Algebra PYQ — Page 42

Let ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ be a $4$-element permutation with ${b}_{i}\in$ ${1,2,3,\ldots \ldots \ldots ,100}$ for $1\leq i\leq 4$ and ${b}_{i}\neq {b}_{j}$ for $i\neq j$, such that either ${b}_{1},{b}_{2},{b}_{3}$ are consecutive integers or ${b}_{2},{b}_{3},{b}_{4}$ are consecutive integers. Then the number of such permutations ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ is equal to ______.

Let $S$ be the set containing all $3\times 3$ matrices with entries from ${-1,0,1}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of ${A}^{T}A$ is $6$ is ______.

The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{{x}^{2}-3x+2}{{x}^{2}+2x+7})$ is

The number of values of $\alpha$ for which the system of equations $x+y+z=\alpha$ $\alpha x+2\alpha y+3z=-1$ $x+3\alpha y+5z=4$ is inconsistent, is

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+(2i-1)=0$. Then, the value of $|{\alpha }^{8}+{\beta }^{8}|$ is equal to

The number of elements in the set {$z=a+ib\in \mathbb{C}:a,b\in \mathbb{Z}$ and $1<|z-3+2i|<4$} is _____.

Let $S={z\in C:|z-2|\leq 1,z(1+i)+\bar{z}(1-i)\leq 2}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at ${z}_{1}\in S$ and ${z}_{2}\in S$. If $5({|{z}_{1}|}^{2}+{|{z}_{2}|}^{2})=\alpha +\beta \sqrt{5}$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha +\beta$ is equal to ______.

Let $A={z\in C:1\leqslant |z-(1+i)|\leqslant 2}$ and $B={z\in A:|z-(1-i)|=1}$. Then, $B$

Let $A=[\begin{matrix}1 & -1 \\ 2 & \alpha \end{matrix}]$ and $B=[\begin{matrix}\beta & 1 \\ 1 & 0\end{matrix}],\alpha ,\beta \in R$. Let ${\alpha }_{1}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={A}^{2}+[\begin{matrix}2 & 2 \\ 2 & 2\end{matrix}]$ and ${\alpha }_{2}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={B}^{2}$. Then $|{\alpha }_{1}-{\alpha }_{2}|$ is equal to

Let $A={1,2,3,4,5,6,7}$. Define $B=${$T\subseteq A$: either $1\notin T$ or $2\in T$} and $C=${$T\subseteq A:T$ the sum of all the elements of $T$ is a prime number.} Then the number of elements in the set $B\cup C$ is _______.

If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is

Let $\alpha ,\beta (\alpha >\beta )$ be the roots of the quadratic equation ${x}^{2}-x-4=0$. If ${P}_{n}={\alpha }^{n}-{\beta }^{n},n\in \mathbb{N}$, then $\frac{{P}_{15}{P}_{16}-{P}_{14}{P}_{16}-{P}_{15}^{2}+{P}_{14}{P}_{15}}{{P}_{13}{P}_{14}}$ is equal to _____.

If for some $p,q,r\in R$, all have positive sign, one of the roots of the equation $({p}^{2}+{q}^{2}){x}^{2}-2q(p+r)x+{q}^{2}+{r}^{2}=0$ is also a root of the equation ${x}^{2}+2x-8=0$, then $\frac{{q}^{2}+{r}^{2}}{{p}^{2}}$ is equal to-

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-4\lambda x+5=0$ and $\alpha ,\gamma$ be the roots of the equation ${x}^{2}-(3\sqrt{2}+2\sqrt{3})x+7+3\lambda \sqrt{3}=0$. If $\beta +\gamma =3\sqrt{2}$, then ${(\alpha +2\beta +\gamma )}^{2}$ is equal to

Let $a,b\in R$ be such that the equation $a{x}^{2}-2bx+15=0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}-2bx+21=0$, then ${\alpha }^{2}+{\beta }^{2}$ is equal to:

The remainder when ${(11)}^{1011}+{(1011)}^{11}$ is divided by $9$ is _____ .

If $z=2+3i$, then ${z}^{5}+{(\bar{z})}^{5}$ is equal to:

Let the minimum value ${v}_{0}$ of $v={|z|}^{2}+{|z-3|}^{2}+{|z-6i|}^{2}$, $z\in \mathbb{C}$ is attained at $z={z}_{0}$. Then ${|2{z}_{0}^{2}-{\bar{z}}_{0}^{3}+3|}^{2}+{v}_{0}^{2}$ is equal to

Let ${S}_{1}={{z}_{1}\in C:|{z}_{1}-3|=\frac{1}{2}}$ and ${S}_{2}={{z}_{2}\in C:|{z}_{2}-|{z}_{2}+1||=|{z}_{2}+|{z}_{2}-1||}.$ Then, for ${z}_{1}\in {S}_{1}$ and ${z}_{2}\in {S}_{2}$, the least value of $|{z}_{2}-{z}_{1}|$ is

For $n\in N$, let ${S}_{n}={z\in C:|z-3+2i|=\frac{n}{4}}$ and ${T}_{n}={z\in C:|z-2+3i|=\frac{1}{n}}$. Then the number of elements in the set ${n\in N:{S}_{n}\cap {T}_{n}=\phi }$ is

Let $(z)$ represent the principal argument of the complex number $z$. The, $|z|=3$ and $\mathrm{arg}(z-1)-\mathrm{arg}(z+1)=\frac{\pi }{4}$ intersect:

Let $A={z\in C:|\frac{z+1}{z-1}|<1}$ and $B={z\in C:\mathrm{arg}(\frac{z-1}{z+1})=\frac{2\pi }{3}}$. Then $A\cap B$ is

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that ${\bar{z}}_{1}=i{\bar{z}}_{2}$ and $\mathrm{arg}(\frac{{z}_{1}}{{\bar{z}}_{2}})=\pi$, then the argument of ${z}_{1}$ is

Let a circle $C$ in complex plane pass through the points ${z}_{1}=3+4i,{z}_{2}=4+3i$ and ${z}_{3}=5i$. If $z(\neq {z}_{1})$ is a point on $C$ such that the line through $z$ and ${z}_{1}$ is perpendicular to the line through ${z}_{2}$ and ${z}_{3}$, then $\mathrm{arg}(z)$ is equal to