Algebra PYQ — Page 32

Let $S={x:x\in \mathbb{R}\text{ and }{(\sqrt{3}+\sqrt{2})}^{{x}^{2}-4}+{(\sqrt{3}-\sqrt{2})}^{{x}^{2}-4}=10}$. Then $n(S)$ is equal to

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is$_______.$

Let $A$ be a $2\times 2$ matrix with real entries such that ${A}^{'}=\alpha A+1,$ where $\alpha \in \mathbb{R}-{-1,1}$., If det $({A}^{2}-A)=4$, the sum of all possible values of $\alpha$ is equal to

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $_____$

If $A=\frac{1}{5!6!7!}[\begin{matrix}5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9!\end{matrix}]$, then $|adj(adj(2A))|$ is equal to

If $Pn-12n+1:Pn2n-1=11:21$, then ${n}^{2}+n+15$ is equal to :

Let $\alpha ,\beta ,\gamma$ be the three roots of the equation ${x}^{3}+bx+c=0$ if $\beta \gamma =1=-\alpha$ then ${b}^{3}+2{c}^{3}-3{\alpha }^{3}-6{\beta }^{3}-8{\gamma }^{3}$ is equal to

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x{)}^{n}$ be in the ratio $2:5:8$. Then the coefficient of the term, which is in the middle of these three terms, is

$\sum _{k=0}^{6}C351-k$ is equal to

The sum to $10$terms of the series $\frac{1}{1+{1}^{2}+{1}^{4}}+\frac{2}{1+{2}^{2}+{2}^{4}}+\frac{3}{1+{3}^{2}+{3}^{4}}+\ldots$is :-

The set of all values of $t\in \mathbb{R}$, for which the matrix $[\begin{matrix}{e}^{t} & {e}^{-t}(\mathrm{sin}t-2\mathrm{cos}t) & {e}^{-t}(-2\mathrm{sin}t-\mathrm{cos}t) \\ {e}^{t} & {e}^{-t}(2\mathrm{sin}t+\mathrm{cos}t) & {e}^{-t}(\mathrm{sin}t-2\mathrm{cos}t) \\ {e}^{t} & {e}^{-t}\mathrm{cos}t & {e}^{-t}\mathrm{sin}t\end{matrix}]$ is invertible, is

Let for $A=[\begin{matrix}1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2\end{matrix}],|A|=2$. If $|2adj(2adj(2A))|={32}^{n}$, then $3n+\alpha$ is equal to

If the domain of the function $f(x)={\mathrm{log}}_{e}(4{x}^{2}+11x+6)+{\mathrm{sin}}^{-1}(4x+3)+{\mathrm{cos}}^{-1}(\frac{10x+6}{3})$ is $(\alpha ,\beta ]$, then $36|\alpha +\beta |$ is equal to

Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in \mathbb{N}$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $\mathrm{a}_1=\mathrm{b}_1=4$ and $\mathrm{r}_1<\mathrm{r}_2$. Let $\mathrm{c}_{\mathrm{k}}=\mathrm{a}_{\mathrm{k}}+\mathrm{b}_{\mathrm{k}}, \mathrm{k} \in \mathbb{N}$. If $\mathrm{c}_2=5$ and $\mathrm{c}_3=\frac{13}{4}$ then $\sum_{\mathrm{k}=1}^{\infty} \mathrm{c}_{\mathrm{k}}-\left(12 \mathrm{a}_6+8 \mathrm{~b}_4\right)$ is equal to

The domain of the function $f(x)=\frac{1}{\sqrt{{[x]}^{2}-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$)

The sum of the coefficients of three consecutive terms in the binomial expansion of ${(1+x)}^{n+2}$, which are in the ratio $1:3:5$, is equal to

Let $A={1,2,3,4}$ and $R$ be a relation on the set $A\times A$ defined by $R={((a,b),(c,d)):2a+3b=4c+5d}$. Then the number of elements in $R$ is _________.

If $A=\frac{1}{2}[\begin{matrix}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{matrix}]$ then,

The mean of the coefficients of $x,{x}^{2},\ldots \ldots ,{x}^{7}$ in the binomial expression of $(2+x{)}^{9}$ is _________

Let $f:(0,1)\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{1}{1-{e}^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements (I) $g$ is an increasing function in $(0,1)$ (II) $g$ is one-one in $(0,1)$ Then,

If the coefficients of $x$ and ${x}^{2}$ in $(1+x{)}^{p}(1-x{)}^{q}$ are $4$ and $-5$ respectively, then $2p+3q$ is equal to

Let $[\begin{matrix}\begin{matrix}2 \\ 1 \\ 0\end{matrix} & \begin{matrix}1 \\ 2 \\ -1\end{matrix} & \begin{matrix}0 \\ -1 \\ 2\end{matrix}\end{matrix}]$. If $|adj(adj(adj2A))|=(16{)}^{n}$, then $n$ is equal to

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Fractional part of the number $\frac{{4}^{2022}}{15}$ is equal to