JEE Main 2023 — Algebra

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 $A={0,3,4,6,7,8,9,10}$ 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 $_________$

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

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

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

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

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

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?

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 _________.

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

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

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

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 $f:R-{2,6}\rightarrow R$ be real valued function defined as $f(x)=\frac{x+2x+1}{{x}^{2}-8x+12}$. Then range of $f$ is

Let 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 :

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

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

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

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 $f:\mathbb{R}\rightarrow \mathbb{R}$ 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 $B=[\begin{matrix}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{matrix}],\alpha >2$ 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

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

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

Fractional part of the number $\frac{{4}^{2022}}{15}$ 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 number of integral terms in the expansion of ${({3}^{\frac{1}{2}}+{5}^{\frac{1}{4}})}^{680}$ is equal to

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 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:

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 $_____$

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 $n\in \mathbb{N}$

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

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 _____ .

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

If $Pn-12n+1:Pn2n-1=11:21$, then ${n}^{2}+n+15$ 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 $(a,b)$:

${50}^{\text{th }}$ 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 ________.

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

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 $K$ 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:

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

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:

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

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

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

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 ?

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

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 $A,B,C$ 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,

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

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?

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

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

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

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

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

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

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 _____ .

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 :

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

Let 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

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

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

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

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

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 $\alpha$ 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 $A$ 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 $\alpha$ 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 $a,b,c$ 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 ______.

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange 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 $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{matrix}]$. Then the sum of the diagonal elements of the matrix ${(A+I)}^{11}$ is equal to:

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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 $A={1,2,3,4,5}$ 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 $R={a,b,c,d,e}$ and $S={1,2,3,4}$. Total number of onto functions $f:R\rightarrow S$ such that $f(a)\neq 1$, is equal to ________.

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

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 _____ .

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

Let $a,b,c$ 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 $5$ 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 _____ .

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?

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

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 _______.

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

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

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

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,

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

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

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 $A$ 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 _________.

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

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 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

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

Let $N$ 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

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

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 :

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

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

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

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

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

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

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

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

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

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

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

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 ______

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

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 $5f(x)+4f(\frac{1}{x})=\frac{1}{x}+3,x>0.$ Then $18{\int }_{1}^{2}f(x)dx$ 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 $A={[{a}_{ij}]}_{2\times 2},$ 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

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

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

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 $\alpha >0$, 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 $p,q\in \mathbb{R}$ 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.

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

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

The 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 _______.

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

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 _____ .

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

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

The sum ${1}^{2}-2.{3}^{2}+3.{5}^{2}-4.{7}^{2}+5.{9}^{2}-\ldots ..+15.{29}^{2}$ 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 symmetric matrices of order 3, with all the entries from the set ${0,1,2,3,4,5,6,7,8,9}$ is

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

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

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 ________

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

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

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

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 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 $S={1,2,3,4,5,6}$. 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

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

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$_______.$

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

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

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

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 _____ .

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 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

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

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 ____.

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 _______.

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

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

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

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

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

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

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

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

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 ________

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 ${\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 $S={\alpha :{\mathrm{log}}_{2}({9}^{2\alpha -4}+13)-{\mathrm{log}}_{2}(\frac{5}{2}\cdot {3}^{2\alpha -4}+1)=2}.$ 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 _____ .

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

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,

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 $w=z\bar{z}+{k}_{1}z+{k}_{2}iz+\lambda (1+i),{k}_{1},{k}_{2}\in \mathbb{R}.$ . 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 $S={z=x+iy:\frac{2z-3i}{4z+2i}$ is a real number $}$. Then which of the following is NOT correct?

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

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 :

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

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

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 _____.

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 $z=1+i$ 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

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:

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

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

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 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

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 ________

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?

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

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

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 ________.

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

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

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 largest natural number $n$ such that $3n$ divides $66!$ 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 _____ .

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

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

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

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

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 :

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

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

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 _____ .

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 ______

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

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$__________.$

Let $[\alpha ]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+.............+[\sqrt{120}]$ 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 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 $0<z<y<x$ 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 $a,b,c$ 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 ${a}_{1}=8,{a}_{2},{a}_{3},\ldots .{a}_{n}$ 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 ${a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ 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

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

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

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

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 :-

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 $a,b,c>1,{a}^{3},{b}^{3}$ 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

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 ${A}_{1},{A}_{2},{A}_{3}$ 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 _____ .

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 _____ .

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 $\left\{a_k\right\}$ 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

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

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 ______