JEE Main 2022 — Algebra

Let $S={x\in [-6,3]-{-2,2}:\frac{|x+3|-1}{|x|-2}\geq 0}$ and $T={x\in Z:{x}^{2}-7|x|+9\leq 0}$. Then the number of elements in $S\cap T$ is

If $1+(2+C149+C249+\ldots .+C4949)(C250+C450+\ldots ..+C5050)$ is equal to ${2}^{n}.m$, where $m$ is odd, then $n+m$ is equal to _____ .

The number of terms in the expansion of (1 + x)²⁰ + (1 - x)²⁰ is:

The value of ¹⁰C₀ + ¹⁰C₁ + ¹⁰C₂ + ... + ¹⁰C₁₀ is:

The number of real solutions of the equation ${e}^{4x}+4{e}^{3x}-58{e}^{2x}+4{e}^{x}+1=0$ is _____.

If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then

Let $f(x)$ and $g(x)$ be two real polynomials of degree $2$ and $1$ respectively. If $f(g(x))=8{x}^{2}-2x$, and $g(f(x))=4{x}^{2}+6x+1$, then the value of $f(2)+g(2)$ is ______.

Let ${R}_{1}={(a,b)\in N\times N:|a-b|\leq 13}$ and ${R}_{2}={(a,b)\in N\times N:|a-b|\neq 13}$ Then on $N$:

Let ${R}_{1}$ and ${R}_{2}$ be relations on the set ${1,2,\ldots ,50}$ such that ${R}_{1}=${$(p,{p}^{n}):p$ is a prime and $n\geq 0$ is an integer} and ${R}_{2}=${$(p,{p}^{n}):p$ is a prime and $n=0$ or $1$}. Then, the number of elements in ${R}_{1}-{R}_{2}$ is ____.

If $\sum _{k=1}^{10}\frac{k}{{k}^{4}+{k}^{2}+1}=\frac{m}{n}$, where $m$ and $n$ are co-prime, then $m+n$ is equal to

The domain of the function ${\mathrm{cos}}^{-1}(\frac{2{\mathrm{sin}}^{-1}(\frac{1}{4{x}^{2}-1})}{\pi })$ is

Let $A=(\begin{matrix}1+i & 1 \\ -i & 0\end{matrix})$ where $i=\sqrt{-1}$. Then, the number of elements in the set ${n\in {1,2,\ldots .,100}:{A}^{n}=A}$ is

If $p$ and $q$ are real number such that $p+q=3,{p}^{4}+{q}^{4}=369$, then the value of ${(\frac{1}{p}+\frac{1}{q})}^{-2}$ is equal to

Let $X=[\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{matrix}],Y=\alpha l+\beta X+\gamma {X}^{2}$ and $Z={\alpha }^{2}I-\alpha \beta X+({\beta }^{2}-\alpha \gamma ){X}^{2},\alpha ,\beta ,\gamma \in \mathbb{R}$. If ${Y}^{-1}=[\begin{matrix}\frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5}\end{matrix}]$, then ${(\alpha -\beta +\gamma )}^{2}$ is equal to ______.

Let $A={x\in R:|x+1|<2}$ and $B={x\in R:|x-1|\geq 2}$. Then which one the following statements is NOT true?

The probability that a randomly chosen one-one function from the set ${a,b,c,d}$ to the set ${1,2,3,4,5}$ satisfied $f(a)+2f(b)-f(c)=f(d)$ is

The ordered pair $(a,b)$, for which the system of linear equations $3x-2y+z=b$ $5x-8y+9z=3$ $2x+y+az=-1$ has no solution, is

Let $A=(\begin{matrix}4 & -2 \\ \alpha & \beta \end{matrix})$. If ${A}^{2}+\gamma A+18I=O$, then $det(A)$ is equal to _______.

Let a set $A={A}_{1}\cup {A}_{2}\cup \ldots \cup {A}_{k}$, where ${A}_{i}\cap {A}_{j}=\phi$ for $i\neq j;1\leq i,j\leq k.$ Define the relation $R$ from $A$ to $A$ by $R=${$(x,y):y\in {A}_{i}$ if and only if $x\in {A}_{i},1\leq i\leq k$}. Then, $R$ is:

The probability that a randomly chosen $2\times 2$ matrix with all the entries from the set of first $10$ primes, is singular, is equal to

The sum of all the elements of the set ${\alpha \in {1,2,\ldots ..100}:HCF(\alpha ,24)=1}$ is

Let ${R}_{1}$ and ${R}_{2}$ be two relations defined on $\mathbb{R}$ by $a{R}_{1}b\Leftrightarrow ab\geq 0$ and $a{R}_{2}b\Leftrightarrow a\geq b$, then

Let $A$ be a $3\times 3$ matrix having entries from the set ${-1,0,1}$. The number of all such matrices $A$ having sum of all the entries equal to $5$, is _____

Let $A$ and $B$ be two $3\times 3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$ then $|adj(Badj(2A))|$ is equal to

Let $S=${$\sqrt{n}:1\leqslant n\leqslant 50$ and $n$ is odd}. Let $a\in S$ and $A=[\begin{matrix}1 & 0 & a \\ -1 & 1 & 0 \\ -a & 0 & 1\end{matrix}]$. If $\underset{a\in S}{\Sigma }det(adjA)=100\lambda$, then $\lambda$ is equal to

The number of real values of $\lambda$, such that the system of linear equations $2x-3y+5z=9$ $x+3y-z=-18$ $3x-y+({\lambda }^{2}-|\lambda |)z=16$ has no solutions, is

Let $a,b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation ${x}^{2}-8ax+2a=0$ and $q$ and $s$ are the roots of the equation ${x}^{2}+12bx+6b=0$, such that $\frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{s}$ are in A.P., then ${a}^{-1}-{b}^{-1}$ is equal to _____ .

Let for the ${9}^{\mathrm{th}}$ term in the binomial expansion of ${(3+6x)}^{n}$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is ${n}_{0}$. If $k$ is the ratio of the coefficient of ${x}^{6}$ to the coefficient of ${x}^{3}$, then $k+{n}_{0}$ is equal to

Let the coefficients of the middle terms in the expansion of ${(\frac{1}{\sqrt{6}}+\beta x)}^{4},{(1-3\beta x)}^{2}$ and ${(1-\frac{\beta }{2}x)}^{6},\beta >0$, respectively form the first three terms of an A.P. If $d$ is the common difference of this A.P., then $50-\frac{2d}{{\beta }^{2}}$ is equal to _____ .

The remainder when ${(2021)}^{2022}+{(2022)}^{2021}$ is divided by $7$ is

If the maximum value of the term independent of $t$ in the expansion of ${({t}^{2}{x}^{\frac{1}{5}}+\frac{{(1-x)}^{{}^{\frac{1}{10}}}}{t})}^{15},x\geq 0$, is $K$, then $8K$ is equal to _____ .

Let the coefficients of ${x}^{-1}$ and ${x}^{-3}$ in the expansion of ${(2{x}^{\frac{1}{5}}-\frac{1}{{x}^{\frac{1}{5}}})}^{15},x>0$, be $m$and $n$ respectively. If $r$ is a positive integer such $m{n}^{2}=Cr.15{2}^{r}$, then the value of $r$ is equal to ______.

If the coefficients of $x$ and ${x}^{2}$ in the expansion of ${(1+x)}^{p}{(1-x)}^{q},p,q\leq 15$, are $-3$ and $-5$ respectively, then the coefficient of ${x}^{3}$ is equal to ______.

Let $n\geq 5$ be an integer. If ${9}^{n}-8n-1=64\alpha$ and ${6}^{n}-5n-1=25\beta$, then $\alpha -\beta$ is equal to:

If the constant term in the expansion of ${(3{x}^{3}-2{x}^{2}+\frac{5}{{x}^{5}})}^{10}$ is ${2}^{k}.l$, where $l$ is an odd integer, then the value of $k$ is equal to

The remainder when ${(2021)}^{2023}$ is divided by $7$ is

If $\frac{1}{2\cdot {3}^{10}}+\frac{1}{{2}^{2}\cdot {3}^{9}}+\ldots +\frac{1}{{2}^{10}\cdot 3}=\frac{K}{{2}^{10}\cdot {3}^{10}}$, then the remainder when $K$ is divided by $6$ is

The remainder when ${3}^{2022}$ is divided by $5$ is

The remainder on dividing $1+3+{3}^{2}+{3}^{3}+\ldots +{3}^{2021}$ by $50$ is _____.

If ${z}^{2}+z+1=0,z\in C$, then $|\sum _{n=1}^{15}{({z}^{n}+(-1{)}^{a}\frac{1}{{z}^{n}})}^{2}|$ is equal to _____.

If the minimum value of $f(x)=\frac{5{x}^{2}}{2}+\frac{\alpha }{{x}^{5}},x>0$, is $14$, then the value of $\alpha$ is equal to

The domain of the function $f(x)={\mathrm{sin}}^{-1}(\frac{{x}^{2}-3x+2}{{x}^{2}+2x+7})$ is

The remainder when ${(11)}^{1011}+{(1011)}^{11}$ is divided by $9$ is _____ .

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R=${$(x,y):3x+\alpha y$ is a multiple of $7$}. The relation $R$ is an equivalence relation if and only if

Let $A={1,2,3,4,5,6,7}$. Define $B=${$T\subseteq A$: either $1\notin T$ or $2\in T$} and $C=${$T\subseteq A:T$ the sum of all the elements of $T$ is a prime number.} Then the number of elements in the set $B\cup C$ is _______.

Let $A={1,2,3,4,5,6,7}$ and $B={3,6,7,9}$. Then the number of elements in the set ${C\subseteq A:C\cap B\neq \phi }$ is ______

Let $O$ be the origin and $A$ be the point ${z}_{1}=1+2i$. If $B$ is the point ${z}_{2},Re({z}_{2})<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true?

Sum of squares of modulus of all the complex numbers $z$ satisfying $\bar{z}=i{z}^{2}+{z}^{2}-z$ is equal to

The remainder when ${7}^{2022}+{3}^{2022}$ is divided by $5$ is

If the sum of the coefficients of all the positive powers of $x$, in the binomial expansion of ${({x}^{n}+\frac{2}{{x}^{5}})}^{7}$ is $939$, then the sum of all the possible integral values of $n$ is

Let $A=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}]$ and $B=[\begin{matrix}{9}^{2} & -{10}^{2} & {11}^{2} \\ {12}^{2} & {13}^{2} & -{14}^{2} \\ -{15}^{2} & {16}^{2} & {17}^{2}\end{matrix}]$, then the value of ${A}^{'}BA$ is;

For a natural number $n$, let ${\alpha }_{n}={19}^{n}-{12}^{n}$. Then, the value of $\frac{31{\alpha }_{9}-{\alpha }_{10}}{57{\alpha }_{8}}$ is ______

Let $R$ be a relation from the set ${1,2,3\ldots \ldots \ldots ,60}$ to itself such that $R=${$(a,b):b=pq$, where $p,q\geq 3$ are prime numbers}. Then, the number of elements in $R$ is

Consider a matrix $A=[\begin{matrix}\alpha & \beta & \gamma \\ {\alpha }^{2} & {\beta }^{2} & {\gamma }^{2} \\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{matrix}]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{det(adj(adj(\mathrm{adj}(adjA)))}{{(\alpha -\beta )}^{16}{(\beta -\gamma )}^{16}{(\gamma -\alpha )}^{16}}={2}^{32}\times {3}^{16}$, then the number of such $3$- tuples $(\alpha ,\beta ,\gamma )$ is _______.

The sum of the maximum and minimum values of the function $f(x)=|5x-7|+[{x}^{2}+2x]$ in the interval $[\frac{5}{4},2]$, where $[t]$ is the greatest integer $\leq t$, is ______.

