JEE Main 2013 — Algebra

Statement-1: The system of linear equations $$ \begin{aligned} & x+(\sin \alpha) y+(\cos \alpha) z=0 \\ & x+(\cos \alpha) y+(\sin \alpha) z=0 \\ & x-(\sin \alpha) y-(\cos \alpha) z=0 \end{aligned} $$ has a non-trivial solution for only one value of $\alpha$ lying in the interval $\left(0, \frac{\pi}{2}\right)$. Statement-2: The equation in $\alpha$ $$ \left|\begin{array}{ccc} \cos \alpha & \sin \alpha & \cos \alpha \\ \sin \alpha & \cos \alpha & \sin \alpha \\ \cos \alpha & -\sin \alpha & -\cos \alpha \end{array}\right|=0 $$ has only one solution lying in the interval $\left(0, \frac{\pi}{2}\right)$

The sum of the series: $1+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots \ldots$. upto 10 terms, is:

If the system of linear equations : $$ \begin{aligned} & x_1+2 x_2+3 x_3=6 \\ & x_1+3 x_2+5 x_3=9 \\ & 2 x_1+5 x_2+a x_3=b \end{aligned} $$ is consistent and has infinite number of solutions, then :

If $a_1, a_2, a_3, \ldots, a_n, \ldots$. are in A.P. such that $a_4-a_7$ $+a_{10}=m$, then the sum of first 13 terms of this A.P., is :

The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^2+\frac{2}{x}\right)^{15}$ is:

The term independent of $x$ in the expansion of ${(\frac{x+1}{{x}^{2/3}-{x}^{1/3}+1}-\frac{x-1}{x-{x}^{1/2}})}^{10}$ is

If the 7th term in the binomial expansion of $\left(\frac{3}{\sqrt[3]{84}}+\sqrt{3} \ln x\right)^9, x>0$, is equal to 729 , then $x$ can be:

5 - digit numbers are to be formed using $2,3,5,7$, 9 without repeating the digits. If $p$ be the number of such numbers that exceed 20000 and $q$ be the number of those that lie between 30000 and 90000 , then $p: q$ is:

If $P=[\begin{matrix}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{matrix}]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4$, then $\alpha$ is equal to

The matrix $A^2+4 A-5 I$, where $I$ is identity matrix and $A=\left[\begin{array}{cc}1 & 2 \\ 4 & -3\end{array}\right]$, equals :

The number of values of $k$, for which the system of equations : $(k+1)x+8y=4k$ $kx+(k+3)y=3k-1$ has no solution, is :

The sum $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\ldots$. upto 11-terms is:

Let ${T}_{n}$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If ${T}_{n+1}-{T}_{n}=10$, then the value of $n$ is :

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :

Let $z$ satisfy $|z|=1$ and $z=1-\bar{z}$. Statement $1: z$ is a real number. Statement 2 : Principal argument of z is $\frac{\pi}{3}$

The values of ' $a$ ' for which one root of the equation $x^2-(a+1) x+a^2+a-8=0$ exceeds 2 and the other is lesser than 2 , are given by :

Let $R=\{(3,3)(5,5),(9,9),(12,12),(5,12),(3,9)$, $(3,12),(3,5)\}$ be a relation on the $\operatorname{set} A=\{3,5,9,12\}$. Then, $R$ is :

If $a, b, c$ are sides of a scalene triangle, then the value of $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|$ is :

The least integral value $\alpha$ of $x$ such that $\frac{x-5}{x^2+5 x-14}>0$, satisfies :

Let $A$ and $B$ be two sets containing $2$ elements and $4$ elements respectively. The number of subsets of $A\times B$ having $3$ or more elements is :

If for positive integers $r>1, n>2$, the coefficients of the $(3 r)^{\text {th }}$ and $(r+2)^{\text {th }}$ powers of $x$ in the expansion of $(1+x)^{2 n}$ are equal, then $n$ is equal to:

If $Z_1 \neq 0$ and $Z_2$ be two complex numbers such that $\frac{Z_2}{Z_1}$ is a purely imaginary number, then $\left|\frac{2 Z_1+3 Z_2}{2 Z_1-3 Z_2}\right|$ is equal to:

If $z$ is a complex number of unit modulus and argument $\theta$, then arg $(\frac{1+z}{1+\overset{-}{z}})$ can be equal to $(\mathrm{given} z\neq -1)$

The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is :

If $x,y,z$ are positive numbers in $A.P.$ and ${\mathrm{tan}}^{-1}x$, ${\mathrm{tan}}^{-1}y$ and ${\mathrm{tan}}^{-1}z$ are also in $A.P.$, then which of the following is correct.

A common tangent to the conics $x^2=6 y$ and $2 x^2-4 y^2=9$ is:

Let $\mathrm{R}=\left\{(x, y): x, y \in N\right.$ and $\left.x^2-4 x y+3 y^2=0\right\}$, where $N$ is the set of all natural numbers. Then the relation $R$ is :

If $p$ and $q$ are non-zero real numbers and $\alpha^3+\beta^3=-p, \alpha \beta=q$, then a quadratic equation whose roots are $\frac{\alpha^2}{\beta}, \frac{\beta^2}{\alpha}$ is :

The sum of the series : $(2)^2+2(4)^2+3(6)^2+\ldots$ upto 10 terms is :

Given a sequence of 4 numbers, first three of which are in G.P. and the last three are in A.P. with common difference six. If first and last terms of this sequence are equal, then the last term is :

Let $a_1, a_2, a_3, \ldots$ be an A.P, such that $\frac{a_1+a_2+\ldots+a_p}{a_1+a_2+a_3+\ldots+a_q}=\frac{p^3}{q^3} ; p \neq q$. Then $\frac{a_6}{a_{21}}$ is equal to:

The sum of first $20$ terms of the sequence $0.7,0.77,0.777,......,$ is :

If $\alpha$ and $\beta$ are roots of the equation $x^2+p x+\frac{3 p}{4}=0$, such that $|\alpha-\beta|=\sqrt{10}$, then $p$ belongs to the set :

If the equations ${x}^{2} + 2 x + 3 = 0$ and $a{x}^{2} +bx+c=0,$ $a,b,c\in R,$ have a common root, then $a:b:c$ is:

If $p, q, r$ are 3 real numbers satisfying the matrix equation, $[p q r]\left[\begin{array}{lll}3 & 4 & 1 \\ 3 & 2 & 3 \\ 2 & 0 & 2\end{array}\right]=\left[\begin{array}{lll}3 & 0 & 1\end{array}\right]$ then $2 p+q-r$ equals :

Given sum of the first $n$ terms of an A.P. is $2 n+$ $3 n^2$. Another A.P. is formed with the same first term and double of the common difference, the sum of $n$ terms of the new A.P. is :

Let $\mathrm{A}$, other than $\mathrm{I}$ or $-\mathrm{I}$, be a $2 \times 2$ real matrix such that $\mathrm{A}^2=\mathrm{I}$, I being the unit matrix. Let $\operatorname{Tr}(\mathrm{A})$ be the sum of diagonal elements of A. Statement-1: $\operatorname{Tr}(\mathrm{A})=0$ Statement-2: $\operatorname{det}(\mathrm{A})=-1$

Let $A=\{1,2,3,4\}$ and $R: A \rightarrow A$ be the relation defined by $R=\{(1,1),(2,3),(3,4),(4,2)\}$. The correct statement is :


If a complex number $z$ statisfies the equation $x+\sqrt{2}|z+1|+i=0$, then $|z|$ is equal to :

Let $a=\operatorname{Im}\left(\frac{1+z^2}{2 i z}\right)$, where $z$ is any non-zero complex number. The set $\mathrm{A}=\{a:|z|=1$ and $z \neq \pm 1\}$ is equal to:

The real number $k$ for which the equation, $2{x}^{3}+3x+k=0$ has two distinct real roots in $[0,1]$ belongs to