Algebra PYQ — Page 48

If the digits are not allowed to repeat in any number formed by using the digits $0,2,4,6,8,$ then the number of all numbers greater than $10,000$ 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 $p$ and $q$ be two positive numbers such that $p+q=2$ and ${p}^{4}+{q}^{4}=272$. Then $p$ and $q$ are roots of the equation:

The value of $4+\frac{1}{5+\frac{1}{4+\frac{1}{5+\frac{1}{4+\ldots \ldots \infty }}}}$ 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 :

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

Let $[x]$ denote the greatest integer $\leq x$, where $x\in R$. If the domain of the real valued function $f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}$ is $(-\infty ,a)\cup [b,c)\cup [4,\infty ),a<b<c$, then the value of $a+b+c$ is:

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

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:

The number of seven digit integers with sum of the digits equal to $10$ and formed by using the digits $1,2$ and $3$ only is:

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

If sum of the first $21$ terms of the series ${\mathrm{log}}_{{9}^{1/2}}x+{\mathrm{log}}_{{9}^{1/3}}x+{\mathrm{log}}_{{9}^{1/4}}x+\ldots ..$ where $x>0$ is $504,$ then $x$ is equal to

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:

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

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 :

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:

All the arrangements, with or without meaning, of the word $\mathrm{FARMER}$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $\mathrm{FARMER}$ in this list is _____ .

Let the system of linear equations $4x+\lambda y+2z=0$ $2x-y+z=0$ $\mu x+2y+3z=0,\lambda ,\mu \in R$ has a non-trivial solution. Then which of the following is true?

For the system of linear equations: $x-2y=1,x-y+kz=-2,ky+4z=6,k\in R$ Consider the following statements: (A) The system has unique solution if $k\neq 2,k\neq -2.$ (B) The system has unique solution if $k=-2.$ (C) The system has unique solution if $k=2.$ (D) The system has no-solution if $k=2.$ (E) The system has infinite number of solutions if $k\neq -2.$ Which of the following statements are correct?

If $\alpha$ and $\beta$ are the distinct roots of the equation ${x}^{2}+{(3)}^{\frac{1}{4}}x+{3}^{\frac{1}{2}}=0$, then the value of ${\alpha }^{96}({\alpha }^{12}-1)+{\beta }^{96}({\beta }^{12}-1)$ is equal to: