Algebra PYQ — Page 38

Let $O$ be the origin and $A$ be the point ${z}_{1}=1+2i$. If $B$ is the point ${z}_{2},Re({z}_{2})<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?

The probability that a randomly chosen one-one function from the set ${a,b,c,d}$ to the set ${1,2,3,4,5}$ satisfied $f(a)+2f(b)-f(c)=f(d)$ is

Let $A$ be a matrix of order $2\times 2$, whose entries are from the set ${0,1,2,3,4,5}$. If the sum of all the entries of $A$ is a prime number $p,2<p<8$, then the number of such matrices $A$ is

Consider a matrix $A=[\begin{matrix}\alpha & \beta & \gamma \\ {\alpha }^{2} & {\beta }^{2} & {\gamma }^{2} \\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{matrix}]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{det(adj(adj(\mathrm{adj}(adjA)))}{{(\alpha -\beta )}^{16}{(\beta -\gamma )}^{16}{(\gamma -\alpha )}^{16}}={2}^{32}\times {3}^{16}$, then the number of such $3$- tuples $(\alpha ,\beta ,\gamma )$ is _______.

The number of matrices $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$, where $a,b,c,d\in {-1,0,1,2,3,\ldots \ldots ,10}$, such that $A={A}^{-1}$, is ______.

Let $A=(\begin{matrix}4 & -2 \\ \alpha & \beta \end{matrix})$. If ${A}^{2}+\gamma A+18I=O$, then $det(A)$ is equal to _______.

The domain of $f(x)=\frac{{\mathrm{cos}}^{-1}(\frac{{x}^{2}-5x+6}{{x}^{2}-9})}{\mathrm{log}({x}^{2}-3x+2)}$ is

Let $S=${$\sqrt{n}:1\leqslant n\leqslant 50$ and $n$ is odd}. Let $a\in S$ and $A=[\begin{matrix}1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1\end{matrix}]$. If $\underset{a\in S}{\Sigma }det(adjA)=100\lambda$, then $\lambda$ is equal to

Let $a,b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation ${x}^{2}-8ax+2a=0$ and $q$ and $s$ are the roots of the equation ${x}^{2}+12bx+6b=0$, such that $\frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s}$ are in A.P., then ${a}^{-1}-{b}^{-1}$ is equal to _____ .

The remainder when ${(2021)}^{2022}+{(2022)}^{2021}$ is divided by $7$ is

If the maximum value of the term independent of $t$ in the expansion of ${({t}^{2}{x}^{\frac{1}{5}}+\frac{{(1-x)}^{{}^{\frac{1}{10}}}}{t})}^{15},x\geq 0$, is $K$, then $8K$ is equal to _____ .

Let the coefficients of ${x}^{-1}$ and ${x}^{-3}$ in the expansion of ${(2{x}^{\frac{1}{5}}-\frac{1}{{x}^{\frac{1}{5}}})}^{15},x>0$, be $m$and $n$ respectively. If $r$ is a positive integer such $m{n}^{2}=Cr.15{2}^{r}$, then the value of $r$ is equal to ______.

If the coefficients of $x$ and ${x}^{2}$ in the expansion of ${(1+x)}^{p}{(1-x)}^{q},p,q\leq 15$, are $-3$ and $-5$ respectively, then the coefficient of ${x}^{3}$ is equal to ______.

Let $A=(\begin{matrix}1+i & 1 \\ -i & 0\end{matrix})$ where $i=\sqrt{-1}$. Then, the number of elements in the set ${n\in {1,2,\ldots .,100}:{A}^{n}=A}$ is

If $1+(2+C149+C249+\ldots .+C4949)(C250+C450+\ldots ..+C5050)$ is equal to ${2}^{n}.m$, where $m$ is odd, then $n+m$ is equal to _____ .

Let $A=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}]$ and $B=[\begin{matrix}{9}^{2} & -{10}^{2} & {11}^{2} \\ {12}^{2} & {13}^{2} & -{14}^{2} \\ -{15}^{2} & {16}^{2} & {17}^{2}\end{matrix}]$, then the value of ${A}^{'}BA$ is;

If $p$ and $q$ are real number such that $p+q=3,{p}^{4}+{q}^{4}=369$, then the value of ${(\frac{1}{p}+\frac{1}{q})}^{-2}$ is equal to

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R=${$(x,y):3x+\alpha y$ is a multiple of $7$}. The relation $R$ is an equivalence relation if and only if

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation ${x}^{4}+{x}^{3}+{x}^{2}+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to

The sum of the cubes of all the roots of the equation ${x}^{4}-3{x}^{3}-2{x}^{2}+3x+1=0$ is _____.

The coefficient of ${x}^{101}$ in the expression ${(5+x)}^{500}+x{(5+x)}^{499}+{x}^{2}{(5+x)}^{498}+\ldots \ldots +{x}^{500},x>0$ is

Let $f(x)$ and $g(x)$ be two real polynomials of degree $2$ and $1$ respectively. If $f(g(x))=8{x}^{2}-2x$, and $g(f(x))=4{x}^{2}+6x+1$, then the value of $f(2)+g(2)$ is ______.

Let $S={1,2,3,4,5,6,7,8,9,10}$. Define $f:S\rightarrow S$ as $f(n)={\begin{matrix}2n, & \mathrm{if}n=1,2,3,4,5 \\ 2n-11 & \mathrm{if}n=6,7,8,9,10\end{matrix}$ Let $g:S\geq S$ be a function such that $\mathrm{fog}(n)={\begin{matrix}n+1 & ,\mathrm{if}n\mathrm{is}\mathrm{odd} \\ n-1 & ,\mathrm{if}n\text{is }\mathrm{even}\end{matrix}$, then $g(10)(g(1)+g(2)+g(3)+g(4)+g(5))$ is equal to

The sum of all real values of $x$ for which $\frac{3{x}^{2}-9x+17}{{x}^{2}+3x+10}=\frac{5{x}^{2}-7x+19}{3{x}^{2}+5x+12}$ is equal to