JEE Main 2011 — Algebra

The number of values of $\mathrm{k}$ for which the linear equations $4 x+k y+2 z=0 ; k x+4 y+z=0 ; 2 x+2 y+z=0$ possess a non-zero solution is

Let $A$ and $B$ be two symmetric matrices of order 3 . This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement $-1$ : $\mathrm{A}(\mathrm{BA})$ and $(\mathrm{AB}) \mathrm{A}$ are symmetric matrices. Statement - 2 : $\quad A B$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after

This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement - 1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${ }^9 \mathrm{C}_3$ Statement-2: The number of ways of choosing any 3 places from 9 different places is ${ }^9 \mathrm{C}_3$.

If $\omega(\neq 1)$ is a cube root of unity, and $(1+\omega)^7=A+B \omega$. Then $(A, B)$ equals

The coefficient of $x^7$ in the expansion of $\left(1-x-x^2+x^3\right)^6$ is

The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is

Let $R$ be the set of real numbers This question has Statement $-1$ and Statement $-2$. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1 : $A=\{(x, y) \in R \times R: y-x$ is an integer $\}$ is an equivalence relation on $R$. Statement-2 : $B=\{(x, y) \in R \times R: x=\alpha y$ for some rational number $\alpha\}$ is an equivalence relation on $\mathrm{R}$.

Let $\alpha, \beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re} z=1$, then it is necessary that