JEE Main 2015 — Algebra

A complex number $z$ is said to be unimodular if $|z|=1$. Let ${z}_{1}$ and ${z}_{2}$ are complex numbers such that $\frac{{z}_{1}-2{z}_{2}}{2-{z}_{1}{z}_{2}}$ is unimodular and ${z}_{2}$ is not unimodular, then the point ${z}_{1}$ lies on a

If in a regular polygon the number of diagonals is $54$, then the number of sides of this polygon is:

If $A=[\begin{matrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{matrix}]$ is a matrix satisfying the equation $A{A}^{T}=9I$ , where $I$is $3\times 3$ identity matrix, then the ordered pair $(a,b)$ is equal to

If $A$ is a $3\times 3$ matrix such that $|5 adjA|=5$, then $|A|$ is equal to

If $z$ is a non-real complex number, then the minimum value of $\frac{Im{z}^{5}}{{(Imz)}^{5}}$ is (Where $Imz=$ Imaginary part of $z$)

If $2+3i$ is one of the roots of the equation $2{x}^{3}-9{x}^{2}+kx-13=0, k\in R,$ then the real root of this equation (where ${i}^{2}=-1$) :

If $m$ is the $A.M.$of two distinct real numbers $I$ and $n$ $(I,n>1)$ and ${G}_{1},{G}_{2}\text{and }{G}_{3}$ are three geometric means between $I\text{and}n$, then ${G}_{1}^{4}+2{G}_{2}^{4}+{G}_{3}^{4}$ equals

If the coefficient of the three successive terms in the binomial expansion of ${(1+x)}^{n}$ are in the ratio $1:7:42$, then the first of these terms in the expansion is

If the two roots of the equation, $(a-1) ({x}^{4}+{x}^{2}+1)+(a+1){({x}^{2}+x+1)}^{2}=0$ are real and distinct, then the set of all values of $a$ is equal to

If $|\begin{matrix}{x}^{2}+x & x+1 & x-2 \\ 2{x}^{2}+3x-1 & 3x & 3x-3 \\ {x}^{2}+2x+3 & 2x-1 & 2x-1\end{matrix}|=ax-12$ , then $a$ is equal to:

If $A=[\begin{matrix}0 \\ 1\end{matrix} \begin{matrix}-1 \\ 0\end{matrix}]$ , then which one of the following statements is not correct?

In a certain town, $25%$ of the families own a phone and $15%$ own a car; $65%$ families own neither a phone nor a car and $2000$ families own both a car and a phone. Consider the following three statements: $(i)$ $5%$ families own both a car and a phone. $(ii)$ $35%$ families own either a car or a phone. $(iii)$ $40000$ families live in the town. Then,

Let $\alpha$ and $\beta$ be the roots of equation ${x}^{2}-6x-2=0$. If ${a}_{n}={\alpha }^{n}-{\beta }^{n},\forall n\geq 1,$ then the value of $\frac{{a}_{10}-2{a}_{8}}{2{a}_{9}}$ is equal to

Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A\times B$, each having at least three elements is

Let $A={{x}_{1},{x}_{2},\ldots ,{x}_{7}} \text{and} B={{y}_{1},{y}_{2},{y}_{3}}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f:A\rightarrow B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f(x)={y}_{2},$ is equal to:

Let the sum of the first three terms of an A.P. be $39$ and the sum of its last four terms be $178$. If the first term of this A.P. is $10,$ then the median of the A.P. is :

The largest value of $r$, for which the region represented by the set ${\omega \in C||\omega -4-i|\leq r}$ is contained in the region represented by the set ${z\in C||z-1|\leq |z+i|}$, is equal to :

The least value of the product $xyz$ (such that $x,y\text{and}z$ are positive real numbers) for which the determinant $|\begin{matrix}x \\ 1 \\ 1\end{matrix} \begin{matrix}1 \\ y \\ 1\end{matrix} \begin{matrix}1 \\ 1 \\ z\end{matrix}|$ is non-negative is

The number of integers greater than $6000$ that can be formed, using the digits $3,5,6,7$ and $8$, without repetition is

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0,0),(0,41)$ and $(41,0)$ is

The number of ways of selecting $15$ teams from $15$ men and $15$ women, such that each team consists of a man and a woman is

The set of all values of $\lambda$ for which the system of linear equations: $2 {x}_{1} - 2 {x}_{2} + {x}_{3} = \lambda {x}_{1}$ $2 {x}_{1} - 3 {x}_{2} + { 2 x }_{3} = \lambda {x}_{2}$ $- {x}_{1} + { 2 x }_{2} = \lambda {x}_{3}$ has a non-trivial solution,

The sum of coefficients of integral powers of $x$ in the binomial expansion of ${(1-2\sqrt{x})}^{\text{50}}$ is

The sum of first $9$ terms of the series $\frac{{1}^{3}}{1}+\frac{{1}^{3}+{2}^{3}}{1+3}+\frac{{1}^{3}+{2}^{3}+{3}^{3}}{1+3+5}+...$ is

The sum of the ${3}^{rd}$ and the ${4}^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000.$ If the first term of this $G.P.$ is positive, then its ${7}^{th}$ term is:

The term independent of $x$ in the binomial expansion of $(1-\frac{1}{x}+3{x}^{5}) {(2{x}^{2}-\frac{1}{x})}^{8}$ is

The value of $\sum _{r=16}^{30}(r+2)(r-3)$ is equal to: