Algebra PYQ — Page 34

The sum to $20$ terms of the series $2\cdot {2}^{2}-{3}^{2}+2\cdot {4}^{2}-{5}^{2}+2\cdot {6}^{2}-............$ is equal to$__________.$

If $C32n:C3n=10:1,$ then the ratio $({n}^{2}+3n):({n}^{2}-3n+4)$ is

If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$ and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

If the domain of the function $f(x)={\mathrm{sec}}^{-1}(\frac{2x}{5x+3})$ is $[\alpha ,\beta )\cup (\gamma ,\delta ]$, then $|3\alpha +10(\beta +\gamma )+21\delta |$ is equal to __________

Let $5$ digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number $42923$ is _____ .

The number of permutations, of the digits $1,2,3,\ldots ,7$ without repetition, which neither contain the string $153$ nor the string $2467$, is _______ .

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is

If $\\alpha$ and $\\beta$ are roots of $x^2 - 5x + 6 = 0$, then $\\alpha + \\beta$ is

Let $S$ be the set of values of $\lambda$, for which the system of equations $6\lambda x-3y+3z=4{\lambda }^{2}$, $2x+6\lambda y+4z=1$ and $3x+2y+3\lambda z=\lambda$ has no solution. Then,$12\underset{\lambda \in S}{\sum }|\lambda |$ is equal to _______.

For the system of linear equations $ax+y+z=1$, $x+ay+z=1,x+y+az=\beta$, which one of the following statements is NOT correct?

Let $a,b,c$ be the three distinct positive real numbers such that ${(2a)}^{{\mathrm{log}}_{e}a}={(bc)}^{{\mathrm{log}}_{e}b}$ and ${b}^{{\mathrm{log}}_{e}2}={a}^{{\mathrm{log}}_{e}c}$ Then $6a+5bc$ is equal to ______.

Consider a function $f:\mathbb{N}\rightarrow \mathbb{R}$, satisfying $f(1)+2f(2)+3f(3)+\ldots +xf(x)=x(x+1)f(x);x\geq 2$ with $f(1)=1$. Then $\frac{1}{f(2022)}+\frac{1}{f(2028)}$ is equal to

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

Let $A={2,3,4}$ and $B={8,9,12}$. Then the number of elements in the relation $R={(({a}_{1},{b}_{1}),({a}_{2},{b}_{2}))\in (A\times B,A\times B):{a}_{1}\text{divides}{b}_{2}\text{and}{a}_{2}\text{divides}{b}_{1}}$ is

Let the number $(22{)}^{2022}+(2022{)}^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $({\alpha }^{2}+{\beta }^{2})$ is equal to

Let $z$ be a complex number such that $|\frac{z-2i}{z+i}|=2,z\neq -i$. Then $z$ lies on the circle of radius $2$ and centre

${50}^{\text{th }}$ 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 ________.

The sum, of the coefficients of the first $50$ terms in the binomial expansion of ${(1-x)}^{100},$ is equal to

Let ${a}_{1},{a}_{2},{a}_{3},....,{a}_{n}$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then $\underset{n\rightarrow \infty }{\mathrm{lim}}\sqrt{\frac{d}{n}}(\frac{1}{\sqrt{{a}_{1}}+\sqrt{{a}_{2}}}+\frac{1}{\sqrt{{a}_{2}}+\sqrt{{a}_{3}}}+\ldots +\frac{1}{\sqrt{{a}_{n-1}}+\sqrt{{a}_{n}}})$ is

The number of ways, in which $5$ girls and $7$ boys can be seated at a round table so that no two girls sit together is

The minimum number of elements that must be added to relation $R={(a,b),(b,c),(b,d)}$ on the set ${a,b,c,d}$, so that it is an equivalence relation is

Let $A$ be a symmetric matrix such that $|A|=2$ and $[\begin{matrix}2 & 1 \\ 3 & \frac{3}{2}\end{matrix}]A=[\begin{matrix}1 & 2 \\ \alpha & \beta \end{matrix}]$ If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{{\alpha }^{2}}$ is equal to _________.

If the solution of the equation ${\mathrm{log}}_{\mathrm{cos}x}(\mathrm{cot}x)+4{\mathrm{log}}_{\mathrm{sin}x}(\mathrm{tan}x)=1,x\in (0,\frac{\pi }{2})$ is ${\mathrm{sin}}^{-1}(\frac{\alpha +\sqrt{\beta }}{2})$, where $\alpha ,\beta$ are integers, then $\alpha +\beta$ is equal to:

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is