Algebra PYQ — Page 27

The number of $3$ digit numbers, that are divisible by either $3$ or $4$ but not divisible by $48$ , is

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

The 4$^{th}$ term of GP is $500$ and its common ratio is $\frac{1}{m},m\in N$. Let ${S}_{n}$ denote the sum of the first $n$ terms of this GP. If ${S}_{6}>{S}_{5}+1$ and ${S}_{7}<{S}_{6}+\frac{1}{2}$, then the number of possible values of $m$ is ______

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 _____ .

Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of ${a}^{5}{b}^{3}{c}^{2}d$ is $3750\beta$, then the value of $\beta$ is

Let $B=[\begin{matrix}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{matrix}],\alpha >2$ be the adjoint of a matrix $A$ and $|A|=2.$ Then $[\begin{matrix}\alpha & -2\alpha & \alpha \end{matrix}]B[\begin{matrix}\alpha \\ -2\alpha \\ \alpha \end{matrix}]$ is equal to

Let $R$ be a relation on $\mathbb{R}$, given by $R={(a,b):3a-3b+\sqrt{7}$ is an irrational number $}$. Then $R$ 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

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{2x}+2=0$ Then ${\alpha }^{14}+{\beta }^{14}$ is equal to

Let $A=[\begin{matrix}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{matrix}]$ and $B=[\begin{matrix}1 & -i \\ 0 & 1\end{matrix}]$, where $i=\sqrt{-1}$. If $M={A}^{T}\mathrm{BA}$, then the inverse of the matrix ${\mathrm{AM}}^{2023}{A}^{T}$ is

Let $R$ be a relation defined on $\mathbb{N}$ as a $R$ b is $2a+3b$ is a multiple of $5,a,b\in \mathbb{N}$. Then $R$ is

Let $a\in R$ and let $\alpha ,\beta$ be the roots of the equation ${x}^{2}+{60}^{\frac{1}{4}}x+a=0$. If ${\alpha }^{4}+{\beta }^{4}=-30$, then the product of all possible values of $a$ 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:

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $_____$

The coefficient of ${x}^{5}$ in the expansion of ${(2{x}^{3}-\frac{1}{3{x}^{2}})}^{5}$ is

The number of $9$ digit numbers, that can be formed using all the digits of the number $123412341$ so that the even digits occupy only even places, is ______

If the set ${Re(\frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}):z\in \mathbb{C},Rez=3}$ is equal to the interval $(\alpha ,\beta ]$, then $24(\beta -\alpha )$ is equal to

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is $840$, then the total numbers of persons, who participated in the tournament, is ________.

Let the system of linear equations $x+y+kz=2$ $2x+3y-z=1$ $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$ $(2k+1)x+|k+5|y=10$ has :