Algebra PYQ — Page 18

Let $\alpha, \beta$ be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

If $\alpha ,\beta$ are the roots of the equation, ${x}^{2}-x-1=0$ and ${S}_{n}=2023{\alpha }^{n}+2024{\beta }^{n}$, then

Let $A={1,2,3,....100}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let ${R}_{1}$ be a symmetric relation on $A$ such that $R\subset {R}_{1}$ and the number of elements in ${R}_{1}$ is $n$. Then the minimum value of $n$ is _______.

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha }}$ lie on the circles ${|z-{z}_{0}|}^{2}=4$ and ${|z-{z}_{0}|}^{2}=16$ respectively, where ${z}_{0}=1+i$. Then, the value of $100|\alpha {|}^{2}$ is__________.

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$. Then a possible value of $p+q$ is :

If the system of equations $\begin{array}{r} 11 x+y+\lambda z=-5 \\ 2 x+3 y+5 z=3 \\ 8 x-19 y-39 z=\mu \end{array}$ has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

Let $\alpha \beta \neq 0$ and $A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]$. If $B=\left[\begin{array}{rrr}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha \\ -2 \alpha & 5 & -2 \beta\end{array}\right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname{det}(A B)$ is equal to :

If the domain of the function $f(x)={\mathrm{log}}_{e}(\frac{2x+3}{4{x}^{2}+x-3})+{\mathrm{cos}}^{-1}(\frac{2x-1}{x+2})$ is $(\alpha ,\beta ]$, then the value of $5\beta -4\alpha$ is equal to

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

Let $S={1,2,3,\ldots ,10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R={(A,B):A\cap B\neq \phi ;A,B\in M}$ is :

If a function $f$ satisfies $f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n} \in \mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum_{k=1}^{2022} f(\lambda+k) \leq(2022)^2$ is equal to _________

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $\mathrm{p}$, then $108 \mathrm{p}$ is equal to

Let $S_1=\{z \in C:|z| \leq 5\}, S_2=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$ and $S_3=\{z \in C: \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :

If $A=[\begin{matrix}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{matrix}],B[\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}],C=AB{A}^{T}$ and $X={A}^{T}{C}^{2}A$, then det $X$ is equal to:

Consider the system of linear equations $x+y+z=5,x+2y+{\lambda }^{2}z=9$ and $x+3y+\lambda z=\mu$, where $\lambda ,\mu \in R$. Then, which of the following statement is NOT correct ?

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|$ is equal to :

The remainder when $428^{2024}$ is divided by 21 is__________

The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is ______.

If the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{2-|x|}{4})+{({\mathrm{log}}_{e}(3-x))}^{-1}$ is $[-\alpha ,\beta )-{\gamma }$, then $\alpha +\beta +\gamma$ is equal to :

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to:

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: \mathrm{A} \rightarrow \mathrm{B}$, such that $f(1)+f(3)=14$, is :

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{matrix}]$ and $|2A{|}^{3}={2}^{21}$ where $\alpha ,\beta \in Z$, Then a value of $\alpha$ is

The coefficient of ${x}^{2012}$ in the expansion of ${(1-x)}^{2008}{(1+x+{x}^{2})}^{2007}$ is equal to _____.

If $f(x)=\frac{4x+3}{6x-4},x\neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g:R-{\frac{2}{3}}\rightarrow R-{\frac{2}{3}}$, then $(gogog)(4)$ is equal to