Algebra PYQ — Page 61

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 ${z}_{1}$ and ${z}_{2}$ be two complex numbers satisfying $|{z}_{1}|=9$ and $|{z}_{2}-3-4i|=4$. Then the minimum value of $|{z}_{1}-{z}_{2}|$ is :

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:

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

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 set of all values of $\lambda$ for which the system of linear equations $x-2y-2z=\lambda x$ $x+2y+z=\lambda y$ $-x-y=\lambda z$ has a non-trivial solution :

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:

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

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 :

If the system of equations $2x+3y-z=0, x+ky-2z=0$ and $2x-y+z=0$ has a non-trivial solution $(x,y,z),$ then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to

If the system of equations $x+y+z=5,$ $x+2y+3z=9,$ $x+3y+\alpha z=\beta$ has inifinitely many solutions, then $\beta -\alpha$ equals

If some three consecutive coefficients in the binomial expansion of ${(x+1)}^{n}$ in powers of $x$ are in the ratio $2:15:70,$ then the average of these three coefficients is:

Let $z\in C$ be such that $|z|<1.$ If $\omega =\frac{5+3z}{5(1-z)},$ then:

Let $A=$ { $x\in R:x$ is not a positive integer} $.$ Define a function $f:A\rightarrow R$ as $f(x)=\frac{2x}{x-1}$ , then $f$ is:

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

The number of real roots of the equation $5+|{2}^{x}-1|={2}^{x}({2}^{x}-2)$ is :

Let $p, q\in Q .$ If $2-\sqrt{3}$ is a root of the quadratic equation ${x}^{2}+px+q=0,$ then

If the system of linear equations $2 x+2 y+3 z=a$ $3 x-y+5 z=b$ $x-3 y+2 z=c$ where, $a, b,$ care non-zero real numbers, has more than onc solution, then

If the fourth term in the Binomial expansion of ${(\frac{2}{x}+{x}^{{log}_{8}x})}^{6},(x>0)$ is $20\times {8}^{7},$ then a value of $x$ is

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