AIEEE 2009 PYQ Mock Test

1: The set $\{x:f(x)=f^{-1}(x)\}=\{0,-1\}$ where $f(x)=(x+1{)}^2-1,x\ge -1$.
2: $f$ is a bijection.

1: $g\circ f$ is differentiable at $x=0$ and its derivative is continuous at that point, where $f(x)=x|x|$ and $g(x)=\sin x$.
2: $g\circ f$ is twice differentiable at $x=0$.

1: The variance of first $n$ even natural numbers is $\frac{n^2-1}{4}$.
2: The sum of first $n$ natural numbers is $\frac{n(n+1)}{2}$ and the sum of squares of first $n$ natural numbers is $\frac{n(n+1)(2n+1)}{6}$.

Statement 1: $\sim (p\leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q$.
Statement 2: $\sim (p\leftrightarrow \sim q)$ is a tautology.

1: $\text{adj}(\text{adj }A)=A$ where $A$ is a $2\times 2$ matrix.
2: $|\text{adj }A|=|A|$.