Algebra PYQ — Page 61

If $A=[\begin{matrix}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{matrix}]$ , then the matrix ${A}^{-50}$ when $\theta =\frac{\pi }{12},$ is equal to:

If $B=[\begin{matrix}5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1\end{matrix}]$ is the inverse of a $3\times 3$ matix $A,$ then the sum of all values of $\alpha$ for which $det(A)+1=0,$ is:

A value of $\theta \in (0,\frac{\pi }{3}),$ for which $|\begin{matrix}1+{cos}^{2}\theta & {sin}^{2}\theta & 4 cos6\theta \\ {cos}^{2}\theta & 1+{sin}^{2}\theta & 4 cos6\theta \\ {cos}^{2}\theta & {sin}^{2}\theta & 1+4 cos6\theta \end{matrix}|=0,$ is

Let $P=[\begin{matrix}1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1\end{matrix}]$ and $Q=[{q}_{ij}]$ be two $3\times 3$ matrices such that $Q-{P}^{5}={I}_{3}$. Then $\frac{{q}_{21}+{q}_{31}}{{q}_{32}}$ is equal to :

The coefficient of ${x}^{18}$ in the product $(1+x){(1-x)}^{10}{(1+x+{x}^{2})}^{9}$ is

The term independent of $x$ in the expansion of $(\frac{1}{60}-\frac{{x}^{8}}{81}).{(2{x}^{2}-\frac{3}{{x}^{2}})}^{6}$ is equal to

If the fourth term in the binomial expansion of ${(\sqrt{{x}^{\frac{1}{1+{log}_{10}x}}}+{x}^{\frac{1}{12}})}^{6}$ is equal to $200,$ and $x>1,$ then the value of $x$ is

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac{3}{x}\right)^{8}$ equals 5670 is :

Let $A=[\begin{matrix}\mathrm{cos}\alpha & -\mathrm{sin}\alpha \\ \mathrm{sin}\alpha & \mathrm{cos}\alpha \end{matrix}],(a\in R)$ such that ${A}^{32}=[\begin{matrix}0 & -1 \\ 1 & 0\end{matrix}].$ Then, a value of $\alpha$ is:

Let $A=[\begin{matrix}2 & b & 1 \\ b & {b}^{2}+1 & b \\ 1 & b & 2\end{matrix}],$ where $b>0$. Then the minimum value of $\frac{\mathrm{det}(A)}{b}$ is:

Let the numbers $2, b, c$ be in an A.P. and $A=[\begin{matrix}1 & 1 & 1 \\ 2 & b & c \\ 4 & {b}^{2} & {c}^{2}\end{matrix}]$ . If $det(A) \in [2,16],$ then $c$ lies in the interval:

Let ${z}_{1}$ and ${z}_{2}$ be any two non-zero complex numbers such that $3|{z}_{1}|=4|{z}_{2}|.$ If $z=\frac{3{z}_{1}}{2{z}_{2}}+\frac{2{z}_{2}}{3{z}_{1}}$ then maximum value of $|z|$ is Note: In actual paper value of $|z|$ was asked. Hence, none of the options given were correct. So we have modified the question as well as options.

Let ${a}_{1}, {a}_{2}, {a}_{3}\ldots ,{a}_{10}$ be in $G.P.$with ${a}_{i}>0$ for $i=1, 2, \ldots , 10$ and $S$ be the set of pairs $(r, k), r, k\in N$ (the set of natural numbers) for which $|\begin{matrix}{\mathrm{log}}_{e}⁡{a}_{1}^{r} {a}_{2}^{k} & {\mathrm{log}}_{e}⁡{a}_{2}^{r}{a}_{3}^{k} & {\mathrm{log}}_{e}⁡{a}_{3}^{r}{a}_{4}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{4}^{r} {a}_{5}^{k} & {\mathrm{log}}_{e}⁡{a}_{5}^{r}{a}_{6}^{k} & {\mathrm{log}}_{e}⁡{a}_{6}^{r}{a}_{7}^{k} \\ {\mathrm{log}}_{e}⁡{a}_{7}^{r}{a}_{8}^{k} & {\mathrm{log}}_{e}⁡{a}_{8}^{r}{a}_{9}^{k} & {\mathrm{log}}_{e}⁡{a}_{9}^{r}{a}_{10}^{k}\end{matrix}|=0$ Then the number of elements in $S,$ is:

If $\alpha ,\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. Such that the equations $\alpha {x}^{2}+2\beta x+\gamma =0$ and ${x}^{2}+x-1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to:

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is $\frac{27}{19}$. Then the common ratio of this series is:

Let $Z$ be the set of integers. If $A={x\in Z :{2}^{(x+2)({x}^{2}-5x+6)}=1}$ and $B={x\in Z :- 3< 2x-1<9}$, then the number of subsets of the set $A\times B$, is :

If $\left|\begin{array}{ccc}a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b\end{array}\right|$ $=(a+b+c)(x+a+b+c)^{2}, x \neq 0$ and $a+b+c \neq 0,$ then $x$ is equal to

If $5, 5r, 5{r}^{2}$ are the lengths of the sides of a triangle, then $r$ can not be equal to:

If the system of linear equations $x-2y+kz=1$ $2x+y+z=2$ $3x-y-kz=3$ has a solution $(x,y,z),z\neq 0,$ then $(x,y)$ lies on the straight line whose equation is:

If $A=[\begin{matrix}{e}^{t} & {e}^{-t}cos t & {e}^{-t}\mathrm{sin}⁡ t \\ {e}^{t} & -{e}^{-t}\mathrm{cos}⁡t-{e}^{-t}\mathrm{sin}⁡t & -{e}^{-t}\mathrm{sin}⁡t+{e}^{-t}\mathrm{cos}⁡t \\ {e}^{t} & {2e}^{-t}\mathrm{sin}⁡t & -2{e}^{-t}\mathrm{cos}⁡t\end{matrix}],$ then $A$ is:

Let $z={(\frac{\sqrt{3}}{2}+\frac{i}{2})}^{5}+{(\frac{\sqrt{3}}{2}-\frac{i}{2})}^{5}.$ If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z,$ then

An ordered pair $(\alpha , \beta )$ for which the system of linear equations $(1+\alpha )x+\beta y+z=2$ $\alpha x+(1+\beta )y+z=3$ $\alpha x+\beta y+2z=2$ has a unique solution, is :

The system of linear equations $x+y+z=2$ $2x+3y+2z=5$ $2x+3y+({a}^{2}-1)z=a+1$

Let $A=\left(\begin{array}{ccc}0 & 2 q & r \\ p & q & -r \\ p & -q & r\end{array}\right)$. If $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}_{3},$ then $|\mathrm{p}|$ is: