Algebra PYQ — Page 25

Let $f:R-{0,1}\rightarrow R$ be a function such that $f(x)+f(\frac{1}{1-x})=1+x$. Then $f(2)$ is equal to :

The number of integral solution $x$ of ${\mathrm{log}}_{(x+\frac{7}{2})}{(\frac{x-7}{2x-3})}^{2}\geq 0$ is

If $gcd(m,n)=1$ and ${1}^{2}-{2}^{2}+{3}^{2}-{4}^{2}+....+{(2021)}^{2}-{(2022)}^{2}+{(2023)}^{2}=1012{m}^{2}n$ then ${m}^{2}-{n}^{2}$ is equal to

The minimum number of elements that must be added to the relation $R=(a,b),(b,c)$ on the set ${a,b,c}$ so that it becomes symmetric and transitive is:

The equation ${e}^{4x}+8{e}^{3x}+13{e}^{2x}-8{e}^{x}+1=0,x\in R$ has :

Let the first term a and the common ratio $r$ of a geometric progression be positive integers. If the sum of squares of its first three terms is $33033$, then the sum of these three terms is equal to

For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations $x-y+z=5$ $2x+2y+\alpha z=8$ $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

Number of $4$-digit numbers (the repetition of digits is allowed) which are made using the digits $1,2,3$ and $5$ , and are divisible by $15$ , is equal to

If all the six digit numbers ${x}_{1}{x}_{2}{x}_{3}{x}_{4}{x}_{5}{x}_{6}$ with $0<{x}_{1}<{x}_{2}<{x}_{3}<{x}_{4}<{x}_{5}<{x}_{6}$ are arranged in the increasing order, then the sum of the digits in the ${72}^{\text{th }}$ number is _______.

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:

Number of $4$-digit numbers that are less than or equal to $2800$ and either divisible by $3$ or by $11$ , is equal to _____ .

If $f(x)=\frac{{2}^{2x}}{{2}^{2x}+2}$, $x\in R$, then $f(\frac{1}{2023})+f(\frac{2}{2023})+f(\frac{3}{2023}).........f(\frac{2022}{2023})$ is equal to

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

If the coefficient of ${x}^{7}$ in ${(ax-\frac{1}{b{x}^{2}})}^{13}$ and the coefficient of ${x}^{-5}$ in ${(ax+\frac{1}{b{x}^{2}})}^{13}$ are equal, then ${a}^{4}{b}^{4}$ is equal to:

If the coefficient of ${x}^{15}$ in the expansion of ${(a{x}^{3}+\frac{1}{b{x}^{\frac{1}{3}}})}^{15}$ is equal to the coefficient of ${x}^{-15}$ in the expansion of ${(a{x}^{\frac{1}{3}}-\frac{1}{b{x}^{3}})}^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:

For $a\in \mathbb{C}$, let $A={z\in \mathbb{C}:Re(a+\bar{z})>Im(\bar{a}+z)}$ and $B={z\in \mathbb{C}:Re(a+\bar{z})<Im(\bar{a}+z)}$. Then among the two statements: $(S1)$ : If $Re(a),Im(a)>0$, then the set $A$ contains all the real numbers $(S2)$ : If $Re(a),Im(a)<0$, then the set $B$ contains all the real numbers,

The number of integral solution $x$ of ${\mathrm{log}}_{(x+\frac{7}{2})}{(\frac{x-7}{2x-3})}^{2}\geq 0$ is

Let $\alpha ,\beta$ be the roots of the quadratic equation ${x}^{2}+\sqrt{6}x+3=0$. Then $\frac{{\alpha }^{23}+{\beta }^{23}+{\alpha }^{14}+{\beta }^{14}}{{\alpha }^{15}+{\beta }^{15}+{\alpha }^{10}+{\beta }^{10}}$ is equal to

The sum of all the roots of the equation $|{x}^{2}-8x+15|-2x+7=0$ is

If the value of real number $\alpha >0$ for which ${x}^{2}-5\alpha x+1=0$ and ${x}^{2}-\alpha x-5=0$ have a common real roots is $\frac{3}{\sqrt{2\beta }}$ then $\beta$ is equal to ________

The number of integral values of $k$, for which one root of the equation $2{x}^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$, is :

Let ${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$${\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{7}$ be the roots of the equation ${x}^{7}+3{x}^{5}-13{x}^{3}-15x=0$ and $|{\alpha }_{1}|\geq |{\alpha }_{2}|\geq \ldots \geq |{\alpha }_{7}|$. Then, ${\alpha }_{1}{\alpha }_{2}-{\alpha }_{3}{\alpha }_{4}+{\alpha }_{5}{\alpha }_{6}$ is equal to _______

Let $R={a,b,c,d,e}$ and $S={1,2,3,4}$. Total number of onto functions $f:R\rightarrow S$ such that $f(a)\neq 1$, is equal to ________.

Let $w=z\bar{z}+{k}_{1}z+{k}_{2}iz+\lambda (1+i),{k}_{1},{k}_{2}\in \mathbb{R}.$ . Let $Re(w)=0$ be the circle $C$ of radius $1$ in the first quadrant touching the line $y=1$ and the $y-$axis. If the curve $Im(w)=0$ intersects $C$ at $A$ and $B,$ then $30(AB{)}^{2}$ is equal to $_______.$