Algebra PYQ — Page 30

Let ${A}_{1}$ and ${A}_{2}$ be two arithmetic means and ${G}_{1},{G}_{2}$ and ${G}_{3}$ be three geometric means of two distinct positive numbers. Then ${G}_{1}^{4}+{G}_{2}^{4}+{G}_{3}^{4}+{G}_{1}^{2}{G}_{3}^{2}$ is equal to

All words, with or without meaning, are made using all the letters of the word $MONDAY$. These words are written as in a dictionary with serial numbers. The serial number of the word $MONDAY$ is

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of ${(1+x)}^{99}$. Let a be the middle term in the expansion of ${(2+\frac{1}{\sqrt{2}})}^{200}$. If $\frac{C99200K}{a}=\frac{{2}^{l}m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(l,n)$ is equal to:

Let $C$ be the circle in the complex plane with centre ${z}_{0}=\frac{1}{2}(1+3i)$ and radius $r=1$. Let ${z}_{1}=1+i$ and the complex number ${z}_{2}$ be outside circle $C$ such that $|{z}_{1}-{z}_{0}||{z}_{2}-{z}_{0}|=1$. If ${z}_{0},{z}_{1}$ and ${z}_{2}$ are collinear, then the smaller value of ${|{z}_{2}|}^{2}$ is equal to

Let $R$ be a relation on $\mathbb{R}$, given by $R={(a,b):3a-3b+\sqrt{7}$ is an irrational number $}$. Then $R$ is

Let ${a}_{1},{a}_{2},{a}_{3},\ldots \ldots$. be an A.P. If ${a}_{7}=3$, the product $({a}_{1}{a}_{4})$ is minimum and the sum of its first $n$ terms is zero then $n!-4{a}_{n(n+2)}$ is equal to

Let $S$ be the set of all values of $\theta \in [-\pi ,\pi ]$ for which the system of linear equations $x+y+\sqrt{3}z=0$ $-x+(\mathrm{tan}\theta )y+\sqrt{7}z=0$ $x+y+(\mathrm{tan}\theta )z=0$ has non-trivial solution.Then $\frac{120}{\pi }\underset{0\in S}{\sum }\theta$ is equal to

The number of seven digits odd numbers, that can be formed using all the seven digits $1,2,2,2,3,3,5$ is

Let $[\alpha ]$ denote the greatest integer $\leq \alpha$. Then $[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+.............+[\sqrt{120}]$ is equal to

Let $0<z<y<x$ be three real numbers such that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in an arithmetic progression and $x,\sqrt{2}y,z$ are in a geometric progression. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z{)}^{2}$ is equal to

If the coefficients of ${x}^{7}$ in ${(a{x}^{2}+\frac{1}{2bx})}^{11}$ and ${x}^{-7}$ in ${(ax-\frac{1}{3b{x}^{2}})}^{11}$ are equal, then

Let ${a}_{1}=8,{a}_{2},{a}_{3},\ldots .{a}_{n}$ be an A.P. If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is _____ .

Let $R$ be a relation defined on $\mathbb{N}$ as a $R$ b is $2a+3b$ is a multiple of $5,a,b\in \mathbb{N}$. Then $R$ is

if the coefficients of three consecutive terms in the expansion of ${(1+x)}^{n}$ are the ratio $1:5:20$then the coefficient of the fourth term is

The 4$^{th}$ term of GP is $500$ and its common ratio is $\frac{1}{m},m\in N$. Let ${S}_{n}$ denote the sum of the first $n$ terms of this GP. If ${S}_{6}>{S}_{5}+1$ and ${S}_{7}<{S}_{6}+\frac{1}{2}$, then the number of possible values of $m$ is ______

If $f(x)=\frac{(\mathrm{tan}{1}^{^{\circ}})x+{\mathrm{log}}_{e}(123)}{x{\mathrm{log}}_{e}(1234)-(\mathrm{tan}{1}^{^{\circ}})},x>0$, then the least value of $f(f(x))+f(f(\frac{4}{x}))$ is

${25}^{190}-{19}^{190}-{8}^{190}+{2}^{190}$ is divisible by

Let $D$ be the domain of the function $f(x)={\mathrm{sin}}^{-1}({\mathrm{log}}_{3x}(\frac{6+2{\mathrm{log}}_{3}x}{-5x}))$. If the range of the function $g:D\rightarrow \mathbb{R}$ defined by $g(x)=x-[x],$ ($[x]$ is the greatest integer function), is $(\alpha ,\beta ),$ then ${\alpha }^{2}+\frac{5}{\beta }$ is equal to

If $A=[\begin{matrix}1 & 5 \\ \lambda & 10\end{matrix}],{A}^{-1}=\alpha A+\beta I$ and $\alpha +\beta =-2$, then $4{\alpha }^{2}+{\beta }^{2}+{\lambda }^{2}$ is equal to :

Let $S={1,2,3,5,7,10,11}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of $3$, is _____ .

Let $A={x\in \mathbb{R}:[x+3]+[x+4]\leq 3},B={x\in \mathbb{R}:{3}^{x}{(\sum _{r=1}^{\infty }\frac{3}{{10}^{r}})}^{x-3}<{3}^{-3x}},$ where $[t]$ denotes greatest integer function. Then,

Let $A=⌊{a}_{\hat{i}\hat{j}}⌋\cdot {a}_{ij}\in Z\cap [0,4],1\leq i,j\leq 2$. The number of matrices $A$ such that the sum of all entries is $a$ prime number $p\in (2,13)$ is _____ .

The coefficient of ${x}^{-6}$, in the expansion of ${(\frac{4x}{5}+\frac{5}{2{x}^{2}})}^{9}$, is

Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x{|}^{2}-2|x|+|\lambda -3|=0$. Then the largest element in the set $S=$ ${x+\lambda :x$ is an integer solution of $E}$ is ______