The number of positive integers $k$ such that the constant term in the binomial expansion of ${(2{x}^{3}+\frac{3}{{x}^{k}})}^{12},x\neq 0$ is ${2}^{8}\cdot l$, where $l$ is an odd integer, is ______.

The number of matrices of order $3\times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is _______.

Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let $A$ and $B$ be two $3\times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

Let $A=[\begin{matrix}1 & -1 \\ 2 & \alpha \end{matrix}]$ and $B=[\begin{matrix}\beta & 1 \\ 1 & 0\end{matrix}],\alpha ,\beta \in R$. Let ${\alpha }_{1}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={A}^{2}+[\begin{matrix}2 & 2 \\ 2 & 2\end{matrix}]$ and ${\alpha }_{2}$ be the value of $\alpha$ which satisfies ${(A+B)}^{2}={B}^{2}$. Then $|{\alpha }_{1}-{\alpha }_{2}|$ is equal to

Let the matrix $A=[\begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{matrix}]$ and the matrix ${B}_{0}={A}^{49}+2{A}^{98}$. If ${B}_{n}=\mathrm{Adj}({B}_{n-1})$ for all $n\geq 1$, then $det({B}_{4})$ is equal to

Let $A=(\begin{matrix}1 & 2 \\ -2 & -5\end{matrix})$. Let $\alpha ,\beta \in \mathbb{R}$ be such that $\alpha {A}^{2}+\beta A=2I$. Then $\alpha +\beta$ is equal to

Let $A$ be a $2 \times 2$ matrix with $\operatorname{det}(A)=-1$ and $\operatorname{det}((A+I)(\operatorname{Adj}(A)+I))=4$. Then the sum of the diagonal elements of $A$ can be:

Let $A=[\begin{matrix}1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1\end{matrix}],a,b\in \mathbb{R}$. If for some $n\in N,{A}^{n}=[\begin{matrix}1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1\end{matrix}]$ then $n+a+b$ is equal to _______.

Let $A=(\begin{matrix}2 & -1 \\ 0 & 2\end{matrix})$. If $B=I-C15(adjA)+C25(adjA{)}^{2}-...-C55{(\mathrm{adj}A)}^{5}$, then the sum of all elements of the matrix $B$ is:

Let $A=[{a}_{ij}]$ be a square matrix of order $3$ such that ${a}_{ij}={2}^{j-i}$, for all $i,j=1,2,3$. Then, the matrix ${A}^{2}+{A}^{3}+\ldots +{A}^{10}$ is equal to

Let $A$ be a $3\times 3$ invertible matrix. If $|\mathrm{adj}(24A)|=$$adj(3adj(2A))|$, then ${|A|}^{2}$ is equal to

The positive value of the determinant of the matrix $A$, whose $Adj(Adj(A))=[\begin{matrix}14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14\end{matrix}]$, is ______.

Let $S={(\begin{matrix}-1 & a \\ 0 & b\end{matrix});a,b\in {1,2,3,\ldots 100}}$ and let ${T}_{n}={A\in S:{A}^{n(n+1)}=I}$. Then the number of elements in $\cap _{n=1}^{100}{T}_{n}$ is _____.

Let $A=[\begin{matrix}0 & -2 \\ 2 & 0\end{matrix}]$. If $M$ and $N$ are two matrices given by $M=\sum _{k=1}^{10}{A}^{2k}$ and $N=\sum _{k=1}^{10}{A}^{2k-1}$ then $M{N}^{2}$ is

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha +\beta$ is equal to

If the system of linear equations $2x+y-z=7$ $x-3y+2z=1$ $x+4y+\delta z=k$, where $\delta ,k\in R$ has infinitely many solutions, then $\delta +k$ is equal to

Let $p$ and $p+2$ be prime numbers and let $\Delta =|\begin{matrix}p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)!\end{matrix}|$ Then the sum of the maximum values of $\alpha$ and $\beta$, such that ${p}^{\alpha }$ and ${(p+2)}^{\beta }$ divide $\Delta$, is _______.

The number of $\theta \in (0,4\pi )$ for which the system of linear equations $3(\mathrm{sin}3\theta )x-y+z=2$ $3(\mathrm{cos}2\theta )x+4y+3z=3$ $6x+7y+7z=9$ has no solution is

If the system of linear equations. $8x+y+4z=-2$ $x+y+z=0$ $\lambda x-3y=\mu$ has infinitely many solutions, then the distance of the point $(\lambda ,\mu ,-\frac{1}{2})$ from the plane $8x+y+4z+2=0$ is:

Let $f(x)=|\begin{matrix}a & -1 & 0 \\ ax & a & -1 \\ a{x}^{2} & ax & a\end{matrix}|,a\in R$. Then the sum of the squares of all the values of a for $2{f}^{'}(10)-{f}^{'}(5)+100=0$ is

Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha ,-\alpha x+y+2z=-\alpha$ be inconsistent. Then $\alpha$ is equal to

The system of equations $-kx+3y-14z=25$ $-15x+4y-kz=3$ $-4x+y+3z=4$ Question: is consistent for all $k$ in the set

Let the system of linear equations $x+y+az=2$ $3x+y+z=4$ $x+2z=1$ have a unique solution $(x,y,z)$. If $(\alpha ,x),(y,\alpha )$ and $(x,-y)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

Let $A=(\begin{matrix}2 & -2 \\ 1 & -1\end{matrix})$ and$B=(\begin{matrix}-1 & 2 \\ -1 & 2\end{matrix})$ . Then the number of elements in the set {$(n,m):n,m\in {1,2,\ldots \ldots .10}$ and $n{A}^{n}+m{B}^{m}=I$} is _____.

Let $x=[\begin{matrix}1 \\ 1 \\ 1\end{matrix}]$ and $A=[\begin{matrix}-1 & 2 & 3 \\ 0 & 1 & 6 \\ 0 & 0 & -1\end{matrix}]$. For $k\in \mathbb{N}$, if ${X}^{'}{A}^{k}X=33$, then $k$ is equal to

Let $f(x)=a{x}^{2}+bx+c$ be such that $f(1)=3,f(-2)=\lambda$ and $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to

The domain of the function $f(x)={\mathrm{sin}}^{-1}[2{x}^{2}-3]+{\mathrm{log}}_{2}({\mathrm{log}}_{\frac{1}{2}}({x}^{2}-5x+5))$, where $[t]$ is the greatest integer function, is

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x)={\mathrm{cos}}^{-1}(\frac{{x}^{2}-4x+2}{{x}^{2}+3})$ is

The coefficient of ${x}^{101}$ in the expression ${(5+x)}^{500}+x{(5+x)}^{499}+{x}^{2}{(5+x)}^{498}+\ldots \ldots +{x}^{500},x>0$ is

The number of bijective function $f(1,3,5,7,\cdots ,99)\rightarrow (2,4,6,8,\cdots ,100)$ if $f(3)>f(5)>f(7)\cdots >f(99)$ is

The total number of functions, $f:{1,2,3,4}\rightarrow {1,2,3,4,5,6}$ such that $f(1)+f(2)=f(3)$, is equal to

Let $f:R\rightarrow R$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to:

Let a function $f:\mathbb{N}\rightarrow \mathbb{N}$ be defined by $f(n)=[\begin{matrix}2n, & n=2,4,6,8,\ldots .. \\ n-1, & n=3,7,11,15,\ldots .. \\ \frac{n+1}{2}, & n=1,5,9,13,\ldots ..\end{matrix}$ then, $f$ is

Let $c,k\in R$. If $f(x)=(c+1){x}^{2}+(1-{c}^{2})x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in R$, then the value of $|2(f(1)+f(2)+f(3)+\ldots \ldots +f(20))|$ is equal to ______.

Let $f:R\rightarrow R$ be a function defined $f(x)=\frac{2{e}^{2x}}{{e}^{2x}+e}$. Then $f(\frac{1}{100})+f(\frac{2}{100})+f(\frac{3}{100})+\ldots +f(\frac{99}{100})$ is equal to ______.

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

Let $f:R\rightarrow R$ be a function defined by $f(x)={(2(1-\frac{{x}^{25}}{2})(2+{x}^{25}))}^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is ______.

Let $f(x)=\frac{x-1}{x+1},x\in R-{0,-1,1)$ . If ${f}^{n+1}(x)=f({f}^{n}(x))$ for all $n\in N$, then ${f}^{6}(6)+{f}^{7}(7)$ is equal to

The domain of $f(x)=\frac{{\mathrm{cos}}^{-1}(\frac{{x}^{2}-5x+6}{{x}^{2}-9})}{\mathrm{log}({x}^{2}-3x+2)}$ is

The number of one-one functions $f:{a,b,c,d}\rightarrow {0,1,2,\ldots ,10}$ such that $2f(a)-f(b)+3f(c)+f(d)=0$ is _____

Let $S={1,2,3,4,5,6,7,8,9,10}$. Define $f:S\rightarrow S$ as $f(n)={\begin{matrix}2n, & \mathrm{if}n=1,2,3,4,5 \\ 2n-11 & \mathrm{if}n=6,7,8,9,10\end{matrix}$ Let $g:S\geq S$ be a function such that $\mathrm{fog}(n)={\begin{matrix}n+1 & ,\mathrm{if}n\mathrm{is}\mathrm{odd} \\ n-1 & ,\mathrm{if}n\text{is }\mathrm{even}\end{matrix}$, then $g(10)(g(1)+g(2)+g(3)+g(4)+g(5))$ is equal to

Let $A$ be a matrix of order $2\times 2$, whose entries are from the set ${0,1,2,3,4,5}$. If the sum of all the entries of $A$ is a prime number $p,2<p<8$, then the number of such matrices $A$ is

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

The number of functions $f$, from the set $A={x\in N:{x}^{2}-10x+9\leq 0}$ to the set $B={{n}^{2}:n\in N}$ such that $f(x)\leq {(x-3)}^{2}+1$, for every $x\in A$, is _______.

The sum of all real values of $x$ for which $\frac{3{x}^{2}-9x+17}{{x}^{2}+3x+10}=\frac{5{x}^{2}-7x+19}{3{x}^{2}+5x+12}$ is equal to

If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of ${(2{x}^{3}+\frac{3}{x})}^{10}$ is ${5}^{10}-\beta \cdot {3}^{9}$, then $\beta$ is equal to _____.

Consider the sequence ${a}_{1},{a}_{2},{a}_{3},\ldots \ldots$ such that ${a}_{1}=1,{a}_{2}=2$ and ${a}_{n+2}=\frac{2}{{a}_{n+1}}+{a}_{n}$ for $n=1,2,3,\ldots$ If $(\frac{{a}_{1}+\frac{1}{{a}_{2}}}{{a}_{3}})\cdot (\frac{{a}_{2}+\frac{1}{{a}_{3}}}{{a}_{4}})\cdot (\frac{{a}_{3}+\frac{1}{{a}_{4}}}{{a}_{5}})\ldots (\frac{{a}_{30}+\frac{1}{{a}_{31}}}{{a}_{32}})={2}^{\alpha }(C3161)$ then $\alpha$ is equal to

Let $A={x\in R:|x+1|<2}$ and $B={x\in R:|x-1|\geq 2}$. Then which one the following statements is NOT true?

Let $M=[\begin{matrix}0 & -\alpha \\ \alpha & 0\end{matrix}]$, where $\alpha$ is a non-zero real number and $N=\sum _{k=1}^{49}{M}^{2k}$. If $(I-{M}^{2})N=-2I$, then the positive integral value of $\alpha$ is ______.

If the system of linear equations $2x-3y=\gamma +5$ $\alpha x+5y=\beta +1$, where $\alpha ,\beta ,\gamma \in R$ has infinitely many solutions, then the value of $|9\alpha +3\beta +5\gamma |$ is equal to

If the system of equations $\alpha x+y+z=5,x+2y+3z=4,x+3y+5z=\beta$. Has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to

If $x=\sum _{n=0}^{\infty }{a}^{n},y=\sum _{n=0}^{\infty }{b}^{n},z=\sum _{n=0}^{\infty }{c}^{n}$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1$, $abc\neq 0$, then

The total number of $5$-digit numbers, formed by using the digits $1,2,3,5,6,7$ without repetition, which are multiple of $6$, is

The term independent of $x$ in the expression of $(1-{x}^{2}+3{x}^{3}){(\frac{5}{2}{x}^{3}-\frac{1}{5{x}^{2}})}^{11},x\neq 0$ is

Let $A$ be a $3\times 3$ real matrix such that $A(\begin{matrix}1 \\ 1 \\ 0\end{matrix})=(\begin{matrix}1 \\ 1 \\ 0\end{matrix});A(\begin{matrix}1 \\ 0 \\ 1\end{matrix})=(\begin{matrix}-1 \\ 0 \\ 1\end{matrix})$ and $A(\begin{matrix}0 \\ 0 \\ 1\end{matrix})=(\begin{matrix}1 \\ 1 \\ 2\end{matrix})$. If $X={[{x}_{1}{x}_{2}{x}_{3}]}^{T}$ and $I$ is an identity matrix of order $3$, then the system $(A-2I)X=(\begin{matrix}4 \\ 1 \\ 1\end{matrix})$ has

The area of the polygon, whose vertices are the non-real roots of the equation $\bar{z}=i{z}^{2}$ is

Let $f:N\rightarrow R$ be a function such that $f(x+y)=2f(x)f(y)$ for natural numbers $x$ and $y$. If $f(1)=2$, then the value of $\alpha$ for which $\sum _{k=1}^{10}f(\alpha +k)=\frac{512}{3}({2}^{20}-1)$ holds, is

Suppose ${a}_{1},{a}_{2},\ldots ,{a}_{n},\ldots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms to the sum of first nine terms of the progression is $5:17$ and $110<{a}_{15}<120$, then the sum of the first ten terms of the progression is equal to

Let $A={n\in N:H.C.F.(n,45)=1}$ and let $B={2k:k\in {1,2,\ldots ,100}}$. Then the sum of all the elements of $A\cap B$ is _____.

If ${a}_{1}(>0),{a}_{2},{a}_{3},{a}_{4},{a}_{5}$ are in a G.P. , ${a}_{2}+{a}_{4}=2{a}_{3}+1$ and $3{a}_{2}+{a}_{3}=2{a}_{4}$, then ${a}_{2}+{a}_{4}+2{a}_{5}$ is equal to _____.

Let $f,g:\mathbb{N}-{1}\rightarrow \mathbb{N}$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that ${p}^{\alpha }$ divides $a$, and $g(a)=a+1$, for all $a\in \mathbb{N}-{1}$. Then, the function $f+g$ is

The number of $3$-digit odd numbers, whose sum of digits is a multiple of $7$, is _____.

Let $A={1,{a}_{1},{a}_{2}\ldots \ldots {a}_{18},77}$ be a set of integers with $1<{a}_{1}<{a}_{2}<\ldots ..<{a}_{18}<77$. Let the set $A+A={x+y:x,y\in A}$ contain exactly $39$ elements. Then, the value of ${a}_{1}+{a}_{2}+\ldots ..+{a}_{18}$ is equal to ______.

Let $A$ be a matrix of order $3\times 3$ and $\mathrm{det}(A)=2$. Then $\mathrm{det}(det(A)adj(5adj({A}^{3}))$ is equal to _____.

Let $3,6,9,12,\ldots$ upto $78$ terms and $5,9,13,17,\ldots$ upto $59$ terms be two series. Then, the sum of the terms common to both the series is equal to ______.

Let $S={z\in \mathbb{C}:{z}^{2}+\bar{z}=0}$. Then $\underset{z\in S}{\sum }(Re(z)+Im(z))$ is equal to _______.

Let the sum of an infinite $G.P.$, whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $\mathrm{AP}$, whose first term is $10ar,{n}^{\mathrm{th}}$ term is ${a}_{n}$ and the common difference is $10{ar}^{2}$, is equal to

The value of $i^{100}$ where $i = \\sqrt{-1}$ is

Let $\alpha$ be a root of the equation $1+{x}^{2}+{x}^{4}=0$. Then the value of ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}$ is equal to:

A class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$, then $b+3g$ is equal to

The number of matrices $A=[\begin{matrix}a & b \\ c & d\end{matrix}]$, where $a,b,c,d\in {-1,0,1,2,3,\ldots \ldots ,10}$, such that $A={A}^{-1}$, is ______.

If the system of linear equations $2x+3y-z=-2$ $x+y+z=4$ $x-y+|\lambda |z=4\lambda -4$ where $\lambda \in \mathbb{R}$, has no solution, then

The total number of $3$-digit numbers, whose greatest common divisor with $36$ is $2$, is ______.

If ${a}_{1},{a}_{2},{a}_{3}\ldots$ and ${b}_{1},{b}_{2},{b}_{3}\ldots .$ are A.P. and ${a}_{1}=2,{a}_{10}=3,{a}_{1}{b}_{1}=1={a}_{10}{b}_{10}$ then ${a}_{4}{b}_{4}$ is equal to

If the coefficient of ${x}^{10}$ in the binomial expansion of ${(\frac{\sqrt{x}}{{5}^{\frac{1}{4}}}+\frac{\sqrt{5}}{{x}^{\frac{1}{3}}})}^{60}$ is ${5}^{k}l$, where $l,k\in N$ and $l$ is coprime to $5$, then $k$ is equal to ______.

Which of the following matrices can NOT be obtained from the matrix $[\begin{matrix}-1 & 2 \\ 1 & -1\end{matrix}]$ by a single elementary row operation?

Let $A=[\begin{matrix}2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0\end{matrix}]$ and $B=A-I$. If $\omega =\frac{\sqrt{3}i-1}{2}$, then the number of elements in the set ${n\in {1,2,\ldots ,100}:{A}^{n}+{(\omega B)}^{n}=A+B}$ is equal to _____ .

Let $S={1,2,3,4}$. Then the number of elements in the set {$f:S\times S\rightarrow S:f$ is onto and $f(a,b)=f(b,a)$ $\geq a\forall (a,b)\in S\times S$} is

Let ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ be a $4$-element permutation with ${b}_{i}\in$ ${1,2,3,\ldots \ldots \ldots ,100}$ for $1\leq i\leq 4$ and ${b}_{i}\neq {b}_{j}$ for $i\neq j$, such that either ${b}_{1},{b}_{2},{b}_{3}$ are consecutive integers or ${b}_{2},{b}_{3},{b}_{4}$ are consecutive integers. Then the number of such permutations ${b}_{1}{b}_{2}{b}_{3}{b}_{4}$ is equal to ______.

Let $S$ be the set containing all $3\times 3$ matrices with entries from ${-1,0,1}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of ${A}^{T}A$ is $6$ is ______.

The number of values of $\alpha$ for which the system of equations $x+y+z=\alpha$ $\alpha x+2\alpha y+3z=-1$ $x+3\alpha y+5z=4$ is inconsistent, is

Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f(x)=\alpha {x}^{5}+\beta {x}^{3}+\gamma x,x\in R$ and $g:R\rightarrow R$ be such that $g(f(x))=x$ for all $x\in R$. If ${a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{n}$ be in arithmetic progression with mean zero, then the value of $f(g(\frac{1}{n}\sum _{i=1}^{n}f({a}_{i})))$ is equal to

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^{n}$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning is $\frac{\alpha }{\sqrt[4]{3}}$, then $\alpha$ is equal to _______.

Let $\alpha$ and $\beta$ be the roots of the equation ${x}^{2}+(2i-1)=0$. Then, the value of $|{\alpha }^{8}+{\beta }^{8}|$ is equal to

The number of elements in the set {$z=a+ib\in \mathbb{C}:a,b\in \mathbb{Z}$ and $1<|z-3+2i|<4$} is _____.

Let $S={x\in [-6,3]-{-2,2}:\frac{|x+3|-1}{|x|-2}\geq 0}$ and $T={x\in Z:{x}^{2}-7|x|+9\leq 0}$. Then the number of elements in $S\cap T$ is

If $\alpha ,\beta$ are the roots of the equation ${x}^{2}-(5+{3}^{\sqrt{{\mathrm{log}}_{3}5}}-{5}^{\sqrt{{\mathrm{log}}_{5}3}})x+3({3}^{{({\mathrm{log}}_{3}5)}^{\frac{1}{3}}}-{5}^{{({\mathrm{log}}_{5}3)}^{\frac{2}{3}}}-1)=0$ then the equation, whose roots are $\alpha +\frac{1}{\beta }$ and $\beta +\frac{1}{\alpha }$,

Let $\alpha ,\beta (\alpha >\beta )$ be the roots of the quadratic equation ${x}^{2}-x-4=0$. If ${P}_{n}={\alpha }^{n}-{\beta }^{n},n\in \mathbb{N}$, then $\frac{{P}_{15}{P}_{16}-{P}_{14}{P}_{16}-{P}_{15}^{2}+{P}_{14}{P}_{15}}{{P}_{13}{P}_{14}}$ is equal to _____.

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-\sqrt{2}x+\sqrt{6}=0$ and $\frac{1}{{\alpha }^{2}}+1,\frac{1}{{\beta }^{2}}+1$ be the roots of the equation ${x}^{2}+ax+b=0$. Then the roots of the equation ${x}^{2}-(a+b-2)x+(a+b+2)=0$ are :

The minimum value of the sum of the squares of the roots of ${x}^{2}+(3-a)x=2a-1$ is

Let $f(x)$ be a quadratic polynomial with leading coefficient $1$ such that $f(0)=p,p\neq 0$, and $f(1)=\frac{1}{3}$. If the equations $f(x)=0$ and $fofofof(x)=0$ have a common real root, then $f(-3)$ is equal to ______.

If for some $p,q,r\in R$, all have positive sign, one of the roots of the equation $({p}^{2}+{q}^{2}){x}^{2}-2q(p+r)x+{q}^{2}+{r}^{2}=0$ is also a root of the equation ${x}^{2}+2x-8=0$, then $\frac{{q}^{2}+{r}^{2}}{{p}^{2}}$ is equal to-

Let $f(x)$ be a quadratic polynomial such that $f(-2)$ $+f(3)=0$. If one of the roots of $f(x)=0$ is $-1$, then the sum of the roots of $f(x)=0$ is equal to

Let $\alpha ,\beta$ be the roots of the equation ${x}^{2}-4\lambda x+5=0$ and $\alpha ,\gamma$ be the roots of the equation ${x}^{2}-(3\sqrt{2}+2\sqrt{3})x+7+3\lambda \sqrt{3}=0$. If $\beta +\gamma =3\sqrt{2}$, then ${(\alpha +2\beta +\gamma )}^{2}$ is equal to

The sum of all real roots of equation $({e}^{2x}-4)(6{e}^{2x}-5{e}^{x}+1)=0$ is

The sum of the cubes of all the roots of the equation ${x}^{4}-3{x}^{3}-2{x}^{2}+3x+1=0$ is _____.

Let $a,b\in R$ be such that the equation $a{x}^{2}-2bx+15=0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation ${x}^{2}-2bx+21=0$, then ${\alpha }^{2}+{\beta }^{2}$ is equal to:

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3{x}^{2}+\lambda x-1=0$ is $15$, then $6{({\alpha }^{3}+{\beta }^{3})}^{2}$ is equal to

If $z=2+3i$, then ${z}^{5}+{(\bar{z})}^{5}$ is equal to:

Let $S={z=x+iy:|z-1+i|\geq |z|,|z|<2,|z+i|=|z-1|}$. Then the set of all values of $x$, for which $w=2x+iy\in S$ for some $y\in \mathbb{R}$, is

Let the minimum value ${v}_{0}$ of $v={|z|}^{2}+{|z-3|}^{2}+{|z-6i|}^{2}$, $z\in \mathbb{C}$ is attained at $z={z}_{0}$. Then ${|2{z}_{0}^{2}-{\bar{z}}_{0}^{3}+3|}^{2}+{v}_{0}^{2}$ is equal to

Let $S$ be the set of all $(\alpha ,\beta ),\pi <\alpha ,\beta <2\pi$, for which the complex number $\frac{1-i\mathrm{sin}\alpha }{1+2i\mathrm{sin}\alpha }$ is purely imaginary and $\frac{1+i\mathrm{cos}\beta }{1-2i\mathrm{cos}\beta }$ is purely real. Let ${Z}_{\alpha \beta }=\mathrm{sin}2\alpha +i\mathrm{cos}2\beta ,(\alpha ,\beta )\in S$. Then $\underset{(\alpha ,\beta )\in S}{\sum }(i{Z}_{\alpha \beta }+\frac{1}{i{\bar{Z}}_{\alpha \beta }})$ is equal to

Let ${S}_{1}={{z}_{1}\in C:|{z}_{1}-3|=\frac{1}{2}}$ and ${S}_{2}={{z}_{2}\in C:|{z}_{2}-|{z}_{2}+1||=|{z}_{2}+|{z}_{2}-1||}.$ Then, for ${z}_{1}\in {S}_{1}$ and ${z}_{2}\in {S}_{2}$, the least value of $|{z}_{2}-{z}_{1}|$ is

Let $z=a+ib,b\neq 0$ be complex numbers satisfying ${z}^{2}=\bar{z}\cdot {2}^{1-|z|}$. Then the least value of $n\in N$, such that ${z}^{n}={(z+1)}^{n}$, is equal to _____ .

For $z\in \mathbb{C}$ if the minimum value of $(|z-3\sqrt{2}|+|z-p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is _______.

For $n\in N$, let ${S}_{n}={z\in C:|z-3+2i|=\frac{n}{4}}$ and ${T}_{n}={z\in C:|z-2+3i|=\frac{1}{n}}$. Then the number of elements in the set ${n\in N:{S}_{n}\cap {T}_{n}=\phi }$ is

If $\alpha ,\beta ,\gamma ,\delta$ are the roots of the equation ${x}^{4}+{x}^{3}+{x}^{2}+x+1=0$, then ${\alpha }^{2021}+{\beta }^{2021}+{\gamma }^{2021}+{\delta }^{2021}$ is equal to

Let $S={z\in C:|z-2|\leq 1,z(1+i)+\bar{z}(1-i)\leq 2}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at ${z}_{1}\in S$ and ${z}_{2}\in S$. If $5({|{z}_{1}|}^{2}+{|{z}_{2}|}^{2})=\alpha +\beta \sqrt{5}$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha +\beta$ is equal to ______.

Let $(z)$ represent the principal argument of the complex number $z$. The, $|z|=3$ and $\mathrm{arg}(z-1)-\mathrm{arg}(z+1)=\frac{\pi }{4}$ intersect:

The number of points of intersection $|z-(4+3i)|=2|$ and $|z|+|z-4|=6,z\in C$ is

Let $A={z\in C:|\frac{z+1}{z-1}|<1}$ and $B={z\in C:\mathrm{arg}(\frac{z-1}{z+1})=\frac{2\pi }{3}}$. Then $A\cap B$ is

Let for some real numbers $\alpha$ and $\beta ,a=\alpha -i\beta$. If the system of equations $4ix+(1+i)y=0$ and $8(\mathrm{cos}\frac{2\pi }{3}+i\mathrm{sin}\frac{2\pi }{3})x+\bar{a}y=0$ has more than one solution then $\frac{\alpha }{\beta }$ is equal to

Let $A={z\in C:1\leqslant |z-(1+i)|\leqslant 2}$ and $B={z\in A:|z-(1-i)|=1}$. Then, $B$

Let $S=${$z\in \mathbb{C}:|z-3|\leq 1$ and$z(4+3i)+\bar{z}(4-3i)\leq 24$}. If $\alpha +i\beta$ is the point in $S$ which is closest to $4i$, then $25(\alpha +\beta )$ is equal to ______.

Let ${z}_{1}$ and ${z}_{2}$ be two complex numbers such that ${\bar{z}}_{1}=i{\bar{z}}_{2}$ and $\mathrm{arg}(\frac{{z}_{1}}{{\bar{z}}_{2}})=\pi$, then the argument of ${z}_{1}$ is

Let a circle $C$ in complex plane pass through the points ${z}_{1}=3+4i,{z}_{2}=4+3i$ and ${z}_{3}=5i$. If $z(\neq {z}_{1})$ is a point on $C$ such that the line through $z$ and ${z}_{1}$ is perpendicular to the line through ${z}_{2}$ and ${z}_{3}$, then $\mathrm{arg}(z)$ is equal to

Let $S={4,6,9}$ and $T={9,10,11,\ldots ,1000}$. If $A={{a}_{1}+{a}_{2}+\ldots +{a}_{k}:k\in N,{a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{k}\in S}$ then the sum of all the elements in the set $T-A$ is equal to _______.

The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by $55$ is _____.

Numbers are to be formed between $1000$ and $3000$, which are divisible by $4$, using the digits $1,2,3,4,5$ and $6$ without repetition of digits. Then the total number of such numbers is _______.

Let $S$ be the set of all passwords which are six to eight characters long, where each character is either an alphabet from ${A,B,C,D,E}$ or a number from ${1,2,3,4,5}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from ${1,2,3,4,5}$ is $\alpha \times {5}^{6}$, then $\alpha$ is equal to

The number of $5$-digit natural numbers, such that the product of their digits is $36$, is

The number of ways, $16$ identical cubes, of which $11$ are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least $2$ blue cubes, is ______.

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

The number of ways to distribute $30$ identical candies among four children ${C}_{1},{C}_{2},{C}_{3}$ and ${C}_{4}$ so that ${C}_{2}$ receives atleast $4$ and atmost $7$ candies, ${C}_{3}$ receives atleast $2$ and atmost $6$ candies, is equal to

There are ten boys ${B}_{1},{B}_{2},\ldots .,{B}_{10}$ and five girls ${G}_{1},{G}_{2},\ldots .{G}_{5}$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both ${B}_{1}$ and ${B}_{2}$ together should not be the members of a group, is _____.

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

In an examination, there are $5$ multiple choice questions with $3$ choices, out of which exactly one is correct. There are $3$ marks for each correct answer, $-2$ marks for each wrong answer and $0$ mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets $5$ marks is _____

The number of $7$-digit numbers which are multiples of $11$ and are formed using all the digits $1,2,3,4,5,7$ and $9$ is _____.

If $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\ldots +$ $\frac{1}{100\times 101\times 102}=\frac{k}{101}$, then $34k$ is equal to _______.

Let ${{{a}_{n}}}_{n=0}^{\infty }$ be a sequence such that ${a}_{0}={a}_{1}=0$ and ${a}_{n+2}=3{a}_{n+1}-2{a}_{n}+1,\forall n\geq 0$. Then ${a}_{25}{a}_{23}-2{a}_{25}{a}_{22}-2{a}_{23}{a}_{24}+4{a}_{22}{a}_{24}$ is equal to

If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\ldots \ldots +$ $\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of $a$ is

Let ${x}_{1},{x}_{2},{x}_{3},\ldots ..,{x}_{20}$ be in geometric progression with ${x}_{1}=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each ${x}_{i}$ by ${({x}_{i}-i)}^{2}$. If $x$ is the mean of new data, then the greatest integer less than or equal to $x$ is

For $p,q\in R$, consider the real valued function $f(x)={(x-p)}^{2}-q,x\in R$ and $q>0$. Let ${a}_{1},{a}_{2},{a}_{3}$ and ${a}_{4}$ be in an arithmetic progression with mean $p$ and positive common difference. If $|f({a}_{i})|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is

Different A.P.'s are constructed with the first term $100$, the last term $199$, And integral common differences. The sum of the common differences of all such, A.P's having at least $3$ terms and at most $33$ terms is.

The series of positive multiples of $3$ is divided into sets : ${3},{6,9,12},{15,18,21,24,27},\ldots$ Then the sum of the elements in the ${11}^{\mathrm{th}}$ set is equal to _______.

Let $f(x)=2{x}^{2}-x-1$ and $S={n\in \mathbb{Z}:|f(n)|\leq 800}$. Then, the value of $\underset{n\in S}{\sum }f(n)$ is equal to _______.

The sum $\sum _{n=1}^{21}\frac{3}{(4n-1)(4n+3)}$ is equal to

Let ${a}_{1}={b}_{1}=1,{a}_{n}={a}_{n-1}+2$ and ${b}_{n}={a}_{n}+{b}_{n-1}$ for every natural number $n\geq 2$. Then $\sum _{n=1}^{15}{a}_{n}\cdot {b}_{n}$ is equal to _____ .

Let ${{{a}_{n}}}_{n=0}^{\infty }$ be a sequence such that ${a}_{0}={a}_{1}=0$ and ${a}_{n+2}=2{a}_{n+1}-{a}_{n}+1$ for all $n\geq 0$. Then, $\sum _{n=2}^{\infty }\frac{{a}_{n}}{{7}^{n}}$ is equal to

Consider two G.Ps. $2,{2}^{2},{2}^{3},\ldots$ and $4,{4}^{2},{4}^{3},\ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is ${(2)}^{\frac{225}{8}}$, then $\sum _{k=1}^{n}k(n-k)$ is equal to:

Let for $n=1,2,\ldots \ldots ,50,{S}_{n}$ be the sum of the infinite geometric progression whose first term is ${n}^{2}$ and whose common ratio is $\frac{1}{{(n+1)}^{2}}$. Then the value of $\frac{1}{26}+\sum _{n=1}^{50}({S}_{n}+\frac{2}{n+1}-n-1)$ is equal to

If $n$ arithmetic means are inserted between a and $100$ such that the ratio of the first mean to the last mean is $1:7$ and $a+n=33$, then the value of $n$ is

Let${A}_{1},{A}_{2},{A}_{3},\ldots \ldots$ be an increasing geometric progression of positive real numbers. If ${A}_{1}{A}_{3}{A}_{5}{A}_{7}=\frac{1}{1296}$ and ${A}_{2}+{A}_{4}=\frac{7}{36}$, then, the value of ${A}_{6}+{A}_{8}+{A}_{10}$ is equal to

If ${{{a}_{i}}}_{i=1}^{n}$, where $n$ is an even integer, is an arithmetic progression with common difference $1$, and $\sum _{i=1}^{n}{a}_{i}=192,\sum _{i=1}^{\frac{n}{2}}{a}_{2i}=120$, then $n$ is equal to

The greatest integer less than or equal to the sum of first $100$ terms of the sequence $\frac{1}{3},\frac{5}{9},\frac{19}{27},\frac{65}{81},\ldots$ is equal to ______

Let $x,y>0$. If ${x}^{3}{y}^{2}={2}^{15}$, then the least value of $3x+2y$ is

If $A=\sum _{n=1}^{\infty }\frac{1}{{(3+{(-1)}^{n})}^{n}}$ and $B=\sum _{n=1}^{\infty }\frac{{(-1)}^{n}}{{(3+{(-1)}^{n})}^{n}}$, then $\frac{A}{B}$ is equal to