Algebra PYQ — Page 16

Let $A=\left[\begin{array}{ccc}2 & 2+p & 2+p+q \\ 4 & 6+2 p & 8+3 p+2 q \\ 6 & 12+3 p & 20+6 p+3 q\end{array}\right]$. If $\operatorname{det}(\operatorname{adj}(\operatorname{adj}(3 \mathrm{~A})))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}, \mathrm{m}, \mathrm{n} \in \mathrm{N}$, then $\mathrm{m}+\mathrm{n}$ is equal to

If $\quad y(x)=\left|\begin{array}{ccc}\sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1\end{array}\right|, x \in \mathbb{R}$, then $\frac{d^2 y}{d x^2}+y$ is equal to

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}$ for all $\mathrm{x} \in \mathbb{R}$. If $\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then $\sum_{r=1}^{10} G\left(r^2\right)$ is equal to

Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} \neq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1, z \in \mathrm{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to ________.

If $\sum_{\mathrm{r}=0}^{10}\left(\frac{10^{\mathrm{r}+1}-1}{10^{\mathrm{r}}}\right) \cdot{ }^{11} \mathrm{C}_{\mathrm{r}+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :

Let $\alpha, \beta(\alpha \neq \beta)$ be the values of m , for which the equations $x+y+z=1 ; x+2 y+4 z=\mathrm{m}$ and $x+4 y+10 z=m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10}\left(n^\alpha+n^\beta\right)$ is equal to :

Let $f: R-\{0\} \rightarrow R$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the $\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in R$, then $\alpha+2 \beta$ is equal to

If the system of equation $\begin{aligned}<br/>& 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4 x+5 y+\mu z=9<br/>\end{aligned}$ has infinitely many solutions, then $\left(\lambda^2+\mu^2\right)$ is equal to :

$\text { In an arithmetic progression, if } S_{40}=1030 \text { and } S_{12}=57 \text {, then } S_{30}-S_{10} \text { is equal to : }$

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to

Let $\alpha ={1}^{2}+{4}^{2}+{8}^{2}+{13}^{2}+{19}^{2}+{26}^{2}+\ldots \ldots .$ upto $10$ terms and $\beta =\sum _{n=1}^{10}{n}^{4}$. If $4\alpha -\beta =55k+40$, then $k$ is equal to _______.

The sum of the square of the modulus of the elements in the set $\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}$ is ________

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n$, then the point $(\mathrm{m}, \mathrm{n})$ lies on the line

Let $S={x\in R:{(\sqrt{3}+\sqrt{2})}^{x}+{(\sqrt{3}-\sqrt{2})}^{x}=10}$. Then the number of elements in $S$ is:

If 2 and 6 are the roots of the equation $a x^2+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :

If $\alpha$ denotes the number of solutions of ${|1-i|}^{x}={2}^{x}$ and $\beta =(\frac{|z|}{\mathrm{arg}(z)})$, where $z=\frac{\pi }{4}{(1+i)}^{4}(\frac{1-\sqrt{\pi }\cdot i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }\cdot i})$, $i=\sqrt{-1}$, then the distance of the point $(\alpha ,\beta )$ from the line $4x-3y=7$ is ______

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :

Let $A$ be a $2 \times 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$ and the determinant of $A$ be 1 . If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha+\beta$ equals _______

Consider the function $f:[\frac{1}{2},1]\rightarrow R$ defined by $f(x)=4\sqrt{2}{x}^{3}-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\mathrm{cos}\frac{\pi }{12}$ Then

Remainder when ${64}^{{32}^{32}}$ is divided by $9$ is equal to _____.

Let $A={1,2,3,\ldots .7}$ and let $P(A)$denote the power set of $A$. If the number of functions $f:A\rightarrow P(A)$ such that $a\in f(a),\forall a\in A$ is ${m}^{n},m$ and $n\in N$ and $m$ is least, then $m+n$ is equal to ______.

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{6}x+3=0$ such that $Im(\alpha )>Im(\beta )$. Let $a,b$ be integers not divisible by $3$and $n$ be a natural number such that $\frac{{\alpha }^{99}}{\beta }+{\alpha }^{98}={3}^{n}(a+ib),i=\sqrt{-1}$. Then $n+a+b$ is equal to ___________.