Algebra PYQ — Page 50

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of ${A}^{2}$ is $1,$ then the possible number of such matrices is:

The domain of the function ${cosec}^{-1}(\frac{1+x}{x})$ is :

Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true ?

If the following system of linear equations $2x+y+z=5$ $x-y+z=3$ $x+y+az=b$ has no solution, then :

For the natural numbers $m,n,$ if $(1-y{)}^{m}(1+y{)}^{n}=1+{a}_{1}y+{a}_{2}{y}^{2}+\ldots .+{a}_{m+n}{y}^{m+n}$ and ${a}_{1}={a}_{2}=10,$ then the value of $m+n,$ is equal to:

Let $A=[\begin{matrix}{a}_{1} \\ {a}_{2}\end{matrix}]$ and $B=[\begin{matrix}{b}_{1} \\ {b}_{2}\end{matrix}]$ be two $2\times 1$matrices with real entries such that $A=XB,$ where $X=\frac{1}{\sqrt{3}}[\begin{matrix}1 & -1 \\ 1 & k\end{matrix}],$ and $k\in R.$ If ${a}_{1}^{2}+{a}_{2}^{2}=\frac{2}{3}({b}_{1}^{2}+{b}_{2}^{2})$ and $({k}^{2}+1){b}_{2}^{2}\neq -2{b}_{1}{b}_{2}$, then the value of $k$ is __________.

The values of $a$ and $b$, for which the system of equations $2x+3y+6z=8$ $x+2y+az=5$ $3x+5y+9z=b$ has no solution, are :

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that $\mathrm{arg}({z}_{1}-{z}_{2})=\frac{\pi }{4}$ and ${z}_{1},{z}_{2}$ satisfy the equation $|z-3|=Re(z)$. Then the imaginary part ${z}_{1}+{z}_{2}$ is equal to

If $A=[\begin{matrix}0 & \mathrm{sin}\alpha \\ \mathrm{sin}\alpha & 0\end{matrix}]$ and $\mathrm{det}({A}^{2}-\frac{1}{2}I)=0,$ then a possible value of $\alpha$ is

There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicketkeepers. The number of ways, a team of $11$ players be selected from them so as to include at least $4$ bowlers, $5$ batsmen and $1$ wicketkeeper, is

If $(\frac{{3}^{6}}{{4}^{4}})k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4}-\frac{12}{{x}^{2}})}^{12},$ then $k$ is equal to

The system of equations $kx+y+z=1,x+ky+z=k$ and $x+y+zk={k}^{2}$ has no solution if $k$ is equal to:

If the sides $AB,BC$ and $CA$ of a triangle $ABC$ have $3,5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

A possible value of $x,$ for which the ninth term in the expansion of ${{{3}^{{\mathrm{log}}_{3}\sqrt{{25}^{x-1}+7}}+{3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}}}^{10}$ in the increasing powers of ${3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}$ is equal to $180,$ is :

Let $S={1,2,3,4,5,6,9}$. Then the number of elements in the set $T={A\subseteq S:A\neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3}$ is

If $a+b+c=1,ab+bc+ca=2$ and $abc=3,$ then the value of ${a}^{4}+{b}^{4}+{c}^{4}$ is equal to:

Team $A''$ consists of $7$ boys and $n$ girls and Team $B''$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:

If the greatest value of the term independent of $x$ in the expansion of ${(x\mathrm{sin}\alpha +a\frac{\mathrm{cos}\alpha }{x})}^{10}$ is $\frac{10!}{{(5!)}^{2}},$ then the value of $a$ is equal to:

Let $S={1,2,3,4,5,6,7}.$ Then the number of possible functions $f:S\rightarrow S$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in S$ and $m\cdot n\in S$, is equal to _____.

The number of $4$-digit numbers which are neither multiple of $7$ nor multiple of $3$ is .

Let $i=\sqrt{-1}.$ If $\frac{{(-1+i\sqrt{3})}^{21}}{{(1-i)}^{24}}+\frac{{(1+i\sqrt{3})}^{21}}{{(1+i)}^{24}}=k,$ and $n=[|k|]$ be the greatest integral part of $|k|.$ Then $\sum _{j=0}^{n+5}{(j+5)}^{2}-\sum _{j=0}^{n+5}(j+5)$ is equal to ________.

The coefficient of ${x}^{256}$ in the expansion of ${(1-x)}^{101}{({x}^{2}+x+1)}^{100}$ is:

If |z - 2| = |z + 2|, then z lies on:

If the real part of the complex number ${(1-\mathrm{cos}\theta +2i\mathrm{sin}\theta )}^{-1}$ is $\frac{1}{5}$ for $\theta \in (0,\pi ),$ then the value of the integral ${\int }_{0}^{\theta }\mathrm{sin}xdx$ is equal to: