Algebra PYQ — Page 26

For two non-zero complex number ${z}_{1}$ and ${z}_{2}$, if $Re({z}_{1}{z}_{2})=0$ and $Re({z}_{1}+{z}_{2})=0$, then which of the following are possible? (A) $Im({z}_{1})>0$ and $Im({z}_{2})>0$ (B) $Im({z}_{1})<0$ and $Im({z}_{2})>0$ (C) $Im({z}_{1})>0$ and $Im({z}_{2})<0$ (D) $Im({z}_{1})<0$ and $Im({z}_{2})<0$ Choose the correct answer from the options given below:

Let ${z}_{1}=2+3i$ and ${z}_{2}=3+4i$. The set $S={z\in C:{|z-{z}_{1}|}^{2}-{|z-{z}_{2}|}^{2}={|{z}_{1}-{z}_{2}|}^{2}}$ represents a

A person forgets his $4$-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is $7$ and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is________.

Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1,2,3,4,5$ with repetition, is ________

The total number of six digit numbers, formed using the digits $4,5,9$ only and divisible by $6$, is _____ .

Let $A=[\begin{matrix}1 & \frac{1}{51} \\ 0 & 1\end{matrix}]$. If $B=[\begin{matrix}1 & 2 \\ -1 & -1\end{matrix}]A[\begin{matrix}-1 & -2 \\ 1 & 1\end{matrix}]$, then the sum of all the elements of the matrix $\sum _{n=1}^{50}{B}^{n}$ is equal to

Let $P$ be a square matrix such that ${P}^{2}=I-P$. For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if ${P}^{\alpha }+{P}^{\beta }=\gamma l-29P$ and ${P}^{\alpha }-{P}^{\beta }=\delta l-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

The number of $3$-digit numbers, that are divisible by either $2$ or $3$ but not divisible by $7$ is _____ .

Let ${a}_{1},{a}_{2},\ldots \ldots ,{a}_{n}$ be in A.P. If ${a}_{5}=2{a}_{7}$ and ${a}_{11}=18$, then $12(\frac{1}{\sqrt{{a}_{10}}+\sqrt{{a}_{11}}}+\frac{1}{\sqrt{{a}_{11}}+\sqrt{{a}_{12}}}+\ldots ..\frac{1}{\sqrt{{a}_{17}}+\sqrt{{a}_{18}}})$ is equal to _____ .

If the sum and product of four positive consecutive terms of a G.P., are $126$ and $1296$, respectively, then the sum of common ratios of all such GPs is

Let ${a}_{1}={b}_{1}=1$ and ${a}_{n}={a}_{n-1}+(n-1),{b}_{n}={b}_{n-1}+{a}_{n-1},\forall n\geq 2$. If $S=\sum _{n=1}^{10}(\frac{{b}_{n}}{{2}^{n}})$ and $T=\sum _{n=1}^{8}\frac{n}{{2}^{n-1}}$ then ${2}^{7}(2S-T)$ is equal to

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1,\forall x$, $y\in \mathbb{R}$. If ${f}^{'}(0)=2$, then $|f(-2)|$ is equal to

If $A$ and $B$ are two non-zero $n\times n$ matrices such that ${A}^{2}+B={A}^{2}B,$ then

Let he sum of the coefficient of first three terms in the expansion of ${(x-\frac{3}{{x}^{2}})}^{n};x=0,n\in N$ be $376$. Then, the coefficient of ${x}^{4}$ is equal to:

The sum of the common terms of the following three arithmetic progressions. $3,7,11,15,\ldots \ldots \ldots \ldots ,399$ $2,5,8,11,.........359$ and $2,7,12,17,\ldots \ldots ,197$, is equal to _____ .

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

If the system of linear equations $7x+11y+\alpha z=13$ $5x+4y+7z=\beta$ $175x+194y+57z=361$ has infinitely many solutions, then $\alpha +\beta +2$ is equal to

Let $P=[\begin{matrix}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{matrix}],A=[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}]$ and $Q=PA{P}^{T}$. If ${P}^{T}{Q}^{2007}P=[\begin{matrix}a & b \\ c & d\end{matrix}]$then $2a+b-3c-4d$ is equal to

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x+y+z=1$ $x+\lambda y+z=1$ $x+y+\lambda z=1$ is inconsistent, then $\underset{\lambda \in S}{\sum }({|\lambda |}^{2}+|\lambda |)$ is equal to

The number of numbers, strictly between $5000$ and $10000$ can be formed using the digits $1,3,5,7,9$ without repetition, is

Let ${(a+bx+c{x}^{2})}^{10}=\sum _{i=10}^{20}{p}_{i}{x}^{i},a,b,c\in \mathbb{N}$. If ${p}_{1}=20$ and ${p}_{2}=210$, then $2(a+b+c)$ is equal to

The number of functions $f:{1,2,3,4}\rightarrow {a\in \mathbb{Z}:|a|\leq 8}$ satisfying $f(n)+\frac{1}{n}f(n+1)=1,\forall n\in {1,2,3}$ is

The number of points, where the curve $f(x)={e}^{8x}-{e}^{6x}-3{e}^{4x}-{e}^{2x}+1,x\in \mathbb{R}$ cuts $x$-axis, is equal to............

For all $z\in C$ on the curve ${C}_{1}:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve ${C}_{2}$. Then