $[\begin{matrix}3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z\end{matrix}]$ is zero, then the value of ${k}^{2}$ is

Let $A={n\in N\mid {n}^{2}\leq n+10,000},B={3k+1\mid k\in N}$ and $C={2k\mid k\in N},$ then the sum of all the elements of the set $A\cap (B-C)$ is equal to ________.

The sum of ${162}^{th}$ power of the roots of the equation ${x}^{3}-2{x}^{2}+2x-1=0$ is ______.

The real valued function $f(x)=\frac{{cosec}^{-1}x}{\sqrt{x-[x]}},$ where $[x]$ denotes the greatest integer less than or equal to $x,$ is defined for all $x$ belonging to: