Algebra PYQ — Page 43

The total number of three-digit numbers, with one digit repeated exactly two times, 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

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 _____.

The number of $\theta \in (0,4\pi )$ for which the system of linear equations $3(\mathrm{sin}3\theta )x-y+z=2$ $3(\mathrm{cos}2\theta )x+4y+3z=3$ $6x+7y+7z=9$ has no solution is

Let $A=(\begin{matrix}2 & -1 \\ 0 & 2\end{matrix})$. If $B=I-C15(adjA)+C25(adjA{)}^{2}-...-C55{(\mathrm{adj}A)}^{5}$, then the sum of all elements of the matrix $B$ 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 a set $A={A}_{1}\cup {A}_{2}\cup \ldots \cup {A}_{k}$, where ${A}_{i}\cap {A}_{j}=\phi$ for $i\neq j;1\leq i,j\leq k.$ Define the relation $R$ from $A$ to $A$ by $R=${$(x,y):y\in {A}_{i}$ if and only if $x\in {A}_{i},1\leq i\leq k$}. Then, $R$ is:

Let ${R}_{1}$ and ${R}_{2}$ be two relations defined on $\mathbb{R}$ by $a{R}_{1}b\Leftrightarrow ab\geq 0$ and $a{R}_{2}b\Leftrightarrow a\geq b$, then

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 _____.

Let $A=(\begin{matrix}2 & -2 \\ 1 & -1\end{matrix})$ and$B=(\begin{matrix}-1 & 2 \\ -1 & 2\end{matrix})$ . Then the number of elements in the set {$(n,m):n,m\in {1,2,\ldots \ldots .10}$ and $n{A}^{n}+m{B}^{m}=I$} is _____.

Let a function $f:\mathbb{N}\rightarrow \mathbb{N}$ be defined by $f(n)=[\begin{matrix}2n, & n=2,4,6,8,\ldots .. \\ n-1, & n=3,7,11,15,\ldots .. \\ \frac{n+1}{2}, & n=1,5,9,13,\ldots ..\end{matrix}$ then, $f$ is

If the system of equations $\alpha x+y+z=5,x+2y+3z=4,x+3y+5z=\beta$. Has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to

Consider two G.Ps. $2,{2}^{2},{2}^{3},\ldots$ and $4,{4}^{2},{4}^{3},\ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is ${(2)}^{\frac{225}{8}}$, then $\sum _{k=1}^{n}k(n-k)$ is equal to:

Let $A=(\begin{matrix}1 & 2 \\ -2 & -5\end{matrix})$. Let $\alpha ,\beta \in \mathbb{R}$ be such that $\alpha {A}^{2}+\beta A=2I$. Then $\alpha +\beta$ is equal to

The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by $55$ is _____.

The number of ways to distribute $30$ identical candies among four children ${C}_{1},{C}_{2},{C}_{3}$ and ${C}_{4}$ so that ${C}_{2}$ receives atleast $4$ and atmost $7$ candies, ${C}_{3}$ receives atleast $2$ and atmost $6$ candies, is equal to

If ${{{a}_{i}}}_{i=1}^{n}$, where $n$ is an even integer, is an arithmetic progression with common difference $1$, and $\sum _{i=1}^{n}{a}_{i}=192,\sum _{i=1}^{\frac{n}{2}}{a}_{2i}=120$, then $n$ is equal to

The remainder when ${7}^{2022}+{3}^{2022}$ is divided by $5$ is

If $\sum _{k=1}^{10}\frac{k}{{k}^{4}+{k}^{2}+1}=\frac{m}{n}$, where $m$ and $n$ are co-prime, then $m+n$ is equal to

Let for the ${9}^{\mathrm{th}}$ term in the binomial expansion of ${(3+6x)}^{n}$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is ${n}_{0}$. If $k$ is the ratio of the coefficient of ${x}^{6}$ to the coefficient of ${x}^{3}$, then $k+{n}_{0}$ is equal to

Let $A=[\begin{matrix}1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1\end{matrix}],a,b\in \mathbb{R}$. If for some $n\in N,{A}^{n}=[\begin{matrix}1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1\end{matrix}]$ then $n+a+b$ is equal to _______.

The sum $\sum _{n=1}^{21}\frac{3}{(4n-1)(4n+3)}$ is equal to

If $A=\sum _{n=1}^{\infty }\frac{1}{{(3+{(-1)}^{n})}^{n}}$ and $B=\sum _{n=1}^{\infty }\frac{{(-1)}^{n}}{{(3+{(-1)}^{n})}^{n}}$, then $\frac{A}{B}$ is equal to

If the system of linear equations. $8x+y+4z=-2$ $x+y+z=0$ $\lambda x-3y=\mu$ has infinitely many solutions, then the distance of the point $(\lambda ,\mu ,-\frac{1}{2})$ from the plane $8x+y+4z+2=0$ is: