JEE Main 2016 — Algebra

A value of $\theta$ for which $\frac{2+3i\mathrm{sin}\theta }{1-2i\mathrm{sin}\theta }$ is purely imaginary, is

For $x \in R, x\neq 0, x\neq 1,$ let ${f}_{0}(x)=\frac{1}{1-x}$ and ${f}_{n+1}(x)={f}_{0}({f}_{n}(x)), n=0, 1, 2,\ldots ..$ Then the value of ${f}_{100}(3)+{f}_{1}(\frac{2}{3})+{f}_{2}(\frac{3}{2})$ is equal to :

If all the words (with or without meaning) having five letters, formed using the letters of the word $SMALL$ and arranged as in a dictionary; then the position of the word $SMALL$ is

If $f(x)+2f(\frac{1}{x})=3x, x\neq 0,$ and $S={x\in R:f(x)=f(-x)}$, then $S$

If $P=[\begin{matrix}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{matrix}], A=[\begin{matrix}1 & 1 \\ 0 & 1\end{matrix}]$ and $Q=PA{P}^{T},$ then ${P}^{T} {Q}^{2015} P$ is :

If $A=[\begin{matrix}5a & -b \\ 3 & 2\end{matrix}]$ and $A.adjA=A {A}^{T}$ , then $5a+b$ is equal to

If $x$ is a solution of the equation $\sqrt{2x+1}- \sqrt{2x-1}=1, (x\geq \frac{1}{2})$ , then $\sqrt{4{x}^{2}-1}$ is equal to :

If the ${2}^{nd}, {5}^{th}$ and ${9}^{th}$ terms of a non-constant arithmetic progression are in geometric progression, then the common ratio of this geometric progression is

If the coefficients of ${x}^{-2}$ and ${x}^{-4}$, in the expansion of ${({x}^{\frac{1}{3}}+\frac{1}{2{x}^{\frac{1}{3}}})}^{18},(x>0)$, are $m$ and $n$ respectively, then $\frac{m}{n}$ is equal to

If the equations ${x}^{2}+bx-1=0$ and ${x}^{2}+x+b=0$ have a common root different from $-1,$ then $|b|$ is equal to :

If the four letter words (need not be meaningful) are to be formed using the letters from the word "MEDITERRANEAN" such that the first letter is $R$ and the fourth letter is $E$, then the total number of all such words is :

If $\frac{{}^{n+2}{C}_{6}}{{}^{n-2}{P}_{2}}=11$, then $n$ satisfies the equation:

If $A=[\begin{matrix}-4 & -1 \\ 3 & 1\end{matrix}]$ , then the determinant of the matrix $({A}^{2016}-2{A}^{2015}-{A}^{2014})$ is :

Let $z=1+ai$, be a complex number, $a>0$, such that ${z}^{3}$ is a real number. Then, the sum $1+z+{z}^{2}+\ldots .+{z}^{11}$ is equal to :

Let $A$, be a $3 \times 3$ matrix, such that ${A}^{2}-5A+7I=O.$ Statement - I : ${A}^{-1}=\frac{1}{7} (5I-A).$ Statement - II : The polynomial ${A}^{3}-2{A}^{2}-3A+I$,can be reduced to $5(A-4I).$ Then :

Let ${a}_{1}, {a}_{2}, {a}_{3},\ldots {a}_{n},\ldots$ ,be in A.P. If ${a}_{3}+{a}_{7}+{a}_{11}+{a}_{15}=72$, then the sum of its first $17$ terms is equal to :

Let $x, y, z$ be positive real numbers such that $x+y+z=12$ and ${x}^{3}{y}^{4}{z}^{5}=(0.1){(600)}^{3}.$ Then ${x}^{3}+{y}^{3}+{z}^{3}$ is equal to

The number of distinct real roots of the equation, $|\begin{matrix}\mathrm{cos}x & \mathrm{sin}x & \mathrm{sin}x \\ \mathrm{sin}x & \mathrm{cos}x & \mathrm{sin}x \\ \mathrm{sin}x & \mathrm{sin}x & \mathrm{cos}x\end{matrix}|=0$ in the interval $[-\frac{\pi }{4},\frac{\pi }{4}]$ is :

The point represented by $2+i$ in the Argand plane moves $1$ unit eastwards, then $2$ units northwards and finally from there $2\sqrt{2}$ units in the south-west wards direction. Then its new position in the Argand plane is at the point represented by :

The sum of all real values of $x$ satisfying the equation ${({x}^{2}-5x+5)}^{{x}^{2}+4x-60}=1$ is

The system of linear equations $x+\lambda y-z=0$ $\lambda x-y-z=0$ $x+y-\lambda z=0$ has a non -trivial solution for

The value of $\sum _{r=1}^{15}{r}^{2}(\frac{{}^{15}{C}_{r}}{{}^{15}{C}_{r-1}})$ is equal to: