Algebra PYQ — Page 45

Let $\frac{1}{16},a$ and $b$ be in G.P. and $\frac{1}{a},\frac{1}{b},6$ be in A.P., where $a,b>0$. Then $72(a+b)$ is equal to _______ .

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

If $\alpha ,\beta \in R$ are such that $1-2i$ (here ${i}^{2}=-1$) is a root of ${z}^{2}+\alpha z+\beta =0$, then $(\alpha -\beta )$ is equal to:

If the equation $a|z{|}^{2}+\bar{\bar{\alpha }z+\alpha \bar{z}}+d=0$ represents a circle where $a,d$ are real constants then which of the following condition is correct?

Let $A$ be a $3\times 3$ matrix with $det(A)=4$. Let ${R}_{i}$ denote the ${i}^{th}$ row of $A$. If a matrix $B$ is obtained by performing the operation ${R}_{2}\rightarrow 2{R}_{2}+5{R}_{3}$ on $2A$, then $det(B)$ is equal to :

Let ${S}_{1},{S}_{2}$ and ${S}_{3}$ be three sets defined as ${S}_{1}={z\in \mathbb{C}:|z-1|\leq \sqrt{2}},$ ${S}_{2}={z\in \mathbb{C}:Re((1-i)z)\geq 1}$ and ${S}_{3}={z\in \mathbb{C}:Im(z)\leq 1}.$ Then, the set ${S}_{1}\cap {S}_{2}\cap {S}_{3}$

If $A=[\begin{matrix}2 & 3 \\ 0 & -1\end{matrix}],$ then the value of $det({A}^{4})+det({A}^{10}-(Adj(2A){)}^{10})$ is equal to ________.

The number of solutions of the equation ${\mathrm{log}}_{(x+1)}(2{x}^{2}+7x+5)+{\mathrm{log}}_{(2x+5)}(x+1{)}^{2}-4=0,x>0,$ is

If $\alpha ,\beta$ are natural numbers such that ${100}^{\alpha }-199\beta =(100)(100)+(99)(101)+(98)(102)+\ldots ..+(1)(199),$ then the slope of the line passing through $(\alpha ,\beta )$ and origin is:

The term independent of $x$ in the expansion of ${[\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}}]}^{10},x\neq 1$, is equal to ___.

If for the complex numbers$z$ satisfying $|z-2-2i|\leq 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to _____ .

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

The domain of the function, $f(x)={\mathrm{sin}}^{-1}(\frac{3{x}^{2}+x-1}{(x-1{)}^{2}})+{\mathrm{cos}}^{-1}(\frac{x-1}{x+1})$ is:

The students ${S}_{1},{S}_{2},\ldots ,{S}_{10}$ are to be divided into $3$ groups $A,B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is __________.

The ratio of the coefficient of the middle term in the expansion of $(1+x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x{)}^{19}$ is .

Let the domain of the function $f(x)={\mathrm{log}}_{4}({\mathrm{log}}_{5}({\mathrm{log}}_{3}(18x-{x}^{2}-77)))$ be $(a,b)$. Then the value of the integral ${\int }_{a}^{b}\frac{{\mathrm{sin}}^{3}x}{({\mathrm{sin}}^{3}x+{\mathrm{sin}}^{3}(a+b-x))}$ is equal to _____.

The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :

Let $A=[\begin{matrix}2 & 3 \\ a & 0\end{matrix}],a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $det(Q)=9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:

Let $z$ and $w$ be two complex numbers such that $w=z\bar{z}-2z+2,|\frac{z+i}{z-3i}|=1$ and $Re(w)$ has minimum value. Then, the minimum value of $n\in N$ for which ${w}^{n}$ is real, is equal to _______.

Let $A$ and $B$ be two $3\times 3$ real matrices such that $({A}^{2}-{B}^{2})$ is invertible matrix. If ${A}^{5}={B}^{5}$ and ${A}^{3}{B}^{2}={A}^{2}{B}^{3},$ then the value of the determinant of the matrix ${A}^{3}+{B}^{3}$ is equal to :

Let a complex number be $w=1-\sqrt{3}i$. Let another complex number $z$ be such that $|zw|=1$ and $\mathrm{arg}(z)-\mathrm{arg}(w)=\frac{\pi }{2}$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$ be an A.P. If $\frac{{a}_{1}+{a}_{2}+\ldots +{a}_{10}}{{a}_{1}+{a}_{2}+\ldots +{a}_{p}}=\frac{100}{{p}^{2}},p\neq 10,$ then $\frac{{a}_{11}}{{a}_{10}}$ is equal to :

If ${\mathrm{log}}_{3}2,{\mathrm{log}}_{3}({2}^{x}-5),{\mathrm{log}}_{3}({2}^{x}-\frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to _____.

Let $A=[{a}_{ij}]$ be a real matrix of order $3\times 3,$ such that ${a}_{i1}+{a}_{i2}+{a}_{i3}=1,$ for $i=1,2,3.$ Then, the sum of all the entries of the matrix ${A}^{3}$ is equal to: