JEE Main 2021 — Algebra

If ${\mathrm{log}}_{3}2,{\mathrm{log}}_{3}({2}^{x}-5),{\mathrm{log}}_{3}({2}^{x}-\frac{7}{2})$ are in an arithmetic progression, then the value of $x$ is equal to _____.

The number of solutions of the equation x² - 5|x| + 6 = 0 is:

The system of linear equations$3x-2y-kz=10$ $2x-4y-2z=6$ $x+2y-z=5m$ is inconsistent if :

Let $A={2,3,4,5,\ldots .,30}$ and $'\simeq '$ be an equivalence relation on $A\times A,$ defined by $(a,b)\simeq (c,d),$ if and only if $ad=bc$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $(4,3)$ is equal to :

Let $A=[\begin{matrix}x & y & z \\ y & z & x \\ z & x & y\end{matrix}],$ where $x,y$ and $z$ are real numbers such that $x+y+z>0$ and $xyz=2$. If ${A}^{2}={I}_{3}$, then the value of ${x}^{3}+{y}^{3}+{z}^{3}$ is

The real valued function $f(x)=\frac{{cosec}^{-1}x}{\sqrt{x-[x]}},$ where $[x]$ denotes the greatest integer less than or equal to $x,$ is defined for all $x$ belonging to:

For real numbers $\alpha$ and $\beta ,$ consider the following system of linear equations: $x+y-z=2,x+2y+\alpha z=1$ and $2x-y+z=\beta .$ If the system has infinite solutions, then $\alpha +\beta$ is equal to ______.

If ${e}^{({\mathrm{cos}}^{2}x+{\mathrm{cos}}^{4}x+{\mathrm{cos}}^{6}x+....\infty ){\mathrm{log}}_{e}2}$ satisfies the equation ${t}^{2}-9t+8=0$, then the value of $\frac{2\mathrm{sin}x}{\mathrm{sin}x+\sqrt{3}\mathrm{cos}x}$, where $0<x<\frac{\pi }{2}$, is equal to

Let $A={n\in N:n$ is a $3-$digit number$}$ $B={9k+2:k\in N}$ and $C={9k+l:k\in N}$ for some $l(0<l<9)$. If the sum of all the elements of the set $A\cap (B\cup C)$ is $274\times 400,$ then $l$ is equal to

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6,3x+5y+5z=26$ and $x+2y+\lambda z=\mu$ has no solution, are:

The term independent of $x$ in the expansion of ${(\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}})}^{10},$ where $x\neq 0,1$ is equal to

The number of elements in the set ${n\in {1,2,3,\ldots ,100}\mid (11{)}^{n}>(10{)}^{n}+(9{)}^{n}}$ is ___________.

Let $A={{a}_{ij}}$ be a $3\times 3$ matrix, where ${a}_{ij}={\begin{matrix}{(-1)}^{j-i}\mathrm{if}i<j \\ 2\mathrm{if}i=j \\ {(-1)}^{i+j}\mathrm{if}i>j\end{matrix}$ then $\mathrm{det}(3Adj(2{A}^{-1}))$ is equal to ________.

Let $g:N\rightarrow N$ be defined as $g(3n+1)=3n+2$ $g(3n+2)=3n+3$ $g(3n+3)=3n+1$, for all $n\geq 0$ Then which of the following statements is true ?

If $x,y,z$ are in arithmetic progression with common difference $d,x\neq 3d,$ and the determinant of the matrix $[\begin{matrix}3 & 4\sqrt{2} & x \\ 4 & 5\sqrt{2} & y \\ 5 & k & z\end{matrix}]$ is zero, then the value of ${k}^{2}$ is

If the equation $a|z{|}^{2}+\bar{\bar{\alpha }z+\alpha \bar{z}}+d=0$ represents a circle where $a,d$ are real constants then which of the following condition is correct?

Let$A=[\begin{matrix}1 & 2 \\ -1 & 4\end{matrix}].$ If ${A}^{-1}=\alpha I+\beta A,\alpha ,\beta \in R,I$ is a $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is equal to :

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A\times B$. Then :

Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in N,{(1-x+{x}^{3})}^{n}=\sum _{j=0}^{3n}{a}_{j}{x}^{j}$, then $\sum _{j=0}^{[\frac{3n}{2}]}{a}_{2j}+4\sum _{j=0}^{[\frac{3n-1}{2}]}{a}_{2j+1}$ is equal to :

Let $R={(P,Q)|P$ and $Q$ are at the same distance from the origin$}$ be a relation, then the equivalence class of $(1,-1)$ is the set

The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :

Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is:

If the matrix $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 3 & 0 & -1\end{matrix}]$ satisfies the equation ${A}^{20}+\alpha {A}^{19}+\beta A=[\begin{matrix}1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1\end{matrix}]$ for some real numbers $\alpha$ and $\beta$, then $\beta -\alpha$ is equal to ______.

The sum of all those terms which are rational numbers in the expansion of ${({2}^{\frac{1}{3}}+{3}^{\frac{1}{4}})}^{12}$ is:

Let $p$ and $q$ be two positive numbers such that $p+q=2$ and ${p}^{4}+{q}^{4}=272$. Then $p$ and $q$ are roots of the equation:

For the natural numbers $m,n,$ if $(1-y{)}^{m}(1+y{)}^{n}=1+{a}_{1}y+{a}_{2}{y}^{2}+\ldots .+{a}_{m+n}{y}^{m+n}$ and ${a}_{1}={a}_{2}=10,$ then the value of $m+n,$ is equal to:

Let $A=[\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{matrix}].$ Then the number of $3\times 3$ matrices $B$ with entries from the set ${1,2,3,4,5}$ and satisfying $AB=BA$ is ________.

$3\times {7}^{22}+2\times {10}^{22}-44$ when divided by $18$ leaves the remainder

If the coefficients of ${x}^{7}$ in ${({x}^{2}+\frac{1}{bx})}^{11}$ and ${x}^{-7}$ in ${(x-\frac{1}{b{x}^{2}})}^{11},b\neq 0,$ are equal, then the value of $b$ is equal to:

If $Prn=Pr+1n$ and $Crn=Cr-1n,$ then the value of $r$ is equal to:

If the greatest value of the term independent of $x$ in the expansion of ${(x\mathrm{sin}\alpha +a\frac{\mathrm{cos}\alpha }{x})}^{10}$ is $\frac{10!}{{(5!)}^{2}},$ then the value of $a$ is equal to:

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity$\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots .+\frac{1}{a-nb}=\alpha n+\beta {n}^{2}+\gamma {n}^{3}$ then the value of $\gamma$ is :

If the constant term, in binomial expansion of ${(2{x}^{r}+\frac{1}{{x}^{2}})}^{10}$ is $180,$ then $r$ is equal to ____________.

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x+\frac{a}{{x}^{2}})}^{n},x\neq 0,$ be in the ratio $12:8:3.$ Then the term independent of $x$ in the expansion, is equal to _______.

The maximum value of the term independent of $t$ in the expansion of ${(t{x}^{\frac{1}{5}}+\frac{{(1-x)}^{\frac{1}{10}}}{t})}^{10}$where $x\in (0,1)$ is:

If $n$ is the number of irrational terms in the expansion of ${({3}^{1/4}+{5}^{1/8})}^{60}$, then $(n-1)$ is divisible by :

Let $f:R\rightarrow R$ be defined as $f(x+y)+f(x-y)=2f(x)f(y),f(\frac{1}{2})=-1.$ Then the value of $\sum _{k=1}^{20}\frac{1}{\mathrm{sin}(k)\mathrm{sin}(k+f(k))}$ is equal to :

Let ${S}_{n}$ denote the sum of first $n$-terms of an arithmetic progression. If ${S}_{10}=530,{S}_{5}=140,$ then ${S}_{20}-{S}_{6}$ is equal to:

Consider function $f:A\rightarrow B$ and $g:B\rightarrow C(A,B,C\subseteq R)$ such that ${(gof)}^{-1}$ exists, then:

If $A=[\begin{matrix}0 & \mathrm{sin}\alpha \\ \mathrm{sin}\alpha & 0\end{matrix}]$ and $\mathrm{det}({A}^{2}-\frac{1}{2}I)=0,$ then a possible value of $\alpha$ is

Team $A''$ consists of $7$ boys and $n$ girls and Team $B''$ has $4$ boys and $6$ girls. If a total of $52$ single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then $n$ is equal to:

Let $f(x)={\mathrm{sin}}^{-1}x$ and $g(x)=\frac{{x}^{2}-x-2}{2{x}^{2}-x-6}$. If $g(2)=\underset{x\rightarrow 2}{\mathrm{lim}}g(x)$, then the domain of the function $fog$ is

Let ${S}_{n}$ be the sum of the first $n$ terms of an arithmetic progression. If ${S}_{3n}=3{S}_{2n}$, then the value of $\frac{{S}_{4n}}{{S}_{2n}}$ is :

Let ${P}_{1},{P}_{2}\ldots ,{P}_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points ${P}_{i},{P}_{j},{P}_{k}$ such that $i+j+k\neq 15,$ is :

The least positive integer $n$ such that $\frac{(2i{)}^{n}}{(1-i{)}^{n-2}},i=\sqrt{-1},$ is a positive integer, is ______.

The sum of the roots of the equation, $x+1-2{\mathrm{log}}_{2}(3+{2}^{x})+2{\mathrm{log}}_{4}(10-{2}^{-x})=0,$ is :

Which of the following is not correct for relation $R$ on the set of real numbers?

Define a relation $R$ over a class of $n\times n$ real matrices $A$ and $B$ as "$ARB$ iff there exists a non-singular matrix $P$ such that $PA{P}^{-1}=B$". Then which of the following is true ?

Out of all the patients in a hospital $89%$ are found to be suffering from heart ailment and $98%$ are suffering from lungs infection. If $K%$ of them are suffering from both ailments, then $K$ can not belong to the set:

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R={(x,y)\in N\times N:{x}^{3}-3{x}^{2}y-x{y}^{2}+3{y}^{3}=0}$. Then the relation $R$ is

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn diagrams can justify the above statement? 

If the remainder when $x$ is divided by $4$ is $3$, then the remainder when ${(2020+x)}^{2022}$ is divided by $8$ is ___ .

The number of the real roots of the equation ${(x+1)}^{2}+|x-5|=\frac{27}{4}$ is ________.

Let ${S}_{1}$ be the sum of first $2n$ terms of an arithmetic progression. Let ${S}_{2}$ be the sum of first $4n$ terms of the same arithmetic progression. If $({S}_{2}-{S}_{1})$ is $1000$, then the sum of the first $6n$ terms of the arithmetic progression is equal to:

If the sum of the coefficients in the expansion of $(x+y{)}^{n}$ is $4096,$ then the greatest coefficient in the expansion is _____.

The number of real roots of the equation ${e}^{4x}-{e}^{3x}-4{e}^{2x}-{e}^{x}+1=0$ is equal to

Let $A=[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0\end{matrix}].$ Then ${A}^{2025}-{A}^{2020}$ is equal to

Let $A={n\in N\mid {n}^{2}\leq n+10,000},B={3k+1\mid k\in N}$ and $C={2k\mid k\in N},$ then the sum of all the elements of the set $A\cap (B-C)$ is equal to ________.

Let ${(1+x+2{x}^{2})}^{20}={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\ldots +{a}_{40}{x}^{40},$ then ${a}_{1}+{a}_{3}+{a}_{5}+\ldots +{a}_{37}$ is equal to

If $A=[\begin{matrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}}\end{matrix}],B=[\begin{matrix}1 & 0 \\ i & 1\end{matrix}],i=\sqrt{-1}$, and $Q={A}^{T}BA$, then the inverse of the matrix $A{Q}^{2021}{A}^{T}$ is equal to:

If $A=[\begin{matrix}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{matrix}]$ and $M=A+{A}^{2}+{A}^{3}+\ldots +{A}^{20},$ then the sum of all the elements of the matrix $M$ is equal to _______.

Let $A$ and $B$ be two $3\times 3$ real matrices such that $({A}^{2}-{B}^{2})$ is invertible matrix. If ${A}^{5}={B}^{5}$ and ${A}^{3}{B}^{2}={A}^{2}{B}^{3},$ then the value of the determinant of the matrix ${A}^{3}+{B}^{3}$ is equal to :

Let $M={A=[\begin{matrix}a & b \\ c & d\end{matrix}]:a,b,c,d\in (\pm 3,\pm 2,\pm 1,0)}$. Define $f:M\rightarrow Z,$ as $f(A)=det(A),$ for all $A\in M$ where $Z$ is set of all integers. Then the number of $A\in M$ such that $f(A)=15$ is equal to .

Let $S={n\in N,{(\begin{matrix}0 & i \\ 1 & 0\end{matrix})}^{n}(\begin{matrix}a & b \\ c & d\end{matrix})=(\begin{matrix}a & b \\ c & d\end{matrix})\forall a,b,c,d\in R},$ where $i=\sqrt{-1}.$ Then the number of $2-$ digit numbers in the set $S$ is

If $P=[\begin{matrix}1 & 0 \\ \frac{1}{2} & 1\end{matrix}],$ then ${P}^{50}$ is:

Let $A=[\begin{matrix}\begin{matrix}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{matrix}\end{matrix}]$ and $B=7{A}^{20}-20{A}^{7}+2I$, where $I$ is an identity matrix of order $3\times 3.$ If $B=[{b}_{ij}]$, then ${b}_{13}$ is equal to

Let $I$ be an identity matrix of order $2\times 2$ and $P=[\begin{matrix}2 & -1 \\ 5 & -3\end{matrix}]$. Then the value of $n\in N$ for which ${P}^{n}=5I-8P$ is equal to ___ .

Let $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$ and $B=[\begin{matrix}\alpha \\ \beta \end{matrix}]\neq [\begin{matrix}0 \\ 0\end{matrix}]$ such that $AB=B$ and $a+d=2021,$ then the value of $ad-bc$ is equal to ______ .

Let $A+2B=[\begin{matrix}1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1\end{matrix}]$ and $2A-B=[\begin{matrix}2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2\end{matrix}].$ If $Tr(A)$ denotes the sum of all diagonal elements of the matrix $A,$ then $Tr(A)-Tr(B)$ has value equal to

Let $A=[\begin{matrix}{a}_{1} \\ {a}_{2}\end{matrix}]$ and $B=[\begin{matrix}{b}_{1} \\ {b}_{2}\end{matrix}]$ be two $2\times 1$matrices with real entries such that $A=XB,$ where $X=\frac{1}{\sqrt{3}}[\begin{matrix}1 & -1 \\ 1 & k\end{matrix}],$ and $k\in R.$ If ${a}_{1}^{2}+{a}_{2}^{2}=\frac{2}{3}({b}_{1}^{2}+{b}_{2}^{2})$ and $({k}^{2}+1){b}_{2}^{2}\neq -2{b}_{1}{b}_{2}$, then the value of $k$ is __________.

The total number of $3\times 3$ matrices $A$ having enteries from the set $(0,1,2,3)$ such that the sum of all the diagonal entries of $A{A}^{T}$ is $9$, is equal to

Let $A=[\begin{matrix}i & -i \\ -i & i\end{matrix}],i=\sqrt{-1}$. Then, the system of linear equations ${A}^{8}[\begin{matrix}x \\ y\end{matrix}]=[\begin{matrix}8 \\ 64\end{matrix}]$ has :

If $A=[\begin{matrix}0 & -\mathrm{tan}(\frac{\theta }{2}) \\ \mathrm{tan}(\frac{\theta }{2}) & 0\end{matrix}]$ and $({I}_{2}+A){({I}_{2}-A)}^{-1}=[\begin{matrix}a & -b \\ b & a\end{matrix}],$ then $13({a}^{2}+{b}^{2})$ is equal to _____ .

Let $P=[\begin{matrix}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{matrix}],$ where $\alpha \in R.$ Suppose $Q=[{q}_{ij}]$ is a matrix satisfying $PQ=k{I}_{3}$ for some non-zero $k\in R.$ If ${q}_{23}=-\frac{k}{8}$ and $|Q|=\frac{{k}^{2}}{2}$, then ${\alpha }^{2}+{k}^{2}$ is equal to_________.

Consider the system of linear equations $-x+y+2z=0$ $3x-ay+5z=1$ $2x-2y-az=7$ Let ${S}_{1}$ be the set of all $a\in R$ for which the system is inconsistent and ${S}_{2}$ be the set of all $a\in R$ for which the system has infinitely many solutions. If $n({S}_{1})$ and $n({S}_{2})$ denote the number of elements in ${S}_{1}$ and ${S}_{2}$ respectively, then

Let $A=[\begin{matrix}[x+1] & [x+2] & [x+3] \\ [x] & [x+3] & [x+3] \\ [x] & [x+2] & [x+4]\end{matrix}]\begin{matrix} \\ \\ \end{matrix}$, where $[x]$ denotes the greatest integer less than or equal to $x$. If $det(A)$$=192$, then the set of values of $x$ is in the interval:

If the following system of linear equations $2x+y+z=5$ $x-y+z=3$ $x+y+az=b$ has no solution, then :

If $\alpha +\beta +\gamma =2\pi ,$ then the system of equations $x+(\mathrm{cos}\gamma )y+(\mathrm{cos}\beta )z=0$ $(\mathrm{cos}\gamma )x+y+(\mathrm{cos}\alpha )z=0$ $(\mathrm{cos}\beta )x+(\mathrm{cos}\alpha )y+z=0$ has :

Let $\theta \in (0,\frac{\pi }{2})$. If the system of linear equations $(1+{\mathrm{cos}}^{2}⁡\theta )x+{\mathrm{sin}}^{2}⁡\theta y+4\mathrm{sin}⁡3\theta z=0$ ${\mathrm{cos}}^{2}⁡\theta x+(1+{\mathrm{sin}}^{2}⁡\theta )y+4\mathrm{sin}⁡3\theta z=0$ ${\mathrm{cos}}^{2}⁡\theta x+{\mathrm{sin}}^{2}⁡\theta y+(1+4\mathrm{sin}⁡3\theta )z=0$ has a non-trivial solution, then the value of $\theta$ is:

If the system of linear equations $2x+y-z=3$ $x-y-z=\alpha$ $3x+3y+\beta z=3$ has infinitely many solutions, then $|\alpha +\beta -\alpha \beta |$ is equal to __________.

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu ,$ and a system of linear equations $x+y+z=5$ $x+2y+3z=\mu$ $x+3y+\lambda z=1$ is constructed. If $p$ is the probability that the system has a unique solution and $q$ is the probability that the system has no solution, then:

Let $f(x)=|\begin{matrix}{\mathrm{sin}}^{2}x & -2+{\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ 2+{\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ {\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & 1+\mathrm{cos}2x\end{matrix}|,x\in [0,\pi ].$ Then the maximum value of $f(x)$ is equal to

The values of $a$ and $b$, for which the system of equations $2x+3y+6z=8$ $x+2y+az=5$ $3x+5y+9z=b$ has no solution, are :

The value of $k\in R,$ for which the following system of linear equations $3x-y+4z=3$ $x+2y-3z=-2$ $6x+5y+kz=-3$ has infinitely many solutions, is:

Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0,\lambda ,\mu \in R$ has a non-trivial solution. Then which of the following is true?

Let $a,b,c,d$ be in arithmetic progression with common difference $\lambda$. If $|\begin{matrix}x+a-c & x+b & x+a \\ x-1 & x+c & x+b \\ x-b+d & x+d & x+c\end{matrix}|=2$, then value of ${\lambda }^{2}$ is equal to________.

Let $\alpha ,\beta ,\gamma$ be the real roots of the equation, ${x}^{3}+a{x}^{2}+bx+c=0,$ ($a,b,c\in R$ and $a,b\neq 0$). If the system of equations (in, $u,v,w$) given by $\alpha u+\beta v+\gamma w=0,\beta u+\gamma v+\alpha w=0,\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\frac{{a}^{2}}{b}$ is

The maximum value of $f(x)=|\begin{matrix}{\mathrm{sin}}^{2}x & 1+{\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ 1+{\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ {\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{sin}2x\end{matrix}|,x\in R$ is

Consider the following system of equations: $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$ where $a,b$ and $c$ are real constants. Then the system of equations :

The value of $|\begin{matrix}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{matrix}|$ is

For the system of linear equations: $x-2y=1,x-y+kz=-2,ky+4z=6,k\in R$ Consider the following statements: (A) The system has unique solution if $k\neq 2,k\neq -2.$ (B) The system has unique solution if $k=-2.$ (C) The system has unique solution if $k=2.$ (D) The system has no-solution if $k=2.$ (E) The system has infinite number of solutions if $k\neq -2.$ Which of the following statements are correct?

If the system of equations$kx+y+2z=1$ $3x-y-2z=2$ $-2x-2y-4z=3$ has infinitely many solutions, then $k$ is equal to ______ .

Let a complex number $z,|z|\neq 1$, satisfy ${\mathrm{log}}_{\frac{1}{\sqrt{2}}}(\frac{|z|+11}{{(|z|-1)}^{2}})\leq 2$. Then, the largest value of $|z|$ is equal to _________.

If the functions are defined as $f(x)=\sqrt{x}$ and $g(x)=\sqrt{1-x},$ then what is the common domain of the following functions: $f+g,f-g,f/g,g/f,g-f$, where $(f\pm g)(x)=f(x)\pm g(x),(f/g)(x)=\frac{f(x)}{g(x)}$

The range of the function $f(x)={\mathrm{log}}_{\sqrt{5}}(3+\mathrm{cos}(\frac{3\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}+x)+\mathrm{cos}(\frac{\pi }{4}-x)-\mathrm{cos}(\frac{3\pi }{4}-x))$ is :

Let $f(x)$ be a polynomial of degree $3$ such that $f(k)=-\frac{2}{k}$ for $k=2,3,4,5.$ Then the value of $52-10f(10)$ is equal to _____ .

The domain of the function ${cosec}^{-1}(\frac{1+x}{x})$ is :

If $[x]$ be the greatest integer less than or equal to $x,$ then $\sum _{n=8}^{100}[\frac{{(-1)}^{n}n}{2}]$ is equal to:

Let $[x]$ denote the greatest integer less than or equal to $x.$ Then, the values of $x\in R$ satisfying the equation ${[{e}^{x}]}^{2}+[{e}^{x}+1]-3=0$ lie in the interval:

If the domain of the function $f(x)=\frac{{\mathrm{cos}}^{-1}\sqrt{{x}^{2}-x+1}}{\sqrt{{\mathrm{sin}}^{-1}(\frac{2x-1}{2})}}$ is the interval $(\alpha ,\beta ],$ then $\alpha +\beta$ is equal to:

Let $f:R-{\frac{\alpha }{6}}\rightarrow R$ be defined by $f(x)=(\frac{5x+3}{6x-\alpha })$. Then the value of $\alpha$ for which $(fof)(x)=x$, for all $x\in R-{\frac{\alpha }{6}},$ is

Let $[x]$ denote the greatest integer $\leq x$, where $x\in R$. If the domain of the real valued function $f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty ),a<b<c$, then the value of $a+b+c$ is:

Let $f:R-{3}\rightarrow R-{1}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:R\rightarrow R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which ${f}^{-1}(x)+{g}^{-1}(x)=\frac{13}{2}$ is equal to

Let $A={1,2,3,\ldots ,10}$ and $f:A\rightarrow A$ be defined as $f(k)={\begin{matrix}k+1 & \mathrm{if}k\mathrm{is}odd \\ k & \mathrm{if}k\mathrm{is}even\end{matrix}$ Then the number of possible functions $g:A\rightarrow A$ such that $gof=f$ is:

The number of elements in the set ${x\in R:(|x|-3)|x+4|=6}$ is equal to

The inverse of $y={5}^{\mathrm{log}x}$ is:

If $a+\alpha =1,b+\beta =2$ and $af(x)+\alpha f(\frac{1}{x})=bx+\frac{\beta }{x},x\neq 0,$ then the value of the expression $\frac{f(x)+f(\frac{1}{x})}{x+\frac{1}{x}}$ is ___________.

A function $f(x)$ is given by $f(x)=\frac{{5}^{x}}{{5}^{x}+5}$, then the sum of the series $f(\frac{1}{20})+f(\frac{2}{20})+f(\frac{3}{20})+\ldots +f(\frac{39}{20})$ is equal to:

Let $f:R\rightarrow R$ be defined as $f(x)=2x-1$ and $g:R-{1}\rightarrow R$. be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition function $f(g(x))$ is:

The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$

The system of equations $kx+y+z=1,x+ky+z=k$ and $x+y+zk={k}^{2}$ has no solution if $k$ is equal to:

If $A=[\begin{matrix}2 & 3 \\ 0 & -1\end{matrix}],$ then the value of $det({A}^{4})+det({A}^{10}-(Adj(2A){)}^{10})$ is equal to ________.

If $(\frac{{3}^{6}}{{4}^{4}})k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4}-\frac{12}{{x}^{2}})}^{12},$ then $k$ is equal to

If $(2021{)}^{3762}$ is divided by $17,$ then the remainder is _______.

The number of elements in the set {$A=[\begin{matrix}a & b \\ 0 & d\end{matrix}]:a,b,d\in {-1,0,1}$ and $(I-A{)}^{3}=I-{A}^{3}$}, where $I$ is $2\times 2$ identity matrix, is .

If the fourth term in the expansion of ${(x+{x}^{{\mathrm{log}}_{2}x})}^{7}$ is $4480,$ then the value of $x$ where $x\in N$ is equal to:

A possible value of $x,$ for which the ninth term in the expansion of ${{{3}^{{\mathrm{log}}_{3}\sqrt{{25}^{x-1}+7}}+{3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}}}^{10}$ in the increasing powers of ${3}^{(-\frac{1}{8}){\mathrm{log}}_{3}({5}^{x-1}+1)}$ is equal to $180,$ is :

If the matrix $A=[\begin{matrix}0 & 2 \\ K & -1\end{matrix}]$ satisfies $A({A}^{3}+3I)=2I,$ then the value of $K$ is

Let $i=\sqrt{-1}.$ If $\frac{{(-1+i\sqrt{3})}^{21}}{{(1-i)}^{24}}+\frac{{(1+i\sqrt{3})}^{21}}{{(1+i)}^{24}}=k,$ and $n=[|k|]$ be the greatest integral part of $|k|.$ Then $\sum _{j=0}^{n+5}{(j+5)}^{2}-\sum _{j=0}^{n+5}(j+5)$ is equal to ________.

The lowest integer which is greater than ${(1+\frac{1}{{10}^{100}})}^{{10}^{100}}$ is

Let $Z$ be the set of all integers, $A={(x,y)\in Z\times Z:(x-2{)}^{2}+{y}^{2}\leq 4}$ $B={(x,y)\in Z\times Z:{x}^{2}+{y}^{2}\leq 4}\text{ and }$ $C={(x,y)\in Z\times Z:(x-2{)}^{2}+(y-2{)}^{2}\leq 4}$ If the total number of relations from $A\cap B$ to $A\cap C$ is ${2}^{p}$, then the value of $p$ is:

If the coefficient of ${a}^{7}{b}^{8}$ in the expansion of $(a+2b+4ab{)}^{10}$ is $K\cdot {2}^{16},$ then $K$ is equal to

The sum of ${162}^{th}$ power of the roots of the equation ${x}^{3}-2{x}^{2}+2x-1=0$ is ______.

If $\alpha ,\beta$ are roots of the equation ${x}^{2}+5(\sqrt{2})x+10=0,\alpha >\beta$ and ${P}_{n}={\alpha }^{n}-{\beta }^{n}$ for each positive integer $n,$ then the value of $(\frac{{P}_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{{P}_{18}{P}_{19}+5\sqrt{2}{P}_{18}^{2}})$ is equal to

Let a complex number be $w=1-\sqrt{3}i$. Let another complex number $z$ be such that $|zw|=1$ and $\mathrm{arg}(z)-\mathrm{arg}(w)=\frac{\pi }{2}$. Then the area of the triangle (in sq. units) with vertices origin, $z$ and $w$ is equal to

The number of rational terms in the binomial expansion of ${({4}^{\frac{1}{4}}+{5}^{\frac{1}{6}})}^{120}$is_______.

Let $A=[{a}_{ij}]$ be a real matrix of order $3\times 3,$ such that ${a}_{i1}+{a}_{i2}+{a}_{i3}=1,$ for $i=1,2,3.$ Then, the sum of all the entries of the matrix ${A}^{3}$ is equal to:

Let $n$ be a positive integer. Let $A=\sum _{k=0}^{n}{(-1)}^{k}\times Ckn[{(\frac{1}{2})}^{k}+{(\frac{3}{4})}^{k}+{(\frac{7}{8})}^{k}+{(\frac{15}{16})}^{k}+{(\frac{31}{32})}^{k}]$. If $63A=1-\frac{1}{{2}^{30}},$ then $n$ is equal to ______ .

The total number of numbers, lying between $100$ and $1000$ that can be formed with the digits $1,2,3,4,5,$ if the repetition of digits is not allowed and numbers are divisible by either $3$ or $5,$ is

If ${x}^{2}+9{y}^{2}-4x+3=0,x,y\in R,$ then $x$ and $y$ respectively lie in the intervals

Let $A={0,1,2,3,4,5,6,7}.$ Then the number of bijective functions $f:A\rightarrow A$ such that $f(1)+f(2)=3-f(3)$ is equal to

The value of $3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots \infty }}}}$ is equal to

The minimum value of $f(x)={a}^{{a}^{x}}+{a}^{1-{a}^{x}}$, where $a,x\in R$ and $a>0$, is equal to:

If ${a}_{r}=\mathrm{cos}\frac{2r\pi }{9}+i\mathrm{sin}\frac{2r\pi }{9},r=1,2,3,\ldots ,i=\sqrt{-1}$, then the determinant $|\begin{matrix}{a}_{1} & {a}_{2} & {a}_{3} \\ {a}_{4} & {a}_{5} & {a}_{6} \\ {a}_{7} & {a}_{8} & {a}_{9}\end{matrix}|$ is equal to :

If the co-efficient of ${x}^{7}$ and ${x}^{8}$ in the expansion of ${(2+\frac{x}{3})}^{n}$ are equal, then the value of $n$ is equal to :

If for the complex numbers$z$ satisfying $|z-2-2i|\leq 1$, the maximum value of $|3iz+6|$ is attained at $a+ib$, then $a+b$ is equal to _____ .

If for the matrix, $A=[\begin{matrix}1 & -\alpha \\ \alpha & \beta \end{matrix}],A{A}^{T}={I}_{2}$, then the value of ${\alpha }^{4}+{\beta }^{4}$ is :

Let ${a}_{1},{a}_{2},\ldots ,{a}_{21}$ be an $A.P.$ such that $\sum _{n=1}^{20}\frac{1}{{a}_{n}{a}_{n+1}}=\frac{4}{9}.$ If the sum of this $A.P.$ is $189,$ then ${a}_{6}{a}_{16}$ is equal to :

If the real part of the complex number ${(1-\mathrm{cos}\theta +2i\mathrm{sin}\theta )}^{-1}$ is $\frac{1}{5}$ for $\theta \in (0,\pi ),$ then the value of the integral ${\int }_{0}^{\theta }\mathrm{sin}xdx$ is equal to:

The number of solutions of the equation ${\mathrm{log}}_{4}(x-1)={\mathrm{log}}_{2}(x-3)$ is ______.

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x)=f({x}^{3})+xg({x}^{3})$ is divisible by ${x}^{2}+x+1$, then $P(1)$ is equal to ___ .

If |z - 2| = |z + 2|, then z lies on:

The coefficient of ${x}^{256}$ in the expansion of ${(1-x)}^{101}{({x}^{2}+x+1)}^{100}$ is:

Let $f,g:N\rightarrow N$ such that $f(n+1)=f(n)+f(1)\forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?

The number of $4$-digit numbers which are neither multiple of $7$ nor multiple of $3$ is .

Let $S={1,2,3,4,5,6,7}.$ Then the number of possible functions $f:S\rightarrow S$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in S$ and $m\cdot n\in S$, is equal to _____.

Let $A$ be a symmetric matrix of order $2$ with integer entries. If the sum of the diagonal elements of ${A}^{2}$ is $1,$ then the possible number of such matrices is:

Let $M$ be any $3\times 3$ matrix with entries from the set ${0,1,2}$. The maximum number of such matrices, for which the sum of diagonal elements of ${M}^{T}M$ is seven, is______.

If $a+b+c=1,ab+bc+ca=2$ and $abc=3,$ then the value of ${a}^{4}+{b}^{4}+{c}^{4}$ is equal to:

The total number of positive integral solutions $(x,y,z)$ such that $xyz=24$ is :

The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1,2$ and $3$ only is:

Let $A$ be a $3\times 3$ matrix with $det(A)=4$. Let ${R}_{i}$ denote the ${i}^{th}$ row of $A$. If a matrix $B$ is obtained by performing the operation ${R}_{2}\rightarrow 2{R}_{2}+5{R}_{3}$ on $2A$, then $det(B)$ is equal to :

If sum of the first $21$ terms of the series ${\mathrm{log}}_{{9}^{1/2}}x+{\mathrm{log}}_{{9}^{1/3}}x+{\mathrm{log}}_{{9}^{1/4}}x+\ldots ..$ where $x>0$ is $504,$ then $x$ is equal to

If $\alpha$ and $\beta$ are the distinct roots of the equation ${x}^{2}+{(3)}^{\frac{1}{4}}x+{3}^{\frac{1}{2}}=0$, then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to:

Let $f:N\rightarrow N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in N.$ If $f(6)=18$ then $f(2)\cdot f(3)$ is equal to :

The number of pairs $(a,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation ${x}^{2}+ax+b=0,{\alpha }^{2}-2$ is also a root of this equation, is :

The term independent of $x$ in the expansion of ${[\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}}]}^{10},x\neq 1$, is equal to ___.

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

Let the domain of the function $f(x)={\mathrm{log}}_{4}({\mathrm{log}}_{5}({\mathrm{log}}_{3}(18x-{x}^{2}-77)))$ be $(a,b)$. Then the value of the integral ${\int }_{a}^{b}\frac{{\mathrm{sin}}^{3}x}{({\mathrm{sin}}^{3}x+{\mathrm{sin}}^{3}(a+b-x))}$ is equal to _____.

Let $z$ and $w$ be two complex numbers such that $w=z\bar{z}-2z+2,|\frac{z+i}{z-3i}|=1$ and $Re(w)$ has minimum value. Then, the minimum value of $n\in N$ for which ${w}^{n}$ is real, is equal to _______.

Let $A=[\begin{matrix}2 & 3 \\ a & 0\end{matrix}],a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $det(Q)=9$, then the modulus of the sum of all possible values of determinant of $P$ is equal to:

Consider an arithmetic series and a geometric series having four initial terms from the set ${11,8,21,16,26,32,4}$. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to _______.

Let ${a}_{1},{a}_{2},{a}_{3},\ldots$ be an A.P. If $\frac{{a}_{1}+{a}_{2}+\ldots +{a}_{10}}{{a}_{1}+{a}_{2}+\ldots +{a}_{p}}=\frac{100}{{p}^{2}},p\neq 10,$ then $\frac{{a}_{11}}{{a}_{10}}$ is equal to :

Let $P=[\begin{matrix}-30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14\end{matrix}]$ and $A=[\begin{matrix}2 & 7 & {\omega }^{2} \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega +1\end{matrix}]$ where $\omega =\frac{-1+i\sqrt{3}}{2}$, and ${I}_{3}$ be the identity matrix of order $3$. If the determinant of the matrix ${({P}^{-1}AP-{I}_{3})}^{2}$ is $\alpha {\omega }^{2}$, then the value of $\alpha$ is equal to _________.

Let $A$ be a $3\times 3$ real matrix. If $det(2Adj(2Adj(Adj(2A))))={2}^{41},$ then the value of $det({A}^{2})$ equals ______.

The domain of the function, $f(x)={\mathrm{sin}}^{-1}(\frac{3{x}^{2}+x-1}{(x-1{)}^{2}})+{\mathrm{cos}}^{-1}(\frac{x-1}{x+1})$ is:

The ratio of the coefficient of the middle term in the expansion of $(1+x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x{)}^{19}$ is .

The number of solutions of the equation ${\mathrm{log}}_{(x+1)}(2{x}^{2}+7x+5)+{\mathrm{log}}_{(2x+5)}(x+1{)}^{2}-4=0,x>0,$ is

If $A={x\in R:|x-2|>1},B={x\in R:\sqrt{{x}^{2}-3}>1},C={x\in R:|x-4|\geqslant 2}$ and $Z$ is the set of all integers, then the number of subsets of the set $(A\cap B\cap C{)}^{c}\cap Z$ is _________.

If for $x\in (0,\frac{\pi }{2}),{\mathrm{log}}_{10}\mathrm{sin}x+{\mathrm{log}}_{10}\mathrm{cos}x=-1$ and ${\mathrm{log}}_{10}(\mathrm{sin}x+\mathrm{cos}x)=\frac{1}{2}({\mathrm{log}}_{10}n-1),n>0$, then the value of $n$ is equal to :

The number of solutions of the equation ${32}^{{\mathrm{tan}}^{2}x}+{32}^{{\mathrm{sec}}^{2}x}=81,0\leq x\leq \frac{\pi }{4}$ is :

The set of all values of $k>-1$, for which the equation ${(3{x}^{2}+4x+3)}^{2}-(k+1)(3{x}^{2}+4x+3)(3{x}^{2}+4x+2)+$ $k{(3{x}^{2}+4x+2)}^{2}=0$ has real roots, is:

Let $\alpha =\underset{x\in R}{\mathrm{max}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}$ and $\beta =\underset{x\in R}{\mathrm{min}}{{8}^{2\mathrm{sin}3x}\cdot {4}^{4\mathrm{cos}3x}}.$ If $8{x}^{2}+bx+c=0$ is a quadratic equation whose roots are ${\alpha }^{1/5}$ and ${\beta }^{1/5},$ then the value of $c-b$ is equal to :

The sum of all integral values of $k(k\neq 0)$ for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is_____.

Let $\lambda \neq 0$ be in $R.$ If $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}-x+2\lambda =0,$ and $\alpha$ and $\gamma$ are the roots of the equation $3{x}^{2}-10x+27\lambda =0,$ then $\frac{\beta \gamma }{\lambda }$ is equal to ________.

Let $\alpha ,\beta$ be two roots of the equation ${x}^{2}+{(20)}^{1/4}x+{(5)}^{1/2}=0$. Then ${\alpha }^{8}+{\beta }^{8}$ is equal to

The probability of selecting integers $a\in [-5,30]$ such that ${x}^{2}+2(a+4)x-5a+64>0$, for all $x\in R$, is:

The number of real solutions of the equation, ${x}^{2}-|x|-12=0$ is:

If $\alpha ,\beta \in R$ are such that $1-2i$ (here ${i}^{2}=-1$) is a root of ${z}^{2}+\alpha z+\beta =0$, then $(\alpha -\beta )$ is equal to:

The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \ldots \infty }}}}$ is:

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha +\beta =1$ and $\alpha \beta =-1$. Let ${p}_{n}={(\alpha )}^{n}+{(\beta )}^{n}$, ${p}_{n-1}=11$ and ${p}_{n+1}=29$ for some integer $n\geqslant 1$. Then, the value of ${p}_{n}^{2}$ is______.

Let $\alpha$ and $\beta$ be the roots of ${x}^{2}-6x-2=0$. If ${a}_{n}={\alpha }^{n}-{\beta }^{n}$ for $n\geqslant 1$, then the value of $\frac{{a}_{10}-2{a}_{8}}{3{a}_{9}}$ is:

The integer $k,$ for which the inequality ${x}^{2}-2(3k-1)x+8{k}^{2}-7>0$ is valid for every $x$ in $R$ is:

If $S={z\in C:\frac{z-i}{z+2i}\in R}$, then

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that $\mathrm{arg}({z}_{1}-{z}_{2})=\frac{\pi }{4}$ and ${z}_{1},{z}_{2}$ satisfy the equation $|z-3|=Re(z)$. Then the imaginary part ${z}_{1}+{z}_{2}$ is equal to

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

A point $z$ moves in the complex plane such that $\mathrm{arg}(\frac{z-2}{z+2})=\frac{\pi }{4},$ then the minimum value of $|z-9\sqrt{2}-2i{|}^{2}$ is equal to

If $(\sqrt{3}+i{)}^{100}={2}^{99}(p+iq),$ then $p$ and $q$ are roots of the equation :

If the real part of the complex number $z=\frac{3+2i\mathrm{cos}\theta }{1-3i\mathrm{cos}\theta },\theta \in (0,\frac{\pi }{2})$ is zero, then the value of ${\mathrm{sin}}^{2}3\theta +{\mathrm{cos}}^{2}\theta$ is equal to ______.

The equation $arg(\frac{z-1}{z+1})=\frac{\pi }{4}$ represents a circle with:

Let $z=\frac{1-i\sqrt{3}}{2},i=\sqrt{-1}$. Then the value of $21+{(z+\frac{1}{z})}^{3}+{({z}^{2}+\frac{1}{{z}^{2}})}^{3}+{({z}^{3}+\frac{1}{{z}^{3}})}^{3}+\ldots +{({z}^{21}+\frac{1}{{z}^{21}})}^{3}$ is______.

The equation of a circle is $Re({z}^{2})+2{(\mathrm{Im}(z))}^{2}+2Re(z)=0,$ where $z=x+iy.$ A line which passes through the centre of the given circle and the vertex of the parabola, ${x}^{2}-6x-y+13=0,$ has $y$-intercept equal to _________.

Let $n$ denote the number of solutions of the equation ${z}^{2}+3\bar{z}=0,$ where $z$ is a complex number. Then the value of $\sum _{k=0}^{\infty }\frac{1}{{n}^{k}}$ is equal to

Let $\mathbb{C}$ be the set of all complex numbers. Let ${S}_{1}={z\in \mathbb{C}:|z-2|\leq 1}$ and ${S}_{2}={z\in \mathbb{C}:z(1+i)+\bar{z}(1-i)\geq 4}.$ Then, the maximum value of ${|z-\frac{5}{2}|}^{2}$ for $z\in {S}_{1}\cap {S}_{2}$ is equal to :

Let $C$ be the set of all complex numbers. Let ${S}_{1}={z\in C|{|z–3–2i|}^{2}=8},$ ${S}_{2}=z\in C|\mathrm{Re}(z)\geq 5$ and ${S}_{3}={z\in C||z–\bar{z}|\geq 8}.$ Then the number of elements in ${S}_{1}\cap {S}_{2}\cap {S}_{3}$ is equal to

If $z$ and $\omega$ are two complex numbers such that $|z\omega |=1$ and $\mathrm{arg}(z)-\mathrm{arg}(\omega )=\frac{3\pi }{2}$, then $\mathrm{arg}(\frac{1-2\bar{z}\omega }{1+3\bar{z}\omega })$ is: (Here $\mathrm{arg}(z)$ denotes the principal argument of complex number $z$)

Let ${S}_{1},{S}_{2}$ and ${S}_{3}$ be three sets defined as ${S}_{1}={z\in \mathbb{C}:|z-1|\leq \sqrt{2}},$ ${S}_{2}={z\in \mathbb{C}:Re((1-i)z)\geq 1}$ and ${S}_{3}={z\in \mathbb{C}:Im(z)\leq 1}.$ Then, the set ${S}_{1}\cap {S}_{2}\cap {S}_{3}$

The area of the triangle with vertices $P(z),Q(iz)$ and $R(z+iz)$ is

Let ${z}_{1},{z}_{2}$ be the roots of the equation ${z}^{2}+az+12=0$ and ${z}_{1},{z}_{2}$ form an equilateral triangle with origin. Then, the value of $|a|$ is

Let $z$ be those complex numbers which satisfy $|z+5|\leq 4$ and $z(1+i)+\bar{z}(1-i)\geqslant -10,i=\sqrt{-1}$. If the maximum value of $|z+1{|}^{2}$ is $\alpha +\beta \sqrt{2}$, then the value of $(\alpha +\beta )$ is

If the least and the largest real values of $\alpha ,$ for which the equation $z+\alpha |z-1|+2i=0$ $(z\in C\text{and}i=\sqrt{-1})$ has a solution, are $p$ and $q$ respectively; then $4({p}^{2}+{q}^{2})$ is equal to_______.

Let the lines $(2-i)z=(2+i)\bar{z}$ and $(2+i)z+(i-2)\bar{z}-4i=0,$ (here ${i}^{2}=-1$) be normal to a circle $C$. If the line $iz+\bar{z}+1+i=0$ is tangent to this circle $C$, then its radius is :

The least value of $|z|$ where $z$ is complex number which satisfies the inequality ${e}^{(\frac{(|z|+3)(|z|-1)}{||z|+1|}{\mathrm{log}}_{e}2)}\geq {\mathrm{log}}_{\sqrt{2}}|5\sqrt{7}+9i|$, $i=\sqrt{-1},$ is equal to :

All the arrangements, with or without meaning, of the word $\mathrm{FARMER}$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $\mathrm{FARMER}$ in this list is _____ .

A number is called a palindrome if it reads the same backward as well as forward. For example $285582$ is a six digit palindrome. The number of six digit palindromes, which are divisible by $55,$ is ________.

Let $S={1,2,3,4,5,6,9}$. Then the number of elements in the set $T={A\subseteq S:A\neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3}$ is

The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come together, is

The number of three-digit even numbers, formed by the digits $0,1,3,4,6,7$ if the repetition of digits is not allowed, is______.

The sum of all $3$-digit numbers less than or equal to $500,$ that are formed without using the digit $1$ and they all are multiple of $11,$ is ______.

There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100k,$ then $k$ is equal to

Let $n$ be a non-negative integer. Then the number of divisors of the form $4n+1$ of the number ${(10)}^{10}\cdot {(11)}^{11}\cdot {(13)}^{13}$ is equal to _____.

There are $15$ players in a cricket team, out of which $6$ are bowlers, $7$ are batsmen and $2$ are wicketkeepers. The number of ways, a team of $11$ players be selected from them so as to include at least $4$ bowlers, $5$ batsmen and $1$ wicketkeeper, is

The sum of all the elements in the set ${n\in {1,2,\ldots \ldots ,100}\mid$ H.C.F. of $n$ and $2040$ is $1$} is equal to __________.

If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8,$ then the number of all numbers greater than $10,000$ is equal to __________.

Consider a rectangle $ABCD$ having $5,6,7,9$ points in the interior of the line segments $AB,BC,CD,DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta -\alpha )$ is equal to

If the sides $AB,BC$ and $CA$ of a triangle $ABC$ have $3,5$ and $6$ interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

The sum of all the $4$-digit distinct numbers that can be formed with the digits $1,2,2$ and $3$ is:

The total number of $4$-digit numbers whose greatest common divisor with $18$ is $3$ is _____.

The total number of two digit numbers $'n'$, such that ${3}^{n}+{7}^{n}$ is a multiple of $10$ , is ___ .

A natural number has prime factorization given by $n={2}^{x}{3}^{y}{5}^{z}$, where $y$ and $z$ are such that $y+z=5$ and ${y}^{-1}+{z}^{-1}=\frac{5}{6},y>z$. Then the number of odd divisors of $n$, including $1$, is:

The students ${S}_{1},{S}_{2},\ldots ,{S}_{10}$ are to be divided into $3$ groups $A,B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is __________.

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is:

Three numbers are in an increasing geometric progression with common ratio $r.$ If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d.$ If the fourth term of GP is $3{r}^{2},$ then ${r}^{2}-d$ is equal to :

If for $x,y\in R,x>0,$ $y={\mathrm{log}}_{10}x+{\mathrm{log}}_{10}{x}^{1/3}+{\mathrm{log}}_{10}{x}^{1/9}+\ldots$upto $\infty$ terms and $\frac{2+4+6+\ldots +2y}{3+6+9+\ldots +3y}=\frac{4}{{\mathrm{log}}_{10}x},$ then the ordered pair $(x,y)$ is equal to

Let ${a}_{1},{a}_{2},\ldots ,{a}_{10}$ be an $A.P.$ with common difference $-3$ and ${b}_{1},{b}_{2},\ldots ,{b}_{10}$ be a $G.P.$ with common ratio $2.$ Let ${c}_{k}={a}_{k}+{b}_{k},k=1,2,\ldots ,10.$ If ${c}_{2}=12$ and ${c}_{3}=13,$ then $\sum _{k=1}^{10}{c}_{k}$ is equal to ______.

If the sum of an infinite $\mathrm{GP}$, $a,ar,a{r}^{2},a{r}^{3},\ldots$ is $15$ and the sum of the squares of its each term is $150,$ then the sum of $ar,2a{r}^{4},a{r}^{6},\ldots$ is:

Let ${{{a}_{n}}}_{n=1}^{\infty }$ be a sequence such that ${a}_{1}=1,{a}_{2}=1$ and ${a}_{n+2}=2{a}_{n+1}+{a}_{n}$ for all $n\geq 1$. Then the value of $47\sum _{n=1}^{\infty }(\frac{{a}_{n}}{{2}^{3n}})$ is equal to ________.

If $\mathrm{tan}(\frac{\pi }{9}),x,\mathrm{tan}(\frac{7\pi }{18})$ are in arithmetic progression and $\mathrm{tan}(\frac{\pi }{9}),y,\mathrm{tan}(\frac{5\pi }{18})$ are also in arithmetic progression, then $|x-2y|$ is equal to :

If $\alpha ,\beta$ are natural numbers such that ${100}^{\alpha }-199\beta =(100)(100)+(99)(101)+(98)(102)+\ldots ..+(1)(199),$ then the slope of the line passing through $(\alpha ,\beta )$ and origin is:

If $1,{\mathrm{log}}_{10}({4}^{x}-2)$ and ${\mathrm{log}}_{10}({4}^{x}+\frac{18}{5})$ are in arithmetic progression for a real number $x$ then the value of the determinant $|\begin{matrix}2(x-\frac{1}{2}) & x-1 & {x}^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{matrix}|$ is equal to:

$\frac{1}{{3}^{2}-1}+\frac{1}{{5}^{2}-1}+\frac{1}{{7}^{2}-1}+\ldots +\frac{1}{(201{)}^{2}-1}$ is equal to

In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is $25.$ Then, the sum of ${4}^{th},{6}^{th}$ and ${8}^{th}$ terms is equal to:

Let $\frac{1}{16},a$ and $b$ be in G.P. and $\frac{1}{a},\frac{1}{b},6$ be in A.P., where $a,b>0$. Then $72(a+b)$ is equal to _______ .

If the arithmetic mean and the geometric mean of the ${p}^{\mathrm{th}}$ and ${q}^{\mathrm{th}}$ terms of the sequence $-16,8,-4,2,\ldots$ satisfy the equation $4{x}^{2}-9x+5=0$, then $p+q$ is equal to _______.

If $0<\theta ,\phi <\frac{\pi }{2},x=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}\theta ,y=\sum _{n=0}^{\infty }{\mathrm{sin}}^{2n}\phi$ and $z=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}\theta \cdot {\mathrm{sin}}^{2n}\phi$ then :

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^{2}+bx+1=0,$ then the value of ${\alpha }^{2}+{\beta }^{2}-\alpha \beta$ is:

The sum of first four terms of a geometric progression $(G.P.)$ is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}.$ If the product of first three terms of the $G.P.$ is $1,$ and the third term is $\alpha ,$ then $2\alpha$ is _________.

Let ${A}_{1},{A}_{2},{A}_{3},\ldots ..$ be squares such that for each $n\geqslant 1,$ the length of the side of ${A}_{n}$ equals the length of diagonal of ${A}_{n+1}$. If the length of ${A}_{1}$ is $12\mathrm{cm}$, then the smallest value of $n$ for which area of ${A}_{n}$ is less than one, is

A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A,B$ and $C$ are placed at the points $(1,1),(2,2)$ and $(4,4)$ respectively. Then which of these stones is$/$are on the path of the man?