Algebra PYQ — Page 39

Let $f(x)=|\begin{matrix}a & -1 & 0 \\ ax & a & -1 \\ a{x}^{2} & ax & a\end{matrix}|,a\in R$. Then the sum of the squares of all the values of a for $2{f}^{'}(10)-{f}^{'}(5)+100=0$ is

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

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 ______.

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 _____ .

If ${z}^{2}+z+1=0,z\in C$, then $|\sum _{n=1}^{15}{({z}^{n}+(-1{)}^{a}\frac{1}{{z}^{n}})}^{2}|$ 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

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 the minimum value of $f(x)=\frac{5{x}^{2}}{2}+\frac{\alpha }{{x}^{5}},x>0$, is $14$, then the value of $\alpha$ is equal to

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

The positive value of the determinant of the matrix $A$, whose $Adj(Adj(A))=[\begin{matrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{matrix}]$, is ______.

Let $x=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}]$ and $A=[\begin{matrix}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{matrix}]$. For $k\in \mathbb{N}$, if ${X}^{'}{A}^{k}X=33$, then $k$ is equal to

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

Let ${R}_{1}$ and ${R}_{2}$ be relations on the set ${1,2,\ldots ,50}$ such that ${R}_{1}=${$(p,{p}^{n}):p$ is a prime and $n\geq 0$ is an integer} and ${R}_{2}=${$(p,{p}^{n}):p$ is a prime and $n=0$ or $1$}. Then, the number of elements in ${R}_{1}-{R}_{2}$ is ____.

The number of ways to distribute $30$ identical candies among four children ${C}_{1},{C}_{2},{C}_{3}$ and ${C}_{4}$ so that ${C}_{2}$ receives atleast $4$ and atmost $7$ candies, ${C}_{3}$ receives atleast $2$ and atmost $6$ candies, is equal to

If the system of linear equations $2x-3y=\gamma +5$ $\alpha x+5y=\beta +1$, where $\alpha ,\beta ,\gamma \in R$ has infinitely many solutions, then the value of $|9\alpha +3\beta +5\gamma |$ is equal to

Let the coefficients of the middle terms in the expansion of ${(\frac{1}{\sqrt{6}}+\beta x)}^{4},{(1-3\beta x)}^{2}$ and ${(1-\frac{\beta }{2}x)}^{6},\beta >0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50-\frac{2d}{{\beta }^{2}}$ is equal to _____ .

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots$ is equal to ______

Let $S$ be the set of all $(\alpha ,\beta ),\pi <\alpha ,\beta <2\pi$, for which the complex number $\frac{1-i\mathrm{sin}\alpha }{1+2i\mathrm{sin}\alpha }$ is purely imaginary and $\frac{1+i\mathrm{cos}\beta }{1-2i\mathrm{cos}\beta }$ is purely real. Let ${Z}_{\alpha \beta }=\mathrm{sin}2\alpha +i\mathrm{cos}2\beta ,(\alpha ,\beta )\in S$. Then $\underset{(\alpha ,\beta )\in S}{\sum }(i{Z}_{\alpha \beta }+\frac{1}{i{\bar{Z}}_{\alpha \beta }})$ is equal to

Let for the ${9}^{\mathrm{th}}$ term in the binomial expansion of ${(3+6x)}^{n}$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is ${n}_{0}$. If $k$ is the ratio of the coefficient of ${x}^{6}$ to the coefficient of ${x}^{3}$, then $k+{n}_{0}$ is equal to

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of ${({x}^{n}+\frac{2}{{x}^{5}})}^{7}$ is $939$, then the sum of all the possible integral values of $n$ is

The sum $\sum _{n=1}^{21}\frac{3}{(4n-1)(4n+3)}$ is equal to

For a natural number $n$, let ${\alpha }_{n}={19}^{n}-{12}^{n}$. Then, the value of $\frac{31{\alpha }_{9}-{\alpha }_{10}}{57{\alpha }_{8}}$ 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