JEE Main 2014 — Algebra

8-digit numbers are formed using the digits 1, 1, 2, $2,2,3,4,4$. The number of such numbers in which the odd digits do no occupy odd places, is:

A relation on the set $\mathrm{A}=\{\mathrm{x}:|\mathrm{x}| < 3, \mathrm{x} \in \mathrm{Z}\}$, where $Z$ is the set of integers is defined by $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=|\mathrm{x}|, \mathrm{x} \neq-1\}$. Then the number of elements in the power set of $R$ is:

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:

For all complex numbers $z$ of the form $1+i\alpha ,\alpha \in R,$ if ${z}^{2}=x+iy,$ then

Given an A.P. whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220.$ If the second term in it is $12$, then its ${4}^{\mathrm{th}}$ term is :

If $\mathrm{f}(\theta)=\left|\begin{array}{ccc}1 & \cos \theta & 1 \\ -\sin \theta & 1 & -\cos \theta \\ -1 & \sin \theta & 1\end{array}\right|$ and $\mathrm{A}$ and $\mathrm{B}$ are respectively the maximum and the minimum values of $f(\theta)$, then $(A, B)$ is equal to:

If $\mathrm{z}_1, \mathrm{z}_2$ and $\mathrm{z}_3, \mathrm{z}_4$ are 2 pairs of complex conjugate numbers, then $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ equals:

If $\alpha$ and $\beta$ are roots of the equation, $x^2-4 \sqrt{2} k x+2 e^{4 \ln k}-1=0$ for some $k$, and $\alpha^2+\beta^2=66$, then $\alpha^3+\beta^3$ is equal to:

If $A=\left[\begin{array}{ccc}1 & 2 & x \\ 3 & -1 & 2\end{array}\right]$ and $B=\left[\begin{array}{c}y \\ x \\ 1\end{array}\right]$ be such that $\mathrm{AB}=\left[\begin{array}{l}6 \\ 8\end{array}\right]$, then:

If $f(x)=x^2-x+5, x>\frac{1}{2}$, and $\mathrm{g}(\mathrm{x})$ is its inverse function, then $\mathrm{g}^{\prime}(7)$ equals:

If $a\in R$ and the equation $-3(x-[x]{)}^{2}+2(x-[x])+{a}^{2}=0$ (where $[x]$ denotes the greatest integer $\leq$$x$) has no integral solution, then all possible values of $a$ lie in the interval

If $\alpha ,\beta \neq 0$, $f(n)={\alpha }^{n}+{\beta }^{n}$ and $|\begin{matrix}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4)\end{matrix}|=K{(1-\alpha )}^{2}{(1-\beta )}^{2}{(\alpha -\beta )}^{2}$, then $K$ is equal to

If $X={{4}^{n}-3n-1:n\in N}$ and $Y={9(n-1):n\in N}$, where $N$ is the set of natural numbers, then $X\cup Y$ is equal to

If $a,b,c$ are non - zero real numbers and if the system of equations $(a-1)x=y+z$ $(b-1)y=x+z$ $(c-1)z=x+y$ has a non - trivial solution, then $ab+bc+ca$ equals :

If $\frac{1}{\sqrt{\alpha }},\frac{1}{\sqrt{\beta }}$ are the roots of the equation $a{x}^{2}+bx+1=0,(a\neq 0,a,b\in R)$, then the equation $x(x+{b}^{3})+({a}^{3}-3abx)=0$ has roots:

If equations $a{x}^{2}+bx+c=0,(a,b,c\in R,a\neq 0)$ and $2{x}^{2}+3x+4=0$ have a common root, then $a:b:c$ equals :

If $1+x^4+x^5=\sum_{i=0}^5 a_i\left(1+x^i\right)$, for all $x$ in $R$, then $a_2$ is:

If $z$ is a complex number such that $|z|\geq 2,$ then the minimum value of $|z+\frac{1}{2}|$ :

If $B$ is a $3\times 3$ matrix such that ${B}^{2}=0$, then $det.[{(I+B)}^{50}-50B]$ is equal to :

If $A$is a $3\times 3$ non-singular matrix such that $A{A}^{'}={A}^{'}A$ and $B={A}^{-1}{A}^{'},$ then $B{B}^{'}$ equals, where ${X}^{'}$ denotes the transpose of the matrix $X$.

If $\left(2+\frac{x}{3}\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are:

If $g$ is the inverse of a function $f$ and ${f}^{'}(x)=\frac{1}{1+{x}^{5}},$ then ${g}^{'}(x)$ is equal to

If the coefficients of ${x}^{3}$ and ${x}^{4}$ in the expansion of $(1+ax+b{x}^{2}){(1-2x)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

If the sum $\frac{3}{{1}^{2}}+\frac{5}{{1}^{2}+{2}^{2}}+\frac{7}{{1}^{2}+{2}^{2}+{3}^{2}}+.....+$ up to $20$ terms is equal to $\frac{k}{21},$ then $k$ is equal to

If $$ \begin{aligned} &\left.\mid \begin{array}{ccc} a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & \left(c+\lambda^2\right) \\ (a-\lambda)^2 & \left(b-\lambda^2\right) & \left(-\lambda^2\right. \end{array}\right) \\ &=k \lambda\left|\begin{array}{ccc} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{array}\right|, \lambda \neq 0 \end{aligned} $$ then $\mathrm{k}$ is equal to:

If ${\Delta }_{r}=|\begin{matrix}r & 2r-1 & 3r-2 \\ \frac{n}{2} & n-1 & a \\ \frac{1}{2}n(n-1) & {(n-1)}^{2} & \frac{1}{2}(n-1)(3n+4)\end{matrix}|$, then the value of $\sum _{r=1}^{n-1}{\Delta }_{r}$

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49 , and the sum of the first and the third term is 35. Then the first term of this geometric progression is:

Let A be a $3 \times 3$ matrix such that $$ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$ Then $\mathrm{A}^{-1}$ is:

Let $A$ and $B$ be any two $3 \times 3$ matrices. If $A$ is symmetric and $B$ is skew symmetric, then the matrix $AB-BA$ is

Let $w(Imw\neq 0)$ be a complex number. Then, the set of all complex numbers $z$ satisfying the equation $w-\bar{w}z=k(1-z)$, for some real number $k$, is

Let $\mathrm{f}$ be an odd function defined on the set of real numbers such that for $\mathrm{x} \geq 0$, $f(x)=3 \sin x+4 \cos x$. Then $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=-\frac{11 \pi}{6}$ is equal to:

Let $z \neq-i$ be any complex number such that $\frac{\mathrm{z}-\mathrm{i}}{\mathrm{z}+\mathrm{i}}$ is a purely imaginary number. Then $\mathrm{z}+\frac{1}{\mathrm{z}}$ is:

Let $f:R\rightarrow R$ be defined by $f(x)=\frac{|x|-1}{|x|+1},$ then $f$ is

Let $\mathrm{G}$ be the geometric mean of two positive numbers $\mathrm{a}$ and $\mathrm{b}$, and $\mathrm{M}$ be the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$. If $\frac{1}{\mathrm{M}}: \mathrm{G}$ is $4: 5$, then $\mathrm{a}: \mathrm{b}$ can be:

Let $P$ be the relation defined on the set of all real numbers such that $P={(a,b):{\mathrm{sec}}^{2}a-{\mathrm{tan}}^{2}b=1}$. Then, $P$ is

Let $\alpha \text{ and } \beta$ be the roots of equation $p{x}^{2}+qx+r=0$, $p\neq 0$. If $p,q,r$are in A.P. and $\frac{ 1 }{ \alpha } + \frac{ 1 }{ \beta } = 4$, then the value of $| \alpha - \beta |$ is

Let for $\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})$ be a polynomial of degree 2 in $x, p^{\prime}{ }_i(x)$ and $p^{\prime \prime}{ }_i(x)$ be the first and second order derivatives of $p_i(x)$ respectively. Let, $$ \left.\mathrm{A}(\mathrm{x})=\left[\begin{array}{lll} \mathrm{p}_1(\mathrm{x}) & \mathrm{p}_1^{\prime}(\mathrm{x}) & \mathrm{p}_1^{\prime \prime} \mathrm{x}( \\ \mathrm{p}_2(\mathrm{x}) & \mathrm{p}_2^{\prime}(\mathrm{x}) & \mathrm{p}_2^{\prime \prime}( \\ \mathrm{p}_3(\mathrm{x}) & \mathrm{p}_3^{\prime}(\mathrm{x}) & \mathrm{p}_3^{\prime \prime}(\mathrm{x} \end{array}\right]\right) $$ and $\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{\mathrm{T}} \mathrm{A}(\mathrm{x})$. Then determinant of $\mathrm{B}(\mathrm{x})$ :

Let $f⁡(n)=[\frac{1}{3}+\frac{3n}{100}]n$, where $[n]$ denotes the greatest integer less than or equal to $n$. Then $\sum _{n=1}^{56}f(n)$ is equal to

The coefficient of $x^{50}$ in the binomial expansion of $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots$ $+x^{1000}$ is:

The coefficient of ${ x⁡ }^{ 1 0 1 2 }$ in the expansion of ${(1+{x}^{n}+{x}^{253})}^{10},$ (where $n\leq 22$ is any positive integer), is

The equation $\sqrt{3{x}^{2}+x+5}=x-3$, where $x⁡$ is real, has

The function $f(x)=|sin4x|+|cos2x|,$ is a periodic function with a fundamental period

The least positive integer $\mathrm{n}$ such that $1-\frac{2}{3}-\frac{2}{3^2}-\ldots .-\frac{2}{3^{n-1}} < \frac{1}{100}$, is:

The number of terms in an $A.P.$ is even, the sum of the odd terms in it is $24$ and that the even terms is $30.$ If the last term exceeds the first term by $10\frac{1}{2},$ then the number of terms in the $A.P.$ is

The number of terms in the expansion of ${(1+x)}^{101}{(1-x+{x}^{2})}^{100}$ in powers of $x$ is

The sum of the digits in the unit's place of all the $4$ - digit numbers formed by using the numbers $3,4,5$ and$6$, without repetition is :

The sum of the first 20 terms common between the series $3+7+11+15+$ and $1+6+11+$ $16+\ldots .$. is

The sum of the roots of the equation, $\mathrm{x}^2+|2 \mathrm{x}-3|-4=0$, is:

Three positive numbers form an increasing $G.P.$ If the middle term in this $G.P.$ is doubled, the new numbers are in $A.P.$ Then the common ratio of the $G.P.$ is :

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by$66$, then the number of men who participated in the tournament lies in the interval