JEE Main 2024 — Algebra

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50^{\text {th }}$ word is :

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

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $\mathrm{m}$ is equal to:

All the letters of the word $GTWENTY$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $GTWENTY$ IS

An arithmetic progression is written in the following way  The sum of all the terms of the $10^{\text {th }}$ row is_______

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

Consider the function $f:[\frac{1}{2},1]\rightarrow R$ defined by $f(x)=4\sqrt{2}{x}^{3}-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\mathrm{cos}\frac{\pi }{12}$ Then

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to ______

Consider the matrices : $A=\left[\begin{array}{ll}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $A X=B$ has a negative solution (i.e., $x < 0$ and $y < 0$ ), be the interval $(a, b)$. Then $8 \int_a^b|A| d m$ is equal to_________

Consider the matrix $f(x)=[\begin{matrix}\mathrm{cos}x & -\mathrm{sin}x & 0 \\ \mathrm{sin}x & \mathrm{cos}x & 0 \\ 0 & 0 & 1\end{matrix}]$. Given below are two statements : Statement I:$f(-x)$ is the inverse of the matrix $f(x)$. Statement II: $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below

Consider the relations ${R}_{1}$ and ${R}_{2}$ defined as $a{R}_{1}b\Leftrightarrow {a}^{2}+{b}^{2}=1$ for all $a,b,\in R$ and $(a,b){R}_{2}(c,d)\Leftrightarrow a+d=b+c$ for all $(a,b),(c,d)\in N\times N$. Then

Consider the system of linear equation $x+y+z=$ $4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^{2}+15$, where $\lambda ,\mu \in R$. Which one of the following statements is NOT correct?

Consider the system of linear equations $x+y+z=5,x+2y+{\lambda }^{2}z=9$ and $x+3y+\lambda z=\mu$, where $\lambda ,\mu \in R$. Then, which of the following statement is NOT correct ?

For a differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, suppose $f^{\prime}(x)=3 f(x)+\alpha$, where $\alpha \in \mathbb{R}$, $f(0)=1$ and $\lim _{x \rightarrow-\infty} f(x)=7$. Then $9 f\left(-\log _{\mathrm{e}} 3\right)$ is equal to_________

For $\alpha, \beta \in \mathbb{R}$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$. Then

For $0<c<b<a$, let $(a+b–2c){x}^{2}+(b+c–2a)x+(c+a–2b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$. (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$.

For $x \geqslant 0$, the least value of $\mathrm{K}$, for which $4^{1+x}+4^{1-x}, \frac{\mathrm{K}}{2}, 16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to :

If 2 and 6 are the roots of the equation $a x^2+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is :

If a function $f$ satisfies $f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n} \in \mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum_{k=1}^{2022} f(\lambda+k) \leq(2022)^2$ is equal to _________

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

If $A=[\begin{matrix}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{matrix}],B[\begin{matrix}1 & 0 \\ 1 & 1\end{matrix}],C=AB{A}^{T}$ and $X={A}^{T}{C}^{2}A$, then det $X$ is equal to:

If $\alpha \neq \mathrm{a}, \beta \neq \mathrm{b}, \gamma \neq \mathrm{c}$ and $\left|\begin{array}{lll}\alpha & \mathrm{b} & \mathrm{c} \\ \mathrm{a} & \beta & \mathrm{c} \\ \mathrm{a} & \mathrm{b} & \gamma\end{array}\right|=0$, then $\frac{\mathrm{a}}{\alpha-\mathrm{a}}+\frac{\mathrm{b}}{\beta-\mathrm{b}}+\frac{\gamma}{\gamma-\mathrm{c}}$ is equal to:

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=n$, then the point $(\mathrm{m}, \mathrm{n})$ lies on the line

If $\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0$, and $(60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}$, where $a, b \in N$, then $(a+b)$ equal to ______

If $f(x)=\frac{4x+3}{6x-4},x\neq \frac{2}{3}$ and $(fof)(x)=g(x)$, where $g:R-{\frac{2}{3}}\rightarrow R-{\frac{2}{3}}$, then $(gogog)(4)$ is equal to

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

If $\alpha ,\beta$ are the roots of the equation, ${x}^{2}-x-1=0$ and ${S}_{n}=2023{\alpha }^{n}+2024{\beta }^{n}$, then

If $z_1, z_2$ are two distinct complex number such that $\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$, then

If $\alpha$ denotes the number of solutions of ${|1-i|}^{x}={2}^{x}$ and $\beta =(\frac{|z|}{\mathrm{arg}(z)})$, where $z=\frac{\pi }{4}{(1+i)}^{4}(\frac{1-\sqrt{\pi }\cdot i}{\sqrt{\pi }+i}+\frac{\sqrt{\pi }-i}{1+\sqrt{\pi }\cdot i})$, $i=\sqrt{-1}$, then the distance of the point $(\alpha ,\beta )$ from the line $4x-3y=7$ is ______

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

If $f(x)=|\begin{matrix}{x}^{3} & 2{x}^{2}+1 & 1+3x \\ 3{x}^{2}+2 & 2x & {x}^{3}+6 \\ {x}^{3}-x & 4 & {x}^{2}-2\end{matrix}|$ for all $x\in \mathbb{R}$, then $2f(0)+{f}^{'}(0)$ is equal to

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

If $z$ is a complex number such that $|z|\leq 1$, then the minimum value of $|z+\frac{1}{2}(3+4i)|$ is:

If $z$ is a complex number, then the number of common roots of the equation ${z}^{1985}+{z}^{100}+1=0$ and ${z}^{3}+2{z}^{2}+2z+1=0$, is equal to :

If $A$ is a square matrix of order 3 such that $\operatorname{det}(A)=3$ and $\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\right)\right)\right)\right)=2^{\mathrm{m}} 3^{\mathrm{n}}$, then $\mathrm{m}+2 \mathrm{n}$ is equal to :

If $z=\frac{1}{2}-2i$, is such that $|z+1|=\alpha z+\beta (1+i),i=\sqrt{-1}$ and $\alpha ,\beta \in R,$ then $\alpha +\beta$ is equal to

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to:

If $R$ is the smallest equivalence relation on the set ${1,2,3,4}$ such that ${(1,2),(1,3)}\subset R$, then the number of elements in $R$ is ______.

If $\alpha$ satisfies the equation ${x}^{2}+x+1=0$ and $(1+\alpha {)}^{7}=A+B\alpha +C{\alpha }^{2},A,B,C\geq 0$, then $5(3A-2B-C)$ is equal to

If $z=x+iy,xy\neq 0$, satisfies the equation ${z}^{2}+i\bar{z}=0$, then $|{z}^{2}|$ is equal to :

If the coefficient of ${x}^{30}$ in the expansion of ${(1+\frac{1}{x})}^{6}{(1+{x}^{2})}^{7}{(1-{x}^{3})}^{8};x\neq 0$ is $\alpha$, then $|\alpha |$ equals _________.

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :

If the constant term in the expansion of $\left(1+2 x-3 x^3\right)\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $\mathrm{p}$, then $108 \mathrm{p}$ is equal to

If the domain of the function $f(x)={\mathrm{log}}_{e}(\frac{2x+3}{4{x}^{2}+x-3})+{\mathrm{cos}}^{-1}(\frac{2x-1}{x+2})$ is $(\alpha ,\beta ]$, then the value of $5\beta -4\alpha$ is equal to

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

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to:

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

If the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{2-|x|}{4})+{({\mathrm{log}}_{e}(3-x))}^{-1}$ is $[-\alpha ,\beta )-{\gamma }$, then $\alpha +\beta +\gamma$ is equal to :

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

If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is 64 and the common ratio is $\frac{\alpha}{\beta}$, is equal to________

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135,30 and $\frac{10}{3}$, respectively, then $6\left(n^3+x^2+y\right)$ is equal to _______

If the set $R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1-i^{n !}\right)=x+i y$, where $i=\sqrt{-1}$, then the value of $m+x+y$ is

If the sum of the series $\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}$ is equal to 5 , then $50 \mathrm{~d}$ is equal to :

If the system of equations $\begin{aligned} & x+(\sqrt{2} \sin \alpha) y+(\sqrt{2} \cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x+(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned}$ has a non-trivial solution, then $\alpha \in\left(0, \frac{\pi}{2}\right)$ is equal to :

If the system of equations $x+4 y-z=\lambda, 7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu+3 \lambda)$ is equal to :

If the system of equations $\begin{array}{r} 11 x+y+\lambda z=-5 \\ 2 x+3 y+5 z=3 \\ 8 x-19 y-39 z=\mu \end{array}$ has infinitely many solutions, then $\lambda^4-\mu$ is equal to :

If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha \beta$ is equal to

If the system of equations $\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$ has infinitely many solutions, then $(\lambda-\mu)$ is equal to________

If the system of linear equations $x-2y+z=-4$ $2x+\alpha y+3z=5$ $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha +13\beta$ is equal to

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

If $S=z\in C:|z-i|=|z+i|=|z-1|$, then, $n(S)$ is:

If $\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots \ldots+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots . .+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}$, then $\alpha$ is equal to________

If $f(x)=|\begin{matrix}2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & 3+{\mathrm{sin}}^{2}2x \\ 3+2{\mathrm{cos}}^{4}x & 2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x \\ 2{\mathrm{cos}}^{4}x & 3+2{\mathrm{sin}}^{4}x & {\mathrm{sin}}^{2}2x\end{matrix}|$ then $\frac{1}{5}{f}^{'}(0)$ is equal to ________.

If $f(x)={\begin{matrix}2+2x,-1\leq x<0 \\ 1-\frac{x}{3},0\leq x\leq 3\end{matrix};g(x)={\begin{matrix}-x,-3\leq x\leq 0 \\ x,0<x\leq 1\end{matrix}$, then range of $(f\circ g(x))$ is

If three successive terms of a G.P. with common ratio $r(r>1)$ are the length of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to r, then $3[r]+[-r]$ is equal to:

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$, where $\mathrm{a}$ and $\mathrm{b}$ are integers with $\operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$, then $11 \mathrm{a}+18 \mathrm{~b}$ is equal to ______

If $S=\{a \in \mathbf{R}:|2 a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72 \sum_{a \in S} a$ is equal to ______

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $\mathrm{m}$ and $\mathrm{n}$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ______

In an A.P., the sixth term ${a}_{6}=2$. If the ${a}_{1}{a}_{4}{a}_{5}$ is the greatest, then the common difference of the A.P., is equal to

In an examination of Mathematics paper, there are $20$ questions of equal marks and the question paper is divided into three sections : $A,B$ and $C$. A student is required to attempt total $15$ questions taking at least $4$ questions from each section. If section $A$ has $8$ questions, section $B$ has $6$ questions and section $C$ has $6$ questions, then the total number of ways a student can select $15$ questions is _________.

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49 . Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to :

In the expansion of $(1+x)(1-{x}^{2}){(1+\frac{3}{x}+\frac{3}{{x}^{2}}+\frac{1}{{x}^{3}})}^{5},x\neq 0$, the sum of the coefficient of ${x}^{3}$ and ${x}^{-13}$ is equal to ______

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

Let a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements: (I) $\mathrm{R}$ is reflexive but not symmetric. (II) $R$ is transitive Then which one of the following is true?

Let \(\alpha, \beta \in\) be roots of equation \(x^2-70 x+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin\). If \(\lambda\) assumes the minimum possible value, then \(\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{\boldsymbol{|\alpha-\beta|}}\) is equal to :

Let $f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x \in \mathbf{R}$. Then $\frac{g(7)}{g^{\prime}(7)}$ is equal to :

Let $B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$ and $A$ be a $2 \times 2$ matrix such that $A B^{-1}=A^{-1}$. If $B C B^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2 \beta-\alpha$ is equal to

Let $A={1,2,3,4}$ and $R={(1,2),(2,3),(1,4)}$ be a relation on $A$. Let $S$ be the equivalence relation on $A$ such that $R\subset S$ and the number of elements in $S$ is $n$. Then, the minimum value of $n$ is _______

Let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{\mathrm{log}}_{e}x, & x>0 \\ {e}^{-x}, & x\leq 0\end{matrix}$ and $g(x)={\begin{matrix}x, & x\geq 0 \\ {e}^{x}, & x<0\end{matrix}$. Then, $gof:R\rightarrow R$ is:

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

Let $\alpha$ and $\beta$ be the roots of the equation $p{x}^{2}+qx-r=0$, where $p\neq 0$. If $p,q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the value of ${(\alpha -\beta )}^{2}$ is:

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$, $z \in$ C. Then $4\left(\alpha^2+\beta^2\right)$ is equal to :

Let $3,7,11,15,..,403$ and $2,5,8,11,...,404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to_________

Let ${z}_{1}$ and ${z}_{2}$ be two complex number such that ${z}_{1}+{z}_{2}=5$ and ${z}_{1}^{3}+{z}_{2}^{3}=20+15i$. Then $|{z}_{1}^{4}+{z}_{2}^{4}|$ equals-

Let $a$ and $b$ be two distinct positive real numbers. Let ${11}^{\text{th }}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to ${p}^{\text{th }}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|\mathrm{A}^{\mathrm{T}} \mathrm{A}(\operatorname{adj}(2 \mathrm{~A}))^{-1}(\operatorname{adj}(4 \mathrm{~B}))(\operatorname{adj}(\mathrm{AB}))^{-1} \mathrm{AA}^{\mathrm{T}}\right|$ is equal to :

Let $\alpha \beta \neq 0$ and $A=\left[\begin{array}{rrr}\beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2 \alpha\end{array}\right]$. If $B=\left[\begin{array}{rrr}3 \alpha & -9 & 3 \alpha \\ -\alpha & 7 & -2 \alpha \\ -2 \alpha & 5 & -2 \beta\end{array}\right]$ is the matrix of cofactors of the elements of $A$, then $\operatorname{det}(A B)$ is equal to :

Let $\alpha \in(0, \infty)$ and $A=\left[\begin{array}{lll}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{array}\right]$. If $\operatorname{det}\left(\operatorname{adj}\left(2 A-A^T\right) \cdot \operatorname{adj}\left(A-2 A^T\right)\right)=2^8$, then $(\operatorname{det}(A))^2$ is equal to:

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let $R$ be a relation defined on $A \times B$ by $\left(a_1, b_1\right) R\left(a_2, b_2\right)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in $R$ is _________

Let $A=\{2,3,6,8,9,11\}$ and $B=\{1,4,5,10,15\}$. Let $R$ be a relation on $A \times B$ defined by $(a, b) R(c, d)$ if and only if $3 a d-7 b c$ is an even integer. Then the relation $R$ is

Let $A={1,2,3,\ldots .7}$ and let $P(A)$denote the power set of $A$. If the number of functions $f:A\rightarrow P(A)$ such that $a\in f(a),\forall a\in A$ is ${m}^{n},m$ and $n\in N$ and $m$ is least, then $m+n$ is equal to ______.

Let $P={z\in \mathbb{C}:|z+2-3i|\leq 1}$ and $Q={z\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\leq -8}$. Let in $P\cap Q,|z-3+2i|$ be maximum and minimum at ${z}_{1}$ and ${z}_{2}$ respectively. If ${|{z}_{1}|}^{2}+2{|z|}^{2}=\alpha +\beta \sqrt{2},$ where $\alpha ,\beta$ are integers, then $\alpha +\beta$ equals __________

Let $r$ and $\theta$ respectively be the modulus and amplitude of the complex number $z=2-i(2\mathrm{tan}\frac{5\pi }{8})$, then $(r,\theta )$ is equal to

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

Let $A=[\begin{matrix}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{matrix}]$ and $P=[\begin{matrix}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{matrix}]$. The sum of the prime factors of $|{P}^{-1}\mathrm{AP}-2I|$ is equal to

Let $\alpha =\frac{(4!)!}{(4!{)}^{3!}}$ and $\beta =\frac{(5!)!}{(5!{)}^{4!}}$. Then :

Let $S_1=\{z \in C:|z| \leq 5\}, S_2=\left\{z \in C: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$ and $S_3=\{z \in C: \operatorname{Re}(z) \geq 0\}$. Then the area of the region $S_1 \cap S_2 \cap S_3$ is :

Let $A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}$ and $B=\{x:(x, y) \in A\}$. Then the number of one-one functions from $A$ to $B$ is equal to _______

Let $A=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $B=I+\operatorname{adj}(A)+(\operatorname{adj} A)^2+\ldots+(\operatorname{adj} A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: \mathrm{A} \rightarrow \mathrm{B}$, such that $f(1)+f(3)=14$, is :

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{matrix}]$ and $|2A{|}^{3}={2}^{21}$ where $\alpha ,\beta \in Z$, Then a value of $\alpha$ is

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{Im}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is

Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|z-(6+8 i)|$ is equal to

Let $f:R\rightarrow R$ be a function defined $f(x)=\frac{x}{{(1+{x}^{4})}^{1/4}}$ and $g(x)=f(f(f(f(x))))$ then $18{\int }_{0}^{\sqrt{2\sqrt{5}}}{x}^{2}g(x)dx$

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to ;

Let $A$ be a $3\times 3$ matrix and $\mathrm{det}(A)=2$. If $n=\mathrm{det}(\underset{2024-\mathrm{times}}{\underset{⏟}{adj(adj(....(adjA))))}})$, then the remainder when $n$ is divided by $9$ is equal to __________.

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

Let $A$ be a non-singular matrix of order 3 . If $\operatorname{det}(3 \operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A)))=3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(3 \operatorname{adj}(2 \mathrm{~A}))=2^{\mathrm{m}} \cdot 3^{\mathrm{n}}$, then $|3 \mathrm{~m}+2 \mathrm{n}|$ is equal to $\qquad$

Let $R=(\begin{matrix}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{matrix})$ be a non-zero $3\times 3$ matrix, where $x\mathrm{sin}\theta =y\mathrm{sin}(\theta +\frac{2\pi }{3})=z\mathrm{sin}(\theta +\frac{4\pi }{3})$ $\neq 0,\theta \in (0,2\pi )$. For a square matrix $M$, let Trace$(M)$ denote the sum of all the diagonal entries of $M.$ Then, among the statements: $(I)$ Trace$(R)=0$ $(\mathrm{II})$ If Trace$(adj(adj(R))=0$, then $R$ has exactly one non-zero entry.

Let $A$ be a $2\times 2$ real matrix and $I$ be the identity matrix of order $2.$ If the roots of the equation $|A-\mathrm{xI}|=0$ be $-1$ and $3,$ then the sum of the diagonal elements of the matrix ${A}^{2}$ is _____.

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

Let $A$ be a square matrix such that ${\mathrm{AA}}^{T}=I$. Then $\frac{1}{2}A[{(A+{A}^{T})}^{2}+{(A-{A}^{T})}^{2}]$ is equal to

Let $A$ be a $2 \times 2$ symmetric matrix such that $A\left[\begin{array}{l}1 \\ 1\end{array}\right]=\left[\begin{array}{l}3 \\ 7\end{array}\right]$ and the determinant of $A$ be 1 . If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2 \times 2$, then $\alpha+\beta$ equals _______

Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :

Let $a, a r, a r^2$, $\qquad$ be an infinite G.P. If $\sum_{n=0}^{\infty} a r^n=57$ and $\sum_{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as : $f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, \text { MARA } & x \geq 0 \\ x+1, & x \leq 0\end{cases}$ Then the function $f(g(x))$ is

Let $a_1, a_2, a_3, \ldots$ be in an arithmetic progression of positive terms. Let $\mathrm{A}_{\mathrm{k}}=\mathrm{a}_1^2-\mathrm{a}_2^2+\mathrm{a}_3^2-\mathrm{a}_4^2+\ldots+\mathrm{a}_{2 \mathrm{k}-1}^2-\mathrm{a}_{2 \mathrm{k}}^2$. If $\mathrm{A}_3=-153, \mathrm{~A}_5=-435$ and $\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=66$, then $\mathrm{a}_{17}-\mathrm{A}_7$ is equal to______

Let $3,a,b,c$ be in $A.P.$ and $3,a-1,b+1,c+9$ be in $G.P.$ Then, the arithmetic mean of $a,b$ and $c$ is:

Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^n$, then $\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}$ is equal to______

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

Let $\alpha, \beta$ be the distinct roots of the equation $x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}$ and $a_n=\alpha^n+\beta^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

Let $a,b,c$ be the length of three sides of a triangle satisfying the condition $({a}^{2}+{b}^{2}){x}^{2}-2b(a+c)$ $x+({b}^{2}+{c}^{2})=0$. If the set of all possible values of $x$ is in the interval $(\alpha ,\beta ),$ then $12({\alpha }^{2}+{\beta }^{2})$ is equal to _______.

Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{6}x+3=0$ such that $Im(\alpha )>Im(\beta )$. Let $a,b$ be integers not divisible by $3$and $n$ be a natural number such that $\frac{{\alpha }^{99}}{\beta }+{\alpha }^{98}={3}^{n}(a+ib),i=\sqrt{-1}$. Then $n+a+b$ is equal to ___________.

Let $\alpha, \beta$ be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-x+2=0$ with $Im(\alpha )>Im(\beta )$. Then ${\alpha }^{6}+{\alpha }^{4}+{\beta }^{4}-5{\alpha }^{2}$ is equal to

Let $S$ be the set of positive integral values of $a$ for which $\frac{a{x}^{2}+2(a+1)x+9a+4}{{x}^{2}-8x+32}<0,\forall x\in \mathbb{R}$. Then, the number of elements in $S$ is:

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$. Then the value of $m$ is

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

Let ${S}_{a}$ denote the sum of first $n$ terms an arithmetic progression. If ${S}_{20}=790$ and ${S}_{10}=145$, then ${S}_{15}-$ ${S}_{5}$ is :

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

Let for any three distinct consecutive terms $a,b,c$ of an A.P, the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha ,\beta )$ be a point such that the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$ and $x+2y+3z=4$, has infinitely many solutions. Then $(PQ{)}^{2}$ is equal to _______.

Let $f(x)={2}^{x}-{x}^{2},x\in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y={f}^{'}(x)$ intersects the $x-$axis, then the value of $m+n$ is

Let $\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$. If $x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$ for some $x, y, z \in \mathbb{R}, x y z \neq 0$, then $6 \alpha+4 \beta+\gamma$ is equal to _______

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

Let $\lambda, \mu \in \mathbf{R}$. If the system of equations $\begin{aligned} & 3 x+5 y+\lambda z=3 \\ & 7 x+11 y-9 z=2 \\ & 97 x+155 y-189 z=\mu \end{aligned}$ has infinitely many solutions, then $\mu+2 \lambda$ is equal to :

Let $0 \leq \mathrm{r} \leq \mathrm{n}$. If ${ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}+1}:{ }^n \mathrm{C}_{\mathrm{r}}:{ }^{\mathrm{n}-1} \mathrm{C}_{\mathrm{r}-1}=55: 35: 21$, then $2 \mathrm{n}+5 \mathrm{r}$ is equal to:

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$ is the identity matrix of order $3 \times 3$, then $2 a+3 b$ is equal to

Let $A=\{n \in[100,700] \cap \mathbb{N}: n$ is neither a multiple of 3 nor a multiple of 4$\}$. Then the number of elements in $A$ is

Let $A={1,2,3,...20}$. Let ${R}_{1}$ and ${R}_{2}$ two relation on $A$ such that ${R}_{1}={(a,b):b$ is divisible by $a$} ${R}_{2}={(a,b):a$ is an integral multiple of $b$} Then, number of elements in ${R}_{1}-{R}_{2}$ is equal to __________.

Let $A={1,2,3,....100}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $2x=3y$. Let ${R}_{1}$ be a symmetric relation on $A$ such that $R\subset {R}_{1}$ and the number of elements in ${R}_{1}$ is $n$. Then the minimum value of $n$ is _______.

Let $A=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x \mathrm{R} y$ if and only if $4 x \leq 5 \mathrm{y}$. Let $\mathrm{m}$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Let $S={z\in C:|z-1|=1\mathrm{and}(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}}$. Let ${z}_{1},{z}_{2}\in S$ be such that $|{z}_{1}|=\underset{z\in s}{\mathrm{max}}|z|$ and $|{z}_{2}|=\underset{z\in s}{\mathrm{min}}|z|$. Then ${|\sqrt{2}{z}_{1}-{z}_{2}|}^{2}$ equals:

Let $S={1,2,3,\ldots ,10}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R={(A,B):A\cap B\neq \phi ;A,B\in M}$ is :

Let the complex numbers $\alpha$ and $\frac{1}{\bar{\alpha }}$ lie on the circles ${|z-{z}_{0}|}^{2}=4$ and ${|z-{z}_{0}|}^{2}=16$ respectively, where ${z}_{0}=1+i$. Then, the value of $100|\alpha {|}^{2}$ is__________.

Let the first term of a series be $T_1=6$ and its $r^{\text {th }}$ term $T_r=3 T_{r-1}+6^r, r=2,3$, $\qquad$ $n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \cdot 3^n+1\right)$, then $n$ is equal to______

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $n^{\text {th }}$ term of the A.P., then $n$ is equal to:

Let the positive integers be written in the form :  If the $k^{\text {th }}$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is _______

Let the range of the function $f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R}$ be $[a, b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac{\alpha}{\beta}$ is equal to

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3, \ldots, 20\}$ be given by $R_1=\{(x, y): 2 x-3 y=2\}$ and $R_2=\{(x, y):-5 x+4 y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M+N$ equals

Let the set $S=\{2,4,8,16, \ldots, 512\}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\mathrm{S}$ and $\mathrm{A} \cap \mathrm{B}=\mathrm{B} \cap \mathrm{C}=\mathrm{A} \cap \mathrm{C}=\phi$. The maximum number of such possible partitions of $S$ is equal to:

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

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Let the system of equations $x+2y+3z=5,2x+3y+z=9,4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to:

Let $S={x\in R:{(\sqrt{3}+\sqrt{2})}^{x}+{(\sqrt{3}-\sqrt{2})}^{x}=10}$. Then the number of elements in $S$ is:

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is

Let $\alpha ={1}^{2}+{4}^{2}+{8}^{2}+{13}^{2}+{19}^{2}+{26}^{2}+\ldots \ldots .$ upto $10$ terms and $\beta =\sum _{n=1}^{10}{n}^{4}$. If $4\alpha -\beta =55k+40$, then $k$ is equal to _______.

Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0 < x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}>0$ and $\mathrm{g}(x)=(f(x \mid)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is

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

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

Number of integral terms in the expansion of ${{{7}^{(\frac{1}{2})}+{11}^{(\frac{1}{6})}}}^{824}$ is equal to ______.

Number of ways of arranging $8$ identical books into $4$ identical shelves where any number of shelves may remain empty is equal to

Remainder when ${64}^{{32}^{32}}$ is divided by $9$ is equal to _____.

Suppose $28-p,p,70-\alpha ,\alpha$ are the coefficient of four consecutive terms in the expansion of $(1+x{)}^{n}$. Then the value of $2\alpha -3p$ equals

The area (in sq. units) of the region $S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{Im}(z) \geq 0\}$ is

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$. Then a possible value of $p+q$ is :

The coefficient of ${x}^{2012}$ in the expansion of ${(1-x)}^{2008}{(1+x+{x}^{2})}^{2007}$ is equal to _____.

The function $f:N-{1}\rightarrow N$; defined by $f(n)=$ the highest prime factor of $n$, is :

The function $\text { f: R->R, }$ $f(x)=\frac{x^2+2 x-15}{x^2-4 x+9}, x \in \mathbb{R}$ is

The lines ${L}_{1},{L}_{2},...,{L}_{20}$ are distinct. For $n=1,2,3,...,10$ all the lines ${L}_{2n-1}$ are parallel to each other and all the lines ${L}_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set ${{L}_{1},{L}_{2},...,{L}_{20}}$ is equal to:

The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7 , when the repetition of digits is not allowed, and which are not divisible by 3 , is equal to__________

The number of common terms in the progressions $4,9,14,19,\ldots \ldots$, up to ${25}^{\text{th }}$ term and $3,6,9,12$,.... up to ${37}^{\text{th }}$ term is :

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is_______

The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is

The number of elements in the set $S={(x,y,z):x,y,z\in Z,x+2y+3z=42,x,y,z\geq 0}$ equals ________

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is _________

The number of real solutions of the equation \(x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0\) is ______.

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is_________

The number of solutions, of the equation ${e}^{\mathrm{sin}x}-2{e}^{-\mathrm{sin}x}=2$ is

The number of symmetric relations defined on the set ${1,2,3,4}$ which are not reflexive is _______.

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to :

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

The number of ways of getting a sum 16 on throwing a dice four times is______

The remainder when $428^{2024}$ is divided by 21 is__________

The sum of all possible values of $\theta \in[-\pi, 2 \pi]$, for which $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely imaginary, is equal

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is :

The sum of the coefficient of $x^{2 / 3}$ and $x^{-2 / 5}$ in the binomial expansion of $\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$ is

The sum of the series $\frac{1}{1-3\cdot {1}^{2}+{1}^{4}}+\frac{2}{1-3\cdot {2}^{2}+{2}^{4}}+\frac{3}{1-3\cdot {3}^{2}+{3}^{4}}+....$ up to $10$ terms is

The sum of the square of the modulus of the elements in the set $\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}$ is ________

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

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

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

The values of $\alpha$, for which $|\begin{matrix}1 & \frac{3}{2} & \alpha +\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha +\frac{1}{3} \\ 2\alpha +3 & 3\alpha +1 & 0\end{matrix}|=0$, lie in the interval

The values of $m, n$, for which the system of equations $\begin{aligned} & x+y+z=4, \\ & 2 x+5 y+5 z=17, \\ & x+2 y+\mathrm{m} z=\mathrm{n} \end{aligned}$ has infinitely many solutions, satisfy the equation:

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _____

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $A B$, excluding $A$ and $B$, of a triangle $A B C$. Similarly there are 6 points $\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}$ on the side $\mathrm{BC}$ and 7 points $\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}$ on the side $C A$ of the triangle. The number of triangles, that can be formed using the points $\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}$ as vertices, is :