JEE Main 2023 — Algebra

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

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________.

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

Among the relations $S={(a,b):a,b\in R-{0},2+\frac{a}{b}>0}$ and $T={(a,b):a,b\in R,{a}^{2}-{b}^{2}\in Z}$,

Among the statements : $(S1):{2023}^{2022}-{1999}^{2022}$ is divisible by $8$. $(S2):13(13{)}^{n}-11n-13$ is divisible by $144$ for infinitely many $n\in \mathbb{N}$

An organization awarded $48$ medals in event $A''$, $25$ in event $B''$ and $18$ in event $C''$. If these medals went to total $60$ men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

If the sum of first n terms of an AP is 3n² + 5n, then the common difference is:

Consider a function $f:\mathbb{N}\rightarrow \mathbb{R}$, satisfying $f(1)+2f(2)+3f(3)+\ldots +xf(x)=x(x+1)f(x);x\geq 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

Consider the following system of questions $\alpha x+2y+z=1$ $2\alpha x+3y+z=1$ $3x+\alpha y+2z=\beta$ For some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct.

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number $77777$ has serial number $1$. Then the serial number of $35337$ is

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

For $\alpha ,\beta ,z\in \mathbb{C}$ and $\lambda >1$, if $\sqrt{\lambda -1}$ is the radius of the circle ${|z-\alpha |}^{2}+{|z-\beta |}^{2}=2\lambda ,$ then $|\alpha -\beta |$ is equal to _____.

For $a\in \mathbb{C}$, let $A={z\in \mathbb{C}:Re(a+\bar{z})>Im(\bar{a}+z)}$ and $B={z\in \mathbb{C}:Re(a+\bar{z})<Im(\bar{a}+z)}$. Then among the two statements: $(S1)$ : If $Re(a),Im(a)>0$, then the set $A$ contains all the real numbers $(S2)$ : If $Re(a),Im(a)<0$, then the set $B$ contains all the real numbers,

For some $a,b,c\in \mathbb{N}$, let $f(x)=ax-3$ and $g(x)={x}^{b}+c,x\in \mathbb{R}$. If ${(fog)}^{-1}(x)={(\frac{x-7}{2})}^{\frac{1}{3}}$, then $(f\circ g)(ac)+(g\circ f)(b)$ is equal to _____ .

For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha z=8$ $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

For the differentiable function $f:\mathbb{R}-{0}-\mathbb{R},$ let $3f(x)+2f(\frac{1}{x})=\frac{1}{x}-10,$ then $|f(3)+{f}^{'}(\frac{1}{4})|$ is equal to

For the system of equations $x+y+z=6$ $x+2y+\alpha z=10$ $x+3y+5z=\beta$, which one of the following is NOT true?

For the system of linear equations $2x-y+3z=5$ $3x+2y-z=7$ $4x+5y+\alpha z=\beta$, which of the following is NOT correct?

For the system of linear equations $x+y+z=6$ $\alpha x+\beta y+7z=3$ $x+2y+3z=14$ which of the following is NOT true ?

For the system of linear equations $2x+4y+2az=b$ $x+2y+3z=4$ $2x+5y+2z=8$ which of the following is $\mathrm{NOT}$ correct?

For the system of linear equations $ax+y+z=1$, $x+ay+z=1,x+y+az=\beta$, which one of the following statements is NOT correct?

For the two positive numbers $a,b$, if $a,b$ and $\frac{1}{18}$ are in a geometric progression, while $\frac{1}{a},10$ and $\frac{1}{b}$ are in an arithmetic progression, then, $16a+12b$ is equal to _____ .

For three positive integers $p,q,r$, ${x}^{p{q}^{2}}={y}^{qr}={z}^{{p}^{2}r}$ and $r=pq+1$ such that $3,3{\mathrm{log}}_{y}x,3{\mathrm{log}}_{z}y,7{\mathrm{log}}_{x}z$are in A.P. with common difference $\frac{1}{2}$. The $r-p-q$is equal to

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:

For $x\in \mathbb{R},$ two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $fog(x)=x+3-\sqrt{x}.$ Then $f(0)$ is equal to

Fractional part of the number $\frac{{4}^{2022}}{15}$ is equal to

If a point $P(\alpha ,\beta ,\gamma )$ satisfying $(\begin{matrix}\alpha & \beta & \gamma \end{matrix})(\begin{matrix}2 & 10 & 8 \\ 9 & 3 & 8 \\ 8 & 4 & 8\end{matrix})=(\begin{matrix}0 & 0 & 0\end{matrix})$ lies on the plane $2x+4y+3z=5$, then $6\alpha +9\beta +7\gamma$ is equal to

If all the six digit numbers ${x}_{1}{x}_{2}{x}_{3}{x}_{4}{x}_{5}{x}_{6}$ with $0<{x}_{1}<{x}_{2}<{x}_{3}<{x}_{4}<{x}_{5}<{x}_{6}$ are arranged in the increasing order, then the sum of the digits in the ${72}^{\text{th }}$ number is _______.

If $a$ and $b$ are the roots of the equation ${x}^{2}-7x-1=0$, then the value of $\frac{{a}^{21}+{b}^{21}+{a}^{17}+{b}^{17}}{{a}^{19}+{b}^{19}}$ is equal to

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

If $A=[\begin{matrix}1 & 5 \\ \lambda & 10\end{matrix}],{A}^{-1}=\alpha A+\beta I$ and $\alpha +\beta =-2$, then $4{\alpha }^{2}+{\beta }^{2}+{\lambda }^{2}$ is equal to :

If $gcd(m,n)=1$ and ${1}^{2}-{2}^{2}+{3}^{2}-{4}^{2}+....+{(2021)}^{2}-{(2022)}^{2}+{(2023)}^{2}=1012{m}^{2}n$ then ${m}^{2}-{n}^{2}$ is equal to

If domain of the function ${\mathrm{log}}_{e}(\frac{6{x}^{2}+5x+1}{2x-1})+{\mathrm{cos}}^{-1}(\frac{2{x}^{2}-3x+4}{3x-5})$ is $(\alpha ,\beta )\cup (\gamma ,\delta )$, then $18({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}+{\delta }^{2})$ is equal to

If for $z=\alpha +i\beta ,|z+2|=z+4(1+i),$ then $\alpha +\beta$ and $\alpha \beta$ are the roots of the equation

If $A$ is a $3\times 3$ matrix and $|A|=2$, then $|3adj(|3A|{A}^{2})|$ is equal to

If $P$ is a $3\times 3$ real matrix such that ${P}^{T}=aP+(a-1)I$, where $a>1$, then

If the center and radius of the circle $|\frac{z-2}{z-3}|=2$ are respectively $(\alpha ,\beta )$ and $\gamma$, then $3(\alpha +\beta +\gamma )$ is equal to

If the co-efficient of ${x}^{9}$ in ${(\alpha {x}^{3}+\frac{1}{\beta x})}^{11}$and the co-efficient of ${x}^{-9}$ in ${(\alpha x-\frac{1}{\beta {x}^{3}})}^{11}$ are equal, then $(\alpha \beta {)}^{2}$ is equal to

If the coefficient of ${x}^{7}$ in ${(ax-\frac{1}{b{x}^{2}})}^{13}$ and the coefficient of ${x}^{-5}$ in ${(ax+\frac{1}{b{x}^{2}})}^{13}$ are equal, then ${a}^{4}{b}^{4}$ is equal to:

If the coefficient of ${x}^{15}$ in the expansion of ${(a{x}^{3}+\frac{1}{b{x}^{\frac{1}{3}}})}^{15}$ is equal to the coefficient of ${x}^{-15}$ in the expansion of ${(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^{3}})}^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:

If the coefficients of $x$ and ${x}^{2}$ in $(1+x{)}^{p}(1-x{)}^{q}$ are $4$ and $-5$ respectively, then $2p+3q$ is equal to

If the coefficients of ${x}^{7}$ in ${(a{x}^{2}+\frac{1}{2bx})}^{11}$ and ${x}^{-7}$ in ${(ax-\frac{1}{3b{x}^{2}})}^{11}$ are equal, then

if the coefficients of three consecutive terms in the expansion of ${(1+x)}^{n}$ are the ratio $1:5:20$then the coefficient of the fourth term is

If the constant term in the binomial expansion of ${(\frac{{x}^{\frac{5}{2}}}{2}-\frac{4}{{x}^{l}})}^{9}$ is $-84$ and the coefficient of ${x}^{-3l}$ is ${2}^{\alpha }\beta$ where $\beta <0$ is an odd number, then $|\alpha l-\beta |$ is equal to _____ .

If the domain of the function $f(x)={\mathrm{sec}}^{-1}(\frac{2x}{5x+3})$ is $[\alpha ,\beta )\cup (\gamma ,\delta ]$, then $|3\alpha +10(\beta +\gamma )+21\delta |$ is equal to __________

If the domain of the function $f(x)={\mathrm{log}}_{e}(4{x}^{2}+11x+6)+{\mathrm{sin}}^{-1}(4x+3)+{\mathrm{cos}}^{-1}(\frac{10x+6}{3})$ is $(\alpha ,\beta ]$, then $36|\alpha +\beta |$ is equal to

If the domain of the function $f(x)=\frac{[x]}{1+{x}^{2}}$, where $[x]$ is greatest integer $\leq x$, is $[2,6)$, then its range is

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is

If the number of words, with or without meaning. which can be made using all the letters of the word MATHEMATICS in which $C$ and $S$ do not come together, is $(6!)k$ then $k$ is equal to

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^{n}$ is $\sqrt{6}:1$, then the third term from the beginning 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

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:

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:

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

If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

If the system of equations $x+y+az=b$ $2x+5y+2z=6$ $x+2y+3z=3$ has infinitely many solutions, then $2a+3b$ is equal to

If the system of equations $2x+y-z=5$ $2x-5y+\lambda z=\mu$ $x+2y-5z=7$ has infinitely many solutions, then$(\lambda +\mu {)}^{2}+(\lambda -\mu {)}^{2}$ is equal to

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

If the ${1011}^{\mathrm{th}}$ term from the end in the binomial expansion of ${(\frac{4x}{5}-\frac{5}{2x})}^{2022}$ is $1024$ times ${1011}^{\mathrm{th}}$ term from the beginning, then $32|x|$ is equal to

If the term without $x$ in the expansion of ${({x}^{\frac{2}{3}}+\frac{\alpha }{{x}^{3}})}^{22}$ is $7315$, then $|\alpha |$ is equal to _____ .

If the value of real number $\alpha >0$ for which ${x}^{2}-5\alpha x+1=0$ and ${x}^{2}-\alpha x-5=0$ have a common real roots is $\frac{3}{\sqrt{2\beta }}$ then $\beta$ is equal to ________

If $|\begin{matrix}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+{\lambda }^{2}\end{matrix}|=\frac{9}{8}(103x+81)$, then $\lambda ,\frac{\lambda }{3}$ are the roots of the equation

If $f(x)={x}^{3}-{x}^{2}{f}^{'}(1)+x{f}^{"}(2)-{f}^{'''}(3)$, $x\in R$, then

If $f(x)=\frac{{2}^{2x}}{{2}^{2x}+2}$, $x\in R$, then $f(\frac{1}{2023})+f(\frac{2}{2023})+f(\frac{3}{2023}).........f(\frac{2022}{2023})$ is equal to

If $A=\frac{1}{5!6!7!}[\begin{matrix}5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9!\end{matrix}]$, then $|adj(adj(2A))|$ is equal to

If $\frac{1}{n+1}{}^{n}{C}_{n}+\frac{1}{n}{}^{n}{C}_{n-1}+...+\frac{1}{2}{}^{n}{C}_{1}{+}^{n}{C}_{0}=\frac{1023}{10}$ then $n$ is equal to

If $Pn-12n+1:Pn2n-1=11:21$, then ${n}^{2}+n+15$ is equal to :

If $f(x)=\frac{(\mathrm{tan}{1}^{^{\circ}})x+{\mathrm{log}}_{e}(123)}{x{\mathrm{log}}_{e}(1234)-(\mathrm{tan}{1}^{^{\circ}})},x>0$, then the least value of $f(f(x))+f(f(\frac{4}{x}))$ is

If $C32n:C3n=10:1,$ then the ratio $({n}^{2}+3n):({n}^{2}-3n+4)$ is

If $\frac{{1}^{3}+{2}^{3}+{3}^{3}......\text{upto }n\text{terms}}{1\cdot 3+2\cdot 5+3\cdot 7+....\text{upto }n\text{terms}}=\frac{9}{5}$ then the value of $n$ is

If $A=\frac{1}{2}[\begin{matrix}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{matrix}]$ then,

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25({\beta }^{2}{x}^{2}+{\alpha }^{2}{y}^{2})={\alpha }^{2}{\beta }^{2}$ is

In an examination, $5$ students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is

${25}^{190}-{19}^{190}-{8}^{190}+{2}^{190}$ is divisible by

$\sum _{k=0}^{6}C351-k$ is equal to

If $\\alpha$ and $\\beta$ are roots of $x^2 - 5x + 6 = 0$, then $\\alpha + \\beta$ is

Let $A={1,2,3,4}$ and $R$ be a relation on the set $A\times A$ defined by $R={((a,b),(c,d)):2a+3b=4c+5d}$. Then the number of elements in $R$ is _________.

Let $x$ and $y$ be distinct integers where $1\leq x\leq 25$ and $1\leq y\leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x+y$ is divisible by $5$ , is _____ .

Let $a,b,c>1,{a}^{3},{b}^{3}$ and ${c}^{3}$ be in $A.P.$ and ${\mathrm{log}}_{a}b$, ${\mathrm{log}}_{c}a$ and ${\mathrm{log}}_{b}c$ be in $G.P.$ If the sum of first $20$ terms of an $A.P.$, whose first term is $\frac{a+4b+c}{3}$ and the common difference is $\frac{a-8b+c}{10}$ is $-444$, then $abc$ 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 $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that ${A}^{2}=3A+\alpha I$. If ${A}^{4}=21A+\beta I$, then

Let ${S}_{1}$ and ${S}_{2}$ be respectively the sets of all $a\in R-{0}$ for which the system of linear equations $ax+2ay-3az=1$ $(2a+1)x+(2a+3)y+(a+1)z=2$ $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions. Then

Let $m$ and $n$ be the numbers of real roots of the quadratic equations ${x}^{2}-12x+[x]+31=0$ and ${x}^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\leq x$. Then ${m}^{2}+mn+{n}^{2}$ is equal to

Let $A={0,3,4,6,7,8,9,10}$ and $R$ be the relation defined on $A$ such that $R{(x,y)\in A\times A:x-y\text{is odd positive integer}\text{or}x-y=2}$. The minimum number of elements that must be added to the relation $R,$ so that it is a symmetric relation, is equal to $_________$

Let ${A}_{1}$ and ${A}_{2}$ be two arithmetic means and ${G}_{1},{G}_{2}$ and ${G}_{3}$ be three geometric means of two distinct positive numbers. Then ${G}_{1}^{4}+{G}_{2}^{4}+{G}_{3}^{4}+{G}_{1}^{2}{G}_{3}^{2}$ is equal to

Let $\left\{a_k\right\}$ and $\left\{b_k\right\}, k \in \mathbb{N}$, be two G.P.s with common ratio $r_1$ and $r_2$ respectively such that $\mathrm{a}_1=\mathrm{b}_1=4$ and $\mathrm{r}_1&lt;\mathrm{r}_2$. Let $\mathrm{c}_{\mathrm{k}}=\mathrm{a}_{\mathrm{k}}+\mathrm{b}_{\mathrm{k}}, \mathrm{k} \in \mathbb{N}$. If $\mathrm{c}_2=5$ and $\mathrm{c}_3=\frac{13}{4}$ then $\sum_{\mathrm{k}=1}^{\infty} \mathrm{c}_{\mathrm{k}}-\left(12 \mathrm{a}_6+8 \mathrm{~b}_4\right)$ is equal to

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 $x={(8\sqrt{3}+13)}^{13}$ and $y={(7\sqrt{2}+9)}^{9}$. If $[t]$ denotes the greatest integer $\leq t$, then

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 $A={1,3,4,6,9}$ and $B={2,4,5,8,10}$. Let $R$ be a relation defined on $A\times B$ such that $R={({a}_{1},{b}_{1}),({a}_{2},{b}_{2}):{a}_{1}\leq {b}_{2}\text{ and}{b}_{1}\leq {a}_{2}}$. Then the number of elements in the set $R$ 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 _____ .

Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x{|}^{2}-2|x|+|\lambda -3|=0$. Then the largest element in the set $S=$ ${x+\lambda :x$ is an integer solution of $E}$ is ______

Let $A={-4,-3,-2,0,1,3,4}$ and $R={(a,b)\in A\times A$ : $b=|a|$ or ${b}^{2}=a+1}$ be a relation on $A$. Then the minimum number of elements, that must be added to the relation $R$ so that it becomes reflexive and symmetric, is

Let $A={1,2,3,4,..........10}$ and $B={0,1,2,3,4}.$ The number of elements in the relation $R={(a,b)\in A\times A:2{(a-b)}^{2}+3(a-b)\in B}$ is $__________.$

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

Let $p,q\in \mathbb{R}$ and $(1-\sqrt{3}i{)}^{200}={2}^{199}(p+iq)$, $i=\sqrt{-1}$. Then, $p+q+{q}^{2}$ and $p-q+{q}^{2}$ are roots of the equation.

Let $z=1+i$ and ${z}_{1}=\frac{1+i\bar{z}}{\bar{z}(1-z)+\frac{1}{z}}\cdot$ Then $\frac{12}{\pi }$ $\mathrm{arg}({z}_{1})$ is equal to

Let $x,y,z>1$ and $A=[\begin{matrix}1 & {\mathrm{log}}_{x}y & {\mathrm{log}}_{x}z \\ {\mathrm{log}}_{y}x & 2 & {\mathrm{log}}_{y}z \\ {\mathrm{log}}_{z}x & {\mathrm{log}}_{z}y & 3\end{matrix}]$. Then $|adj(adj{A}^{2})|$ is equal to

Let $\alpha =8-14i,A={z\in \mathbb{C}:\frac{\alpha z-\bar{\alpha }\bar{z}}{{z}^{2}-(\bar{z}{)}^{2}-112i}=1}$ and $B={z\in \mathbb{C}:|z+3i|=4}$ Then, $\underset{z\in A\cap B}{\sum }(Rez-Imz)$ is equal to ________

Let $A={2,3,4}$ and $B={8,9,12}$. Then the number of elements in the relation $R={(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}))\in (A\times B,A\times B):{a}_{1}\text{divides}{b}_{2}\text{and}{a}_{2}\text{divides}{b}_{1}}$ is

Let $A={1,2,3,4,5}$ and $B={1,2,3,4,5,6}$. Then the number of functions $f:A\rightarrow B$ satisfying $f(1)+f(2)=f(4)-1$ is equal to........

Let $f(x)=2{x}^{n}+\lambda ,\lambda \in \mathbb{R},n\in \mathbb{N}$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is

Let $A=[\begin{matrix}m & n \\ p & q\end{matrix}],d=|A|\neq 0$ and $|A-d(AdjA)|=0$. Then

Let $R={a,b,c,d,e}$ and $S={1,2,3,4}$. Total number of onto functions $f:R\rightarrow S$ such that $f(a)\neq 1$, 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 $z$ be a complex number such that $|\frac{z-2i}{z+i}|=2,z\neq -i$. Then $z$ lies on the circle of radius $2$ and centre

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

Let $f:(0,1)\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{1}{1-{e}^{-x}}$, and $g(x)=(f(-x)-f(x))$. Consider two statements (I) $g$ is an increasing function in $(0,1)$ (II) $g$ is one-one in $(0,1)$ Then,

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)={\mathrm{log}}_{\sqrt{m}}{\sqrt{2}(\mathrm{sin}x-\mathrm{cos}x)+m-2}$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is _____ .

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in N$, If $f(1)=3$ and $\sum _{k=1}^{n}f(k)=3279$, then the value of $n$ is

Let $f:R\rightarrow R$ be a function such that $f(x)=\frac{{x}^{2}+2x+1}{{x}^{2}+1}$. Then

Let $f:R-{0,1}\rightarrow R$ be a function such that $f(x)+f(\frac{1}{1-x})=1+x$. Then $f(2)$ is equal to :

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$. be a GP of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then ${a}_{1}{a}_{9}+{a}_{2}{a}_{4}{a}_{9}+{a}_{5}+{a}_{7}$ is equal to

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$. be a G.P. of increasing positive numbers. Let the sum of its ${6}^{\text{th }}$ and ${8}^{\text{th }}$ terms be $2$ and the product of its ${3}^{\text{rd }}$ and ${5}^{\text{th }}$ terms be $\frac{1}{9}$. Then $6({a}_{2}+{a}_{4})({a}_{4}+{a}_{6})$ is equal to

Let $A$ be a $n\times n$ matrix such that $|A|=2$. If the determinant of the matrix $Adj(2.Adj(2{A}^{-1}))$ is ${2}^{84}$, then $n$ is equal to _____ .

Let $A$ be a $3\times 3$ matrix such that $|adj(adj(adj.A))|={12}^{4}$. Then $|{A}^{-1}adjA|$ is equal to

Let $A$ be a $2\times 2$ matrix with real entries such that ${A}^{'}=\alpha A+1,$ where $\alpha \in \mathbb{R}-{-1,1}$., If det $({A}^{2}-A)=4$, the sum of all possible values of $\alpha$ is equal to

Let $\lambda \neq 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14{x}^{2}-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35{x}^{2}-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation :

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

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 $\alpha$ be a root of the equation $(a-c){x}^{2}+(b-a)x+(c-b)=0$ where $a,b,c$ are distinct real numbers such that the matrix $[\begin{matrix}{\alpha }^{2} & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{matrix}]$ is singular. Then the value of $\frac{{(a-c)}^{2}}{(b-a)(c-b)}+\frac{{(b-a)}^{2}}{(a-c)(c-b)}+\frac{{(c-b)}^{2}}{(a-c)(b-a)}$ is

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

Let $A$ be a symmetric matrix such that $|A|=2$ and $[\begin{matrix}2 & 1 \\ 3 & \frac{3}{2}\end{matrix}]A=[\begin{matrix}1 & 2 \\ \alpha & \beta \end{matrix}]$ If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{{\alpha }^{2}}$ is equal to _________.

Let ${a}_{1},{a}_{2},{a}_{3},\ldots \ldots$. be an A.P. If ${a}_{7}=3$, the product $({a}_{1}{a}_{4})$ is minimum and the sum of its first $n$ terms is zero then $n!-4{a}_{n(n+2)}$ is equal to

Let ${a}_{1}=8,{a}_{2},{a}_{3},\ldots .{a}_{n}$ be an A.P. If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is _____ .

Let ${x}_{1},{x}_{2},\ldots ,{x}_{100}$ be in an arithmetic progression, with ${x}_{1}=2$ and their mean equal to $200$ . If ${y}_{i}=i({x}_{i}-i),1\leq i\leq 100$, then the mean of ${y}_{1},{y}_{2},\ldots$, ${y}_{100}$ 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 _____ .

Let $A,B,C$ be $3\times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric. Consider the statements $(S1){A}^{13}{B}^{26}-{B}^{26}{A}^{13}$ is symmetric $(S2){A}^{26}{C}^{13}-{C}^{13}{A}^{26}$ is symmetric Then,

Let ${a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then $\underset{n\rightarrow \infty }{\mathrm{lim}}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{{a}_{1}}+\sqrt{{a}_{2}}}+\frac{1}{\sqrt{{a}_{2}}+\sqrt{{a}_{3}}}+\ldots +\frac{1}{\sqrt{{a}_{n-1}}+\sqrt{{a}_{n}}})$ is

Let $f:R-{2,6}\rightarrow R$ be real valued function defined as $f(x)=\frac{x+2x+1}{{x}^{2}-8x+12}$. Then range of $f$ 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 $C$ be the circle in the complex plane with centre ${z}_{0}=\frac{1}{2}(1+3i)$ and radius $r=1$. Let ${z}_{1}=1+i$ and the complex number ${z}_{2}$ be outside circle $C$ such that $|{z}_{1}-{z}_{0}||{z}_{2}-{z}_{0}|=1$. If ${z}_{0},{z}_{1}$ and ${z}_{2}$ are collinear, then the smaller value of ${|{z}_{2}|}^{2}$ is equal to

Let $\alpha$ be the constant term in the binomial expansion of ${(\sqrt{x}-\frac{6}{{x}^{\frac{3}{2}}})}^{n},n\leq 15.$ If the sum of the coefficients of the remaining terms in the expansion is $649$ and the coefficient of ${x}^{-n}$ is $\lambda \alpha ,$ then $\lambda$ is equal to $________.$

Let $D$ be the domain of the function $f(x)={\mathrm{sin}}^{-1}({\mathrm{log}}_{3x}(\frac{6+2{\mathrm{log}}_{3}x}{-5x}))$. If the range of the function $g:D\rightarrow \mathbb{R}$ defined by $g(x)=x-[x],$ ($[x]$ is the greatest integer function), is $(\alpha ,\beta ),$ then ${\alpha }^{2}+\frac{5}{\beta }$ is equal to

Let ${w}_{1}$ be the point obtained by the rotation of ${z}_{1}=5+4i$ about the origin through a right angle in the anticlockwise direction, and ${w}_{2}$ be the point obtained by the rotation of ${z}_{2}=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument ${w}_{1}-{w}_{2}$ is equal to

Let ${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$ be the roots of the equation ${x}^{7}+3{x}^{5}-13{x}^{3}-15x=0$ and $|{\alpha }_{1}|\geq |{\alpha }_{2}|\geq \ldots \geq |{\alpha }_{7}|$. Then, ${\alpha }_{1}{\alpha }_{2}-{\alpha }_{3}{\alpha }_{4}+{\alpha }_{5}{\alpha }_{6}$ is equal to _______

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

Let $\alpha ,\beta$ be the roots of the quadratic equation ${x}^{2}+\sqrt{6}x+3=0$. Then $\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}$ is equal to

Let $S$ be the set of all values of $\theta \in [-\pi ,\pi ]$ for which the system of linear equations $x+y+\sqrt{3}z=0$ $-x+(\mathrm{tan}\theta )y+\sqrt{7}z=0$ $x+y+(\mathrm{tan}\theta )z=0$ has non-trivial solution.Then $\frac{120}{\pi }\underset{0\in S}{\sum }\theta$ is equal to

Let $S$ be the set of values of $\lambda$, for which the system of equations $6\lambda x-3y+3z=4{\lambda }^{2}$, $2x+6\lambda y+4z=1$ and $3x+2y+3\lambda z=\lambda$ has no solution. Then,$12\underset{\lambda \in S}{\sum }|\lambda |$ is equal to _______.

Let $\alpha >0$, be the smallest number such that the expansion of ${({x}^{\frac{2}{3}}+\frac{2}{{x}^{3}})}^{30}$ has a term $\beta {x}^{-\alpha },\beta \in N$. Then $\alpha$ is equal to _____ .

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of ${(1+x)}^{99}$. Let a be the middle term in the expansion of ${(2+\frac{1}{\sqrt{2}})}^{200}$. If $\frac{C99200K}{a}=\frac{{2}^{l}m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(l,n)$ is equal to:

Let ${A}_{1},{A}_{2},{A}_{3}$ be the three A.P. with the same common difference $d$ and having their first terms as $A,A+1,A+2$, respectively. Let $a,b,c$ be the ${7}^{\text{th }},{9}^{\text{th }},{17}^{\text{th }}$ terms of ${A}_{1},{A}_{2},{A}_{3}$, respectively such that $|\begin{matrix}a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{matrix}|+70=0$. If $a=29$, then the sum of first $20$ terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to _____ .

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.

Let $\alpha ,\beta ,\gamma$ be the three roots of the equation ${x}^{3}+bx+c=0$ if $\beta \gamma =1=-\alpha$ then ${b}^{3}+2{c}^{3}-3{\alpha }^{3}-6{\beta }^{3}-8{\gamma }^{3}$ is equal to

Let $0<z<y<x$ be three real numbers such that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in an arithmetic progression and $x,\sqrt{2}y,z$ are in a geometric progression. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z{)}^{2}$ is equal to

Let $a\neq b$ be two non-zero real numbers. Then the number of elements in the set $X={z\in C:\mathrm{Re}(a{z}^{2}+bz)=a\mathrm{and}\mathrm{Re}(b{z}^{2}+az)=b}$ is equal to

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\frac{1+[a]}{4b}$, where $[t]$ is greatest integer function, is :

Let $[t]$ denote the greatest integer $\leq t$. if the constant term in the expansion of ${(3{x}^{2}-\frac{1}{2{x}^{5}})}^{7}$ is $\alpha$ then $[\alpha ]$ is equal to $_____$

Let $[\alpha ]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+.............+[\sqrt{120}]$ is equal to

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1 2x+Ny+2z=2 3x+3y+Nz=3$ has unique solution is $\frac{k}{6},$ then the sum of value of $k$ and all possible values of $N$ is

Let $P(S)$ denote the power set of $S={1,2,3,\ldots ,10}$. Define the relations ${R}_{1}$ and ${R}_{2}$ on $P(S)$ as $A{R}_{1}B$ if $(A\cap {B}^{c})\cup (B\cap {A}^{c})=\phi$ and $A{R}_{2}B$ if $A\cup {B}^{c}=B\cup {A}^{c},\forall A,B\in P(S)$ . Then :

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

Let $5$ digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is _____ .

Let ${f}^{1}(x)=\frac{3x+2}{2x+3},x\in R-{-\frac{3}{2}}$. For $n\geq 2$, define ${f}^{n}(x)={f}^{1}o{f}^{n-1}(x)$. If ${f}^{5}(x)=\frac{ax+b}{bx+a},\mathrm{gcd}(a,b)=1$, then $a+b$ is equal to ________

Let for $A=[\begin{matrix}1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2\end{matrix}],|A|=2$. If $|2adj(2adj(2A))|={32}^{n}$, then $3n+\alpha$ is equal to

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:

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

Let $[\begin{matrix}\begin{matrix}2 \\ 1 \\ 0\end{matrix} & \begin{matrix}1 \\ 2 \\ -1\end{matrix} & \begin{matrix}0 \\ -1 \\ 2\end{matrix}\end{matrix}]$. If $|adj(adj(adj2A))|=(16{)}^{n}$, then $n$ is equal to

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 $S={z=x+iy:\frac{2z-3i}{4z+2i}$ is a real number $}$. Then which of the following is NOT correct?

Let $w=z\bar{z}+{k}_{1}z+{k}_{2}iz+\lambda (1+i),{k}_{1},{k}_{2}\in \mathbb{R}.$ . Let $Re(w)=0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y=1$ and the $y-$axis. If the curve $Im(w)=0$ intersects $C$ at $A$ and $B,$ then $30(AB{)}^{2}$ is equal to $_______.$

Let ${s}_{1},{s}_{2},{s}_{3}....,{s}_{10}$ respectively be the sum of $12$ terms of $10A.\mathrm{Ps}$ whose first terms are $1,2,3,....,10$ and the common differences are $1,3,5,...,19$ respectively. Then $\sum _{i=1}^{10}{s}_{i}$ is equal to

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x{)}^{n}$ be in the ratio $2:5:8$. Then the coefficient of the term, which is in the middle of these three terms, is

Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+{y}^{2}=0$, then ${y}^{4}+{y}^{2}-y$ is equal to

Let the determinant of a square matrix $A$ of order $m$ be $m-n$, where m and $n$ satisfy $4m+n=22$ and $17m+4n=93$. If $det(nadj(adj(mA)))={3}^{a}{5}^{b}{6}^{c}$, then $a+b+c$ is equal to

Let the digits $a,b,c$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Let the first term a and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to

Let the number $(22{)}^{2022}+(2022{)}^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $({\alpha }^{2}+{\beta }^{2})$ is equal to

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A\times B$ each having at least $3$ and at most $6$ elements is

Let $A=⌊{a}_{\hat{i}\hat{j}}⌋\cdot {a}_{ij}\in Z\cap [0,4],1\leq i,j\leq 2$. The number of matrices $A$ such that the sum of all entries is $a$ prime number $p\in (2,13)$ is _____ .

Let $S={1,2,3,5,7,10,11}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is _____ .

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{[x]-x}},$ where $[x]$ denotes the smallest integer greater than or equal to $x$. Then among the statements $(S1):A\cap B=(1,\infty )-\mathbb{N}$ and $(S2):A\cup B=(1,\infty )$

Let the sixth term in the binomial expansion of ${(\sqrt{{2}^{{\mathrm{log}}_{2}(10-{3}^{x})}}+\sqrt[5]{{2}^{(x-2){\mathrm{log}}_{2}3}})}^{m}$ powers of ${2}^{(x-2){\mathrm{log}}_{2}3}$, be $21$ . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $x$ is _____ .

Let the system of linear equations $–x+2y-9z=7$ $-x+3y+7z=9$ $-2x+y+5z=8$ $-3x+y+13z=\lambda$ has a unique solution $x=\alpha ,y=\beta ,z=\gamma$. Then the distance of the point $(\alpha ,\beta ,\gamma )$ from the plane $2x-2y+z=\lambda$ 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 :

Let $5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3,x>0.$ Then $18{\int }_{1}^{2}f(x)dx$ is equal to

Let $S={x:x\in \mathbb{R}\text{ and }{(\sqrt{3}+\sqrt{2})}^{{x}^{2}-4}+{(\sqrt{3}-\sqrt{2})}^{{x}^{2}-4}=10}$. Then $n(S)$ is equal to

Let $S={z\in \mathbb{C}-{i,2i}:\frac{{z}^{2}+8iz-15}{{z}^{2}-3iz-2}\in \mathbb{R}}$. $\alpha -\frac{13}{11}i\in S,\alpha \in \mathbb{R}-{0}$, then $242{\alpha }^{2}$ is equal to

Let $S={z\in \mathbb{C}:\bar{z}=i({z}^{2}+Re(\bar{z}))}$. Then $\underset{z\in S}{\sum }|z{|}^{2}$ is equal to

Let $S={\alpha :{\mathrm{log}}_{2}({9}^{2\alpha -4}+13)-{\mathrm{log}}_{2}(\frac{5}{2}\cdot {3}^{2\alpha -4}+1)=2}.$ Then the maximum value of $\beta$ for which the equation ${x}^{2}-2{(\underset{\alpha \in s}{\sum }\alpha )}^{2}x+\underset{a\in s}{\sum }{(\alpha +1)}^{2}\beta =0$ has real roots, is _____ .

Let $S={1,2,3,4,5,6}$. Then the number of oneone functions $f:S\rightarrow P(S)$, where $P(S)$ denote the power set of $S$, such that $f(n)\subset f(m)$ where $n<m$ is

Let $A={1,2,3,5,8,9}$. Then the number of possible functions $f:A\rightarrow A$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in A$ with $m\cdot n\in A$ is equal to

Let $A={1,2,3,4,5,6,7}$. Then the relation $R={(x,y)\in A\times A:x+y=7}$ is

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{matrix}]$. Then the sum of the diagonal elements of the matrix ${(A+I)}^{11}$ is equal to:

Let $A={\theta \in (0,2\pi ):\frac{1+2i\mathrm{sin}\theta }{1-i\mathrm{sin}\theta }\mathrm{is}\mathrm{purely}\mathrm{imaginary}}$ Then the sum of the elements is in $A$ is

Let $A={x\in \mathbb{R}:[x+3]+[x+4]\leq 3},B={x\in \mathbb{R}:{3}^{x}{(\sum _{r=1}^{\infty }\frac{3}{{10}^{r}})}^{x-3}<{3}^{-3x}},$ where $[t]$ denotes greatest integer function. Then,

Let $A={[{a}_{ij}]}_{2\times 2},$ where $a\neq ij0$ for all $i,j$ and ${A}^{2}=I$, Let $a$ be the sum of all diagonal elements of $A$ and $b=|A|$ Then $3{a}^{2}+4{b}^{2}$ is equal to

Let $A=[\begin{matrix}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{matrix}]$, where $a,c\in R$. If ${A}^{3}=A$ and the positive value of $a$ belongs to the interval $(n-1,n]$, where $n\in \mathbb{N}$, then $n$ is equal to ____.

Number of $4$-digit numbers that are less than or equal to $2800$ and either divisible by $3$ or by $11$ , is equal to _____ .

Number of $4$-digit numbers (the repetition of digits is allowed) which are made using the digits $1,2,3$ and $5$ , and are divisible by $15$ , is equal to

Number of integral solutions to the equation $x+y+z=21$, where $x\geq 1,y\geq 3,z\geq 4$, is equal to _____ .

If α and β are roots of x² - 5x + 6 = 0, then α³ + β³ is equal to:

${50}^{\text{th }}$ root of a number $x$ is $12$ and ${50}^{\text{th }}$ root of another number $y$ is $18$ . Then the remainder obtained on dividing $(x+y)$ by $25$ is ________.

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 ________.

Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is _____ .

Suppose $f:R\rightarrow (0,\infty )$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y),\forall x,y\in R$, If $f(3)=320$, then $\sum _{n=0}^{5}f(n)$ 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 ______

The absolute difference of the coefficients of ${x}^{10}$ and ${x}^{7}$ in the expansion of ${(2{x}^{2}+\frac{1}{2x})}^{11}$ is equal to

The coefficient of ${x}^{7}$ in ${(1-x+2{x}^{3})}^{10}$ is __________ .

The coefficient of ${x}^{18}$ in the expansion of ${({x}^{4}-\frac{1}{{x}^{3}})}^{15}$ is $____________$

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

The coefficient of ${x}^{-6}$, in the expansion of ${(\frac{4x}{5}+\frac{5}{2{x}^{2}})}^{9}$, is

The ${8}^{\text{th }}$ common term of the series ${S}_{1}=3+7+11+15+19+\ldots$ ${S}_{2}=1+6+11+16+21+\ldots$. is

The complex number $z=\frac{i-1}{\mathrm{cos}\frac{\pi }{3}+i\mathrm{sin}\frac{\pi }{3}}$ is equal to:

The constant term in the expansion of ${(2x+\frac{1}{{x}^{7}}+3{x}^{2})}^{5}$ is _____ .

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

The domain of the function $f(x)=\frac{1}{\sqrt{{[x]}^{2}-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$)

The equation ${e}^{4x}+8{e}^{3x}+13{e}^{2x}-8{e}^{x}+1=0,x\in R$ has :

The equation ${x}^{2}–4x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

The largest natural number $n$ such that $3n$ divides $66!$ is $_______$

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

The mean of the coefficients of $x,{x}^{2},\ldots \ldots ,{x}^{7}$ in the binomial expression of $(2+x{)}^{9}$ is _________

The minimum number of elements that must be added to the relation $R=(a,b),(b,c)$ on the set ${a,b,c}$ so that it becomes symmetric and transitive is:

The minimum number of elements that must be added to relation $R={(a,b),(b,c),(b,d)}$ on the set ${a,b,c,d}$, so that it is an equivalence relation is

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

The number of $4$-letter words, with or without meaning, each consisting of $2$ vowels and $2$ consonants, which can be formed from the letters of the word UNIVERSE without repetition is _____.

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

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

The number of elements in the set ${n\in \mathbb{N}:10\leq n\leq 100$ and ${3}^{n}-3$ is a multiple of $7}$ is _______.

The number of elements in the set ${n\in \mathbb{Z}:|{n}^{2}-10n+19|<6}$ is _______ .

The number of five-digit numbers, greater than $40000$ and divisible by $5$, which can be formed using the digits $0,1,3,5,7$ and $9$ without repetition, 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 integers, greater than $7000$ that can be formed, using the digits $3,5,6,7,8$ without repetition is

The number of integral solution $x$ of ${\mathrm{log}}_{(x+\frac{7}{2})}{(\frac{x-7}{2x-3})}^{2}\geq 0$ is

The number of integral solution $x$ of ${\mathrm{log}}_{(x+\frac{7}{2})}{(\frac{x-7}{2x-3})}^{2}\geq 0$ is

The number of integral terms in the expansion of ${({3}^{\frac{1}{2}}+{5}^{\frac{1}{4}})}^{680}$ is equal to

The number of integral values of $k$, for which one root of the equation $2{x}^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is :

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

The number of permutations, of the digits $1,2,3,\ldots ,7$ without repetition, which neither contain the string $153$ nor the string $2467$, 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............

The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

The number of real roots of the equation $\sqrt{{x}^{2}-4x+3}+\sqrt{{x}^{2}-9}=\sqrt{4{x}^{2}-14x+6}$, is:

The number of real solutions of the equation $3({x}^{2}+\frac{1}{{x}^{2}})-2(x+\frac{1}{x})+5=0$, is

The number of relations, on the set ${1,2,3}$ containing $(1,2)$ and $(2,3)$ which are reflexive and transitive but not symmetric, is _________.

The number of seven digit positive integers formed using the digits $1,2,3$ and $4$ only and sum of the digits equal to $12$ is$_______.$

The number of seven digits odd numbers, that can be formed using all the seven digits $1,2,2,2,3,3,5$ is

The number of square matrices of order $5$ with entries from the set ${0,1}$, such that the sum of all the elements in each row is $1$ and the sum of all the elements in each column is also $1$, is

The number of symmetric matrices of order 3, with all the entries from the set ${0,1,2,3,4,5,6,7,8,9}$ is

The number of triplets $(x,y,z)$ where $x,y,z$ are distinct non negative integers satisfying $x+y+z=15$, is

The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together is

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

The number of ways of selecting two numbers $a$ and $b$, $a\in {2,4,6,\ldots \ldots ,100}$ and $b\in {1,3,5,\ldots \ldots ,99}$ such that $2$ is the remainder when $a+b$ is divided by $23$ is

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____ .

The range of $f(x)=4{\mathrm{sin}}^{-1}(\frac{{x}^{2}}{{x}^{2}+1})$ is

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is

The relation $R={(a,b):gcd(a,b)=1,2a\neq b,a,b\in \mathbb{Z}}$ is:

The remainder on dividing ${5}^{99}$ by $11$ is _____ .

The remainder when ${(2023)}^{2023}$ is divided by $35$ is

The remainder, when ${7}^{103}$ is divided by $17,$ is

The remainder when ${19}^{200}+{23}^{200}$ is divided by $49$, is _____ .

The set of all values of $t\in \mathbb{R}$, for which the matrix $[\begin{matrix}{e}^{t} & {e}^{-t}(\mathrm{sin}t-2\mathrm{cos}t) & {e}^{-t}(-2\mathrm{sin}t-\mathrm{cos}t) \\ {e}^{t} & {e}^{-t}(2\mathrm{sin}t+\mathrm{cos}t) & {e}^{-t}(\mathrm{sin}t-2\mathrm{cos}t) \\ {e}^{t} & {e}^{-t}\mathrm{cos}t & {e}^{-t}\mathrm{sin}t\end{matrix}]$ is invertible, is

The sum ${1}^{2}-2.{3}^{2}+3.{5}^{2}-4.{7}^{2}+5.{9}^{2}-\ldots ..+15.{29}^{2}$ is _____ .

The sum of all the four-digit numbers that can be formed using all the digits $2,1,2,3$ is equal to ____.

The sum of all the roots of the equation $|{x}^{2}-8x+15|-2x+7=0$ is

The sum of all those terms, of the arithmetic progression $3,8,13,...,373$, which are not divisible by $3$, is equal to ________.

The sum, of the coefficients of the first $50$ terms in the binomial expansion of ${(1-x)}^{100},$ is equal to

The sum of the coefficients of three consecutive terms in the binomial expansion of ${(1+x)}^{n+2}$, which are in the ratio $1:3:5$, 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 sum to $10$terms of the series $\frac{1}{1+{1}^{2}+{1}^{4}}+\frac{2}{1+{2}^{2}+{2}^{4}}+\frac{3}{1+{3}^{2}+{3}^{4}}+\ldots$is :-

The sum to $20$ terms of the series $2\cdot {2}^{2}-{3}^{2}+2\cdot {4}^{2}-{5}^{2}+2\cdot {6}^{2}-............$ is equal to$__________.$

The total number of $4$-digit numbers whose greatest common divisor with $54$ is $2$ , is

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

The total number of three-digit numbers, divisible by $3$, which can be formed using the digits $1,3,5,8$, if repetition of digits is allowed, is

The value of ${(\frac{1+\mathrm{sin}\frac{2\pi }{9}+i\mathrm{cos}\frac{2\pi }{9}}{1+\mathrm{sin}\frac{2\pi }{9}-i\mathrm{cos}\frac{2\pi }{9}})}^{3}$ 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 ________