Algebra PYQ — Page 48

If $Prn=Pr+1n$ and $Crn=Cr-1n,$ then the value of $r$ is equal to:

If the functions are defined as $f(x)=\sqrt{x}$ and $g(x)=\sqrt{1-x},$ then what is the common domain of the following functions: $f+g,f-g,f/g,g/f,g-f$, where $(f\pm g)(x)=f(x)\pm g(x),(f/g)(x)=\frac{f(x)}{g(x)}$

Let $f(x)={\mathrm{sin}}^{-1}x$ and $g(x)=\frac{{x}^{2}-x-2}{2{x}^{2}-x-6}$. If $g(2)=\underset{x\rightarrow 2}{\mathrm{lim}}g(x)$, then the domain of the function $fog$ is

Let $f:R-{3}\rightarrow R-{1}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:R\rightarrow R$ be given as $g(x)=2x-3$. Then, the sum of all the values of $x$ for which ${f}^{-1}(x)+{g}^{-1}(x)=\frac{13}{2}$ is equal to

Three numbers are in an increasing geometric progression with common ratio $r.$ If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $d.$ If the fourth term of GP is $3{r}^{2},$ then ${r}^{2}-d$ is equal to :

Let $\alpha ,\beta ,\gamma$ be the real roots of the equation, ${x}^{3}+a{x}^{2}+bx+c=0,$ ($a,b,c\in R$ and $a,b\neq 0$). If the system of equations (in, $u,v,w$) given by $\alpha u+\beta v+\gamma w=0,\beta u+\gamma v+\alpha w=0,\gamma u+\alpha v+\beta w=0$ has non-trivial solution, then the value of $\frac{{a}^{2}}{b}$ is

Let ${a}_{1},{a}_{2},\ldots ,{a}_{10}$ be an $A.P.$ with common difference $-3$ and ${b}_{1},{b}_{2},\ldots ,{b}_{10}$ be a $G.P.$ with common ratio $2.$ Let ${c}_{k}={a}_{k}+{b}_{k},k=1,2,\ldots ,10.$ If ${c}_{2}=12$ and ${c}_{3}=13,$ then $\sum _{k=1}^{10}{c}_{k}$ is equal to ______.

Let $f:N\rightarrow N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in N.$ If $f(6)=18$ then $f(2)\cdot f(3)$ is equal to :

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity$\frac{1}{a-b}+\frac{1}{a-2b}+\frac{1}{a-3b}+\ldots .+\frac{1}{a-nb}=\alpha n+\beta {n}^{2}+\gamma {n}^{3}$ then the value of $\gamma$ is :

The total number of two digit numbers $'n'$, such that ${3}^{n}+{7}^{n}$ is a multiple of $10$ , is ___ .

A number is called a palindrome if it reads the same backward as well as forward. For example $285582$ is a six digit palindrome. The number of six digit palindromes, which are divisible by $55,$ is ________.

Let $R={(P,Q)|P$ and $Q$ are at the same distance from the origin$}$ be a relation, then the equivalence class of $(1,-1)$ is the set

The maximum value of the term independent of $t$ in the expansion of ${(t{x}^{\frac{1}{5}}+\frac{{(1-x)}^{\frac{1}{10}}}{t})}^{10}$where $x\in (0,1)$ is:

Let $\theta \in (0,\frac{\pi }{2})$. If the system of linear equations $(1+{\mathrm{cos}}^{2}⁡\theta )x+{\mathrm{sin}}^{2}⁡\theta y+4\mathrm{sin}⁡3\theta z=0$ ${\mathrm{cos}}^{2}⁡\theta x+(1+{\mathrm{sin}}^{2}⁡\theta )y+4\mathrm{sin}⁡3\theta z=0$ ${\mathrm{cos}}^{2}⁡\theta x+{\mathrm{sin}}^{2}⁡\theta y+(1+4\mathrm{sin}⁡3\theta )z=0$ has a non-trivial solution, then the value of $\theta$ is:

If $P=[\begin{matrix}1 & 0 \\ \frac{1}{2} & 1\end{matrix}],$ then ${P}^{50}$ is:

If $0<\theta ,\phi <\frac{\pi }{2},x=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}\theta ,y=\sum _{n=0}^{\infty }{\mathrm{sin}}^{2n}\phi$ and $z=\sum _{n=0}^{\infty }{\mathrm{cos}}^{2n}\theta \cdot {\mathrm{sin}}^{2n}\phi$ then :

If the greatest value of the term independent of $x$ in the expansion of ${(x\mathrm{sin}\alpha +a\frac{\mathrm{cos}\alpha }{x})}^{10}$ is $\frac{10!}{{(5!)}^{2}},$ then the value of $a$ is equal to:

The maximum value of $f(x)=|\begin{matrix}{\mathrm{sin}}^{2}x & 1+{\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ 1+{\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{cos}2x \\ {\mathrm{sin}}^{2}x & {\mathrm{cos}}^{2}x & \mathrm{sin}2x\end{matrix}|,x\in R$ is

If the equation $a|z{|}^{2}+\bar{\bar{\alpha }z+\alpha \bar{z}}+d=0$ represents a circle where $a,d$ are real constants then which of the following condition is correct?

If $S={z\in C:\frac{z-i}{z+2i}\in R}$, then

The sum of all $3$-digit numbers less than or equal to $500,$ that are formed without using the digit $1$ and they all are multiple of $11,$ is ______.

Out of all the patients in a hospital $89%$ are found to be suffering from heart ailment and $98%$ are suffering from lungs infection. If $K%$ of them are suffering from both ailments, then $K$ can not belong to the set:

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

Let ${S}_{1}$ be the sum of first $2n$ terms of an arithmetic progression. Let ${S}_{2}$ be the sum of first $4n$ terms of the same arithmetic progression. If $({S}_{2}-{S}_{1})$ is $1000$, then the sum of the first $6n$ terms of the arithmetic progression is equal to: