Algebra PYQ — Page 35

Let $A={2,3,4}$ and $B={8,9,12}$. Then the number of elements in the relation $R={(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}))\in (A\times B,A\times B):{a}_{1}\text{divides}{b}_{2}\text{and}{a}_{2}\text{divides}{b}_{1}}$ is

Let the number $(22{)}^{2022}+(2022{)}^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $({\alpha }^{2}+{\beta }^{2})$ is equal to

The coefficient of ${x}^{-6}$, in the expansion of ${(\frac{4x}{5}+\frac{5}{2{x}^{2}})}^{9}$, is

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is$_______.$

The sum, of the coefficients of the first $50$ terms in the binomial expansion of ${(1-x)}^{100},$ is equal to

The total number of $4$-digit numbers whose greatest common divisor with $54$ is $2$ , is

The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together is

The minimum number of elements that must be added to relation $R={(a,b),(b,c),(b,d)}$ on the set ${a,b,c,d}$, so that it is an equivalence relation is

If the solution of the equation ${\mathrm{log}}_{\mathrm{cos}x}(\mathrm{cot}x)+4{\mathrm{log}}_{\mathrm{sin}x}(\mathrm{tan}x)=1,x\in (0,\frac{\pi }{2})$ is ${\mathrm{sin}}^{-1}(\frac{\alpha +\sqrt{\beta }}{2})$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to:

If $C32n:C3n=10:1,$ then the ratio $({n}^{2}+3n):({n}^{2}-3n+4)$ is

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

The coefficient of ${x}^{7}$ in ${(1-x+2{x}^{3})}^{10}$ is __________ .

The number of triplets $(x,y,z)$ where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is

Let $5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3,x>0.$ Then $18{\int }_{1}^{2}f(x)dx$ is equal to

Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set $X={z\in C:\mathrm{Re}(a{z}^{2}+bz)=a\mathrm{and}\mathrm{Re}(b{z}^{2}+az)=b}$ is equal to

The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to ____.

Let $A={[{a}_{ij}]}_{2\times 2},$ where $a\neq ij0$ for all $i,j$ and ${A}^{2}=I$, Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$ Then $3{a}^{2}+4{b}^{2}$ is equal to

The remainder, when ${7}^{103}$ is divided by $17,$ is

The domain of $f(x)=\frac{{\mathrm{log}}_{(x+1)}(x-2)}{{e}^{2{\mathrm{log}}_{e}x}-(2x+3)},x\in R$ is

Let $A$ be a $3\times 3$ matrix such that $|adj(adj(adj.A))|={12}^{4}$. Then $|{A}^{-1}adjA|$ is equal to

Let $P(S)$ denote the power set of $S={1,2,3,\ldots ,10}$. Define the relations ${R}_{1}$ and ${R}_{2}$ on $P(S)$ as $A{R}_{1}B$ if $(A\cap {B}^{c})\cup (B\cap {A}^{c})=\phi$ and $A{R}_{2}B$ if $A\cup {B}^{c}=B\cup {A}^{c},\forall A,B\in P(S)$ . Then :

Let $\alpha >0$, be the smallest number such that the expansion of ${({x}^{\frac{2}{3}}+\frac{2}{{x}^{3}})}^{30}$ has a term $\beta {x}^{-\alpha },\beta \in N$. Then $\alpha$ is equal to _____ .

Let $p,q\in \mathbb{R}$ and $(1-\sqrt{3}i{)}^{200}={2}^{199}(p+iq)$, $i=\sqrt{-1}$. Then, $p+q+{q}^{2}$ and $p-q+{q}^{2}$ are roots of the equation.

The number of real roots of the equation $\sqrt{{x}^{2}-4x+3}+\sqrt{{x}^{2}-9}=\sqrt{4{x}^{2}-14x+6}$, is: