Algebra PYQ — Page 27

Among the relations $S={(a,b):a,b\in R-{0},2+\frac{a}{b}>0}$ and $T={(a,b):a,b\in R,{a}^{2}-{b}^{2}\in Z}$,

Let $\alpha >0$, be the smallest number such that the expansion of ${({x}^{\frac{2}{3}}+\frac{2}{{x}^{3}})}^{30}$ has a term $\beta {x}^{-\alpha },\beta \in N$. Then $\alpha$ is equal to _____ .

Let $f:R\rightarrow R$ be a function such that $f(x)=\frac{{x}^{2}+2x+1}{{x}^{2}+1}$. Then

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{2x}+2=0$ Then ${\alpha }^{14}+{\beta }^{14}$ is equal to

Let $p,q\in \mathbb{R}$ and $(1-\sqrt{3}i{)}^{200}={2}^{199}(p+iq)$, $i=\sqrt{-1}$. Then, $p+q+{q}^{2}$ and $p-q+{q}^{2}$ are roots of the equation.

If $A$ is a $3\times 3$ matrix and $|A|=2$, then $|3adj(|3A|{A}^{2})|$ is equal to

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____ .

Let $A=[\begin{matrix}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{matrix}]$ and $B=[\begin{matrix}1 & -i \\ 0 & 1\end{matrix}]$, where $i=\sqrt{-1}$. If $M={A}^{T}\mathrm{BA}$, then the inverse of the matrix ${\mathrm{AM}}^{2023}{A}^{T}$ is

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in N$, If $f(1)=3$ and $\sum _{k=1}^{n}f(k)=3279$, then the value of $n$ is

Let $a\in R$ and let $\alpha ,\beta$ be the roots of the equation ${x}^{2}+{60}^{\frac{1}{4}}x+a=0$. If ${\alpha }^{4}+{\beta }^{4}=-30$, then the product of all possible values of $a$ is _____ .

The complex number $z=\frac{i-1}{\mathrm{cos}\frac{\pi }{3}+i\mathrm{sin}\frac{\pi }{3}}$ is equal to:

Let $z=1+i$ and ${z}_{1}=\frac{1+i\bar{z}}{\bar{z}(1-z)+\frac{1}{z}}\cdot$ Then $\frac{12}{\pi }$ $\mathrm{arg}({z}_{1})$ is equal to

The absolute difference of the coefficients of ${x}^{10}$ and ${x}^{7}$ in the expansion of ${(2{x}^{2}+\frac{1}{2x})}^{11}$ is equal to

If the solution of the equation ${\mathrm{log}}_{\mathrm{cos}x}(\mathrm{cot}x)+4{\mathrm{log}}_{\mathrm{sin}x}(\mathrm{tan}x)=1,x\in (0,\frac{\pi }{2})$ is ${\mathrm{sin}}^{-1}(\frac{\alpha +\sqrt{\beta }}{2})$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to:

The number of real solutions of the equation $3({x}^{2}+\frac{1}{{x}^{2}})-2(x+\frac{1}{x})+5=0$, is

Among the statements : $(S1):{2023}^{2022}-{1999}^{2022}$ is divisible by $8$. $(S2):13(13{)}^{n}-11n-13$ is divisible by $144$ for infinitely many $n\in \mathbb{N}$

The number of $3$ digit numbers, that are divisible by either $3$ or $4$ but not divisible by $48$ , is

Let $A=[\begin{matrix}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{matrix}]$, where $a,c\in R$. If ${A}^{3}=A$ and the positive value of $a$ belongs to the interval $(n-1,n]$, where $n\in \mathbb{N}$, then $n$ is equal to ____.

If the set ${Re(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}):z\in \mathbb{C},Rez=3}$ is equal to the interval $(\alpha ,\beta ]$, then $24(\beta -\alpha )$ is equal to

For three positive integers $p,q,r$, ${x}^{p{q}^{2}}={y}^{qr}={z}^{{p}^{2}r}$ and $r=pq+1$ such that $3,3{\mathrm{log}}_{y}x,3{\mathrm{log}}_{z}y,7{\mathrm{log}}_{x}z$are in A.P. with common difference $\frac{1}{2}$. The $r-p-q$is equal to

Let $S={\alpha :{\mathrm{log}}_{2}({9}^{2\alpha -4}+13)-{\mathrm{log}}_{2}(\frac{5}{2}\cdot {3}^{2\alpha -4}+1)=2}.$ Then the maximum value of $\beta$ for which the equation ${x}^{2}-2{(\underset{\alpha \in s}{\sum }\alpha )}^{2}x+\underset{a\in s}{\sum }{(\alpha +1)}^{2}\beta =0$ has real roots, is _____ .

Let the system of linear equations $x+y+kz=2$ $2x+3y-z=1$ $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$ $(2k+1)x+|k+5|y=10$ has :

The remainder on dividing ${5}^{99}$ by $11$ is _____ .

The sum of all those terms, of the arithmetic progression $3,8,13,...,373$, which are not divisible by $3$, is equal to ________.