The set of all $a \in R$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root is:

Let $f(x) = \frac{x^2 - 4}{x^2 + 4}$ for $|x| > 2$. Then the function $f: (- \infty, -2] \cup [2, \infty) \to (-1, 1)$ is

Let $f(x) = x^{12} - x^9 + x^4 - x + 1$. Which of the following is true?

If $h(x) = [\ln(x/e)] + [\ln(e/x)]$,where $[.]$ denotes the greatest integer function,then which of the following is false?

Suppose $f: [-2, 2] \rightarrow R$ is defined by $f(x) = \begin{cases} -1, & -2 \leq x \leq 0 \\ x - 1, & 0 < x \leq 2 \end{cases}$. Then the set $\{x \in [-2, 2] : x \leq 0 \text{ and } f(|x|) = x\}$ is equal to

Let $\alpha, \beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x, x \in \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \dots, a_n$ are in arithmetic progression with mean zero,then the value