Algebra PYQ — Page 23

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______

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 :

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

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

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

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

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_________

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

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______

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

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

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:

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_________

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 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

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:

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 ________

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 ;

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