Algebra PYQ — Page 22

Let $m\text{and}n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of ${(\frac{1}{3}{x}^{\frac{1}{3}}+\frac{1}{2{x}^{\frac{2}{3}}})}^{18}$. Then ${(\frac{n}{m})}^{\frac{1}{3}}$ is:

If ${\mathrm{log}}_{e}a,{\mathrm{log}}_{e}b,{\mathrm{log}}_{e}c$ are in an $A.P.$ and ${\mathrm{log}}_{e}a-{\mathrm{log}}_{e}2b,{\mathrm{log}}_{e}2b-{\mathrm{log}}_{e}3c,{\mathrm{log}}_{e}3c-{\mathrm{log}}_{e}a$ are also in an $A.P.$, then $a:b:c$ is equal to

Let A be a $3\times 3$ real matrix such that $A(\begin{matrix}1 \\ 0 \\ 1\end{matrix})=2(\begin{matrix}1 \\ 0 \\ 1\end{matrix}),A(\begin{matrix}-1 \\ 0 \\ 1\end{matrix})=4(\begin{matrix}-1 \\ 0 \\ 1\end{matrix}),A(\begin{matrix}0 \\ 1 \\ 0\end{matrix})=2(\begin{matrix}0 \\ 1 \\ 0\end{matrix})$. Then, the system $(A-3I)(\begin{matrix}x \\ y \\ z\end{matrix})=(\begin{matrix}1 \\ 2 \\ 3\end{matrix})$ has

If the function $f:(-\infty ,-1]\rightarrow (a,b)]$ defined by $f(x)={e}^{{x}^{3}-3x+1}$ is one-one and onto, then the distance of the point $P(2b+4,a+2)$ from the line $x+{e}^{-3}y=4$ is:

Let ${S}_{n}$ be the sum to n-terms of an arithmetic progression $3,7,11,\ldots \ldots$, if $40<(\frac{6}{n(n+1)}\sum _{k=1}^{n}{S}_{k})<42$, then $n$ equals ____________.

Let $R$ be a relation on $Z\times Z$ defined by $(a,b)R(c,d)$ if and only if $ad-bc$ is divisible by $5$ . Then $R$ is

If the domain of the function $f(x)=\frac{\sqrt{{x}^{2}-25}}{(4-{x}^{2})}+{\mathrm{log}}_{10}({x}^{2}+2x-15)$ is $(-\infty ,\alpha )\cup [\beta ,\infty ),$ then ${\alpha }^{2}+{\beta }^{3}$ is equal to:

The ${20}^{\text{th }}$ term from the end of the progression $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\ldots ,-129\frac{1}{4}$ is :-

Let $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then $n$ is equal to_________

If each term of a geometric progression ${a}_{1},{a}_{2},{a}_{3},\ldots$ with ${a}_{1}=\frac{1}{8}$ and ${a}_{2}\neq {a}_{1}$, is the arithmetic mean of the next two terms and ${S}_{n}={a}_{1}+{a}_{2}+\ldots +{a}_{n}$, then ${S}_{20}-{S}_{18}$ is equal to

Let $A={I}_{2}-2M{M}^{T},$ where $M$ is real matrix of order $2\times 1$ such that the relation ${M}^{T}M={I}_{1}$ holds. If $\lambda$ is a real number such that the relation $AX=\lambda X$ holds for some non-zero real matrix $X$ of order $2\times 1,$ then the sum of squares of all possible values of $\lambda$ is equal to:

Let ${2}^{\mathrm{nd}},{8}^{\mathrm{th}}$ and ${44}^{\mathrm{th}}$, terms of a non-constant $A.P.$ be respectively the ${1}^{\mathrm{st}},{2}^{\mathrm{nd}}$ and ${3}^{\mathrm{rd}}$ terms of $G.P.$ If the first term of A.P. is $1$ then the sum of first $20$ terms is equal to-

The value of $\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots .+100^2 \times 101}$ is

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples, is

The total number of words (with or without meaning) that can be formed out of the letters of the word "DISTRIBUTION" taken four at a time, is equal to ______.

If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $\mathbf{R}-(\alpha, \beta)$, then $12 \alpha \beta$ is equal to :

Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is ______

Let $f:R-{\frac{-1}{2}}\rightarrow R$ and $g:R-{\frac{-5}{2}}\rightarrow R$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$. Then the domain of the function $\mathrm{fog}$ is :

A group of $40$ students appeared in an examination of $3$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $20$ students passed in Mathematics, $25$ students passed in Physics, $16$ students passed in Chemistry, at most $11$ students passed in both Mathematics and Physics, at most $15$ students passed in both Physics and Chemistry, at most $15$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.

Consider the following two statements : Statement I : For any two non-zero complex numbers $z_1, z_2$, $\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right) \text {, and }$ Statement II : If $x, y, z$ are three distinct complex numbers and $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three positive real numbers such that $\frac{\mathrm{a}}{|y-z|}=\frac{\mathrm{b}}{|z-x|}=\frac{\mathrm{c}}{|x-y|}$, then $\frac{\mathrm{a}^2}{y-z}+\frac{\mathrm{b}^2}{z-x}+\frac{\mathrm{c}^2}{x-y}=1 .$ Between the above two statements,

Let the set $C={(x,y)\mid {x}^{2}-{2}^{y}=2023,x,y\in \mathbb{N}}$. Then $\underset{(x,y)\in C}{\sum }(x+y)$ is equal to _______.

Let ${S}_{n}$ denote the sum of the first n terms of an arithmetic progression. If ${S}_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$, then ${S}_{15}-{S}_{5}$ is equal to:

Let $A=[\begin{matrix}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{matrix}],B=[\begin{matrix}{B}_{1} & {B}_{2} & {B}_{3}\end{matrix}]$, where ${B}_{1}$, ${B}_{2},{B}_{3}$ are column matrices, and ${\mathrm{AB}}_{1}=[\begin{matrix}1 \\ 0 \\ 0\end{matrix}]$, ${\mathrm{AB}}_{2}=[\begin{matrix}2 \\ 3 \\ 0\end{matrix}],{\mathrm{AB}}_{3}=[\begin{matrix}3 \\ 2 \\ 1\end{matrix}]$ If $\alpha =|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then ${\alpha }^{3}+{\beta }^{3}$ is equal to