Algebra PYQ — Page 53

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

Let $I$ be an identity matrix of order $2\times 2$ and $P=[\begin{matrix}2 & -1 \\ 5 & -3\end{matrix}]$. Then the value of $n\in N$ for which ${P}^{n}=5I-8P$ 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:

If $A=[\begin{matrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{matrix}]$ and $M=A+{A}^{2}+{A}^{3}+\ldots +{A}^{20},$ then the sum of all the elements of the matrix $M$ is equal to _______.

The total number of $3\times 3$ matrices $A$ having enteries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $A{A}^{T}$ is $9$, is equal to

If $A=[\begin{matrix}0 & -\mathrm{tan}(\frac{\theta }{2}) \\ \mathrm{tan}(\frac{\theta }{2}) & 0\end{matrix}]$ and $({I}_{2}+A){({I}_{2}-A)}^{-1}=[\begin{matrix}a & -b \\ b & a\end{matrix}],$ then $13({a}^{2}+{b}^{2})$ is equal to _____ .

Let $P=[\begin{matrix}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{matrix}],$ where $\alpha \in R.$ Suppose $Q=[{q}_{ij}]$ is a matrix satisfying $PQ=k{I}_{3}$ for some non-zero $k\in R.$ If ${q}_{23}=-\frac{k}{8}$ and $|Q|=\frac{{k}^{2}}{2}$, then ${\alpha }^{2}+{k}^{2}$ is equal to_________.

If $Prn=Pr+1n$ and $Crn=Cr-1n,$ then the value of $r$ is equal to:

Let $f(x)=|\begin{matrix}{\mathrm{sin}}^{2}x & -2+{\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ 2+{\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ {\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & 1+\mathrm{cos}2x\end{matrix}|,x\in [0,\pi ].$ Then the maximum value of $f(x)$ is equal to

If the system of linear equations $2x+y-z=3$ $x-y-z=\alpha$ $3x+3y+\beta z=3$ has infinitely many solutions, then $|\alpha +\beta -\alpha \beta |$ is equal to __________.

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu ,$ and a system of linear equations $x+y+z=5$ $x+2y+3z=\mu$ $x+3y+\lambda z=1$ is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then:

Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $|\begin{matrix}x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c\end{matrix}|=2$, then value of ${\lambda }^{2}$ is equal to________.

The value of $|\begin{matrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{matrix}|$ is

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity$\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots .+\frac{1}{a-nb}=\alpha n+\beta {n}^{2}+\gamma {n}^{3}$ then the value of $\gamma$ is :

The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :

Let $Z$ be the set of all integers, $A={(x,y)\in Z\times Z:(x-2{)}^{2}+{y}^{2}\leq 4}$ $B={(x,y)\in Z\times Z:{x}^{2}+{y}^{2}\leq 4}\text{ and }$ $C={(x,y)\in Z\times Z:(x-2{)}^{2}+(y-2{)}^{2}\leq 4}$ If the total number of relations from $A\cap B$ to $A\cap C$ is ${2}^{p}$, then the value of $p$ is:

Let $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$ and $B=[\begin{matrix}\alpha \\ \beta \end{matrix}]\neq [\begin{matrix}0 \\ 0\end{matrix}]$ such that $AB=B$ and $a+d=2021,$ then the value of $ad-bc$ is equal to ______ .

Let $f(x)$ be a polynomial of degree $3$ such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5.$ Then the value of $52-10f(10)$ is equal to _____ .

Let $[x]$ denote the greatest integer less than or equal to $x.$ Then, the values of $x\in R$ satisfying the equation ${[{e}^{x}]}^{2}+[{e}^{x}+1]-3=0$ lie in the interval:

Let $f:R-{\frac{\alpha }{6}}\rightarrow R$ be defined by $f(x)=(\frac{5x+3}{6x-\alpha })$. Then the value of $\alpha$ for which $(fof)(x)=x$, for all $x\in R-{\frac{\alpha }{6}},$ is

The inverse of $y={5}^{\mathrm{log}x}$ is:

Consider the system of linear equations $-x+y+2z=0$ $3x-ay+5z=1$ $2x-2y-az=7$ Let ${S}_{1}$ be the set of all $a\in R$ for which the system is inconsistent and ${S}_{2}$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $n({S}_{1})$ and $n({S}_{2})$ denote the number of elements in ${S}_{1}$ and ${S}_{2}$ respectively, then

A number is called a palindrome if it reads the same backward as well as forward. For example $285582$ is a six digit palindrome. The number of six digit palindromes, which are divisible by $55,$ is ________.