Algebra PYQ — Page 19

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__________.

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 $x^{70}$ 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 $A=[\begin{matrix}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{matrix}],B[\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}],C=AB{A}^{T}$ and $X={A}^{T}{C}^{2}A$, then det $X$ is equal to:

Let $R=(\begin{matrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{matrix})$ be a non-zero $3\times 3$ matrix, where $x\mathrm{sin}\theta =y\mathrm{sin}(\theta +\frac{2\pi }{3})=z\mathrm{sin}(\theta +\frac{4\pi }{3})$ $\neq 0,\theta \in (0,2\pi )$. For a square matrix $M$, let Trace$(M)$ denote the sum of all the diagonal entries of $M.$ Then, among the statements: $(I)$ Trace$(R)=0$ $(\mathrm{II})$ If Trace$(adj(adj(R))=0$, then $R$ has exactly one non-zero entry.

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:

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 $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as : $f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, \text { MARA } & x \geq 0 \\ x+1, & x \leq 0\end{cases}$ Then the function $f(g(x))$ is

If $\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0$, and $(60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}$, where $a, b \in N$, then $(a+b)$ equal to ______

Let $\lambda, \mu \in \mathbf{R}$. If the system of equations $\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}$ has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha \beta$ 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 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 range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to________

If $S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to ______

If the function $f:(-\infty ,-1]\rightarrow (a,b)]$ defined by $f(x)={e}^{{x}^{3}-3x+1}$ is one-one and onto, then the distance of the point $P(2b+4,a+2)$ from the line $x+{e}^{-3}y=4$ is:

Let $f:R\rightarrow R$ be a function defined $f(x)=\frac{x}{{(1+{x}^{4})}^{1/4}}$ and $g(x)=f(f(f(f(x))))$ then $18{\int }_{0}^{\sqrt{2\sqrt{5}}}{x}^{2}g(x)dx$

Let $f(x)={2}^{x}-{x}^{2},x\in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y={f}^{'}(x)$ intersects the $x-$axis, then the value of $m+n$ is