JEE Main 2009 — Algebra

For real $x$, let $f(x)=x^3+5 x+1$, then

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is

If $A, B$ and $C$ are three sets such that $A \cap B=A \cap C$ and $A \cup B=A \cup C$, then

If the roots of the equation $b x^2+c x+a=0$ be imaginary, then for all real values of $x$, the expression $3 b^2 x^2+6 b c x+2 c^2$ is

If $\left|z-\frac{4}{z}\right|=2$, then the maximum value of $|z|$ is equal to

Let A be a $2 \times 2$ matrix Statement-1 $: \operatorname{adj}(\operatorname{adj} A)=A$ Statement-2 : $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|$

Let $A$ and $B$ denote the statements A: $\cos \alpha+\cos \beta+\cos \gamma=0$ B: $\sin \alpha+\sin \beta+\sin \gamma=0$ If $\cos (\beta-\gamma)+\cos (\gamma-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}$, then

Let $a, b, c$ be such that $b(a+c) \neq 0$. If $\left|\begin{array}{ccc}a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1\end{array}\right|+\left|\begin{array}{ccc}a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2} a & (-1)^{n+1} b & (-1)^n c\end{array}\right|=0$, then the value of ' $n$ ' is

Let $f(x)=(x+1)^2-1, x \geq-1$ Statement-1: The set $\left\{x: f(x)=f^{-1}(x)\right\}=\{0,-1\}$ Statement-2 : $\mathrm{f}$ is a bijection.

The remainder left out when $8^{2 n}-(62)^{2 n+1}$ is divided by 9 is