JEE Main 2017 — Algebra

A man $X$ has $7$ friends, $4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends, $3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. Then the total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men, so that $3$ friends of each of $X$ and $Y$ are in this party is:

For any three positive real numbers $a,b$ and $c$. If $9(25{a}^{2}+{b}^{2})+25({c}^{2}-3ac)=15b(3a+c).$ Then

For two $3\times 3$ matrices $A$ and $B$, let $A+B=2{B}^{'}$ and $3A+2B={I}_{3},$ where ${B}^{'}$ is the transpose of $B$ and ${I}_{3}$ is $3\times 3$ identity matrix. Then :

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is:

If, for a positive integer $n$, the quadratic equation, $x(x+1)+(x+1)(x+2)+...+(x+\bar{n-1})(x+n)=10n$ has two consecutive integral solutions, then $n$ is equal to:

If $x=a, y=b, z=c$ is a solution of the system of linear equations $x+8y+7z=0$ $9x+2y+3z=0$ $x+y+z=0$ Such that the point $(a,b,c)$ lies on the plane $x+2y+z=6$ , then $2a+b+c$ equals:

If ${(27)}^{999}$ is divided by $7$, then the remainder is

If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x+y+z=1$ $x+ay+z=1$ $ax+by+z=0$ has no solution, then $S$ is:

If the arithmetic mean of two numbers $a$ and $b, a>b>0$ , is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to:

If the sum of the first $n$ terms of the series $\sqrt{3}+ \sqrt{75}+ \sqrt{243}+ \sqrt{507}+\ldots$ is $435\sqrt{3},$ then $n$ equals:

If $S={x\in [0, 2\pi ] :|\begin{matrix}0 & \mathrm{cos}⁡x & -\mathrm{sin}⁡x \\ \mathrm{sin}⁡x & 0 & \mathrm{cos}⁡x \\ \mathrm{cos}⁡x & \mathrm{sin}⁡x & 0\end{matrix}|=0},$ then $\underset{x \in S}{\sum }\mathrm{tan}(\frac{\pi }{3}+x)$ is equal to:

If $A=[\begin{matrix}2 & -3 \\ -4 & 1\end{matrix}]$ , then $\mathrm{Adj}(3{A}^{2}+12A)$ is equal to:

If three positive numbers $a$, $b$ and $c$ are in A.P. such that $abc=8$, then the minimum possible value of $b$ is:

Let $f(x)={2}^{10}x+1$ and $g(x)={3}^{10}x-1.$ If $(fog)(x)=x,$ then $x$ is equal to:

Let $\omega$ be a complex number such that $2\omega +1=z$ where $z=\sqrt{-3}$ . If $|\begin{matrix}1 & 1 & 1 \\ 1 & -{\omega }^{2}-1 & {\omega }^{2} \\ 1 & {\omega }^{2} & {\omega }^{7}\end{matrix}|=3k,$ Then $k$ can be equal to:

Let $p(x)$ be a quadratic polynomial such that $p(0)=1.$ If $p(x)$ leaves remainder $4$ when divided by $x-1$ and it leaves remainder $6$ when divided by $x+1$ then:

Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true?

Let $a, b, c\in R$ . If $f(x)=a{x}^{2}+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy, \forall x, y\in R$ , then $\sum _{ n=1 }^{ 10 } f(n)$ is equal to:

Let ${S}_{n}=\frac{1}{{1}^{3}}+\frac{1+2}{{1}^{3}+{2}^{3}}+\frac{1+2+3}{{1}^{3}+{2}^{3}+{3}^{3}}+\ldots +\frac{1+2+\ldots ,+n}{{1}^{3}+{2}^{3}+\ldots {n}^{3}}$ . If $100 {S}_{n}=n,$ then $n$ is equal to:

Let $z\in C,$ the set of complex numbers. Then the equation, $2|z+3i|-|z-i|=0$ represents:

The coefficient of ${x}^{-5}$ in the binomial expansion of ${(\frac{x+1}{{x}^{\frac{2}{3}}-{x}^{\frac{1}{3}}+1 }-\frac{x-1}{x-{x}^{\frac{1}{2}}})}^{10}$ where $x\neq 0,1$ is

The equation $Im(\frac{iz-2}{z-i})+1=0, z\in C, z\neq i$ represents a part of a circle having radius equal to :

The function $f :R\rightarrow [-\frac{1}{2},\frac{1}{2}]$ defined as $f(x)=\frac{x}{1+{x}^{2}},$ is:

The function $f :N\rightarrow I$ defined by $f(x)=x-5[\frac{x}{5}]$ , where $N$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to $x$, is:

The number of real values of $\lambda$ for which the system of linear equations, $2x+4y-\lambda z=0$, $4x+\lambda y+2z=0$ and $\lambda x+2y+2z=0$, has infinitely many solutions, is:

The number of ways in which$5$ boys and $3$ girls can be seated on a round table if a particular boy ${B}_{1}$ and a particular girl ${G}_{1}$ never sit adjacent to each other, is:

The sum of all the real values of $x$ Satisfying the equation ${2}^{(x-1)({x}^{2}+5x-50)}=1$ is: