Algebra PYQ — Page 32

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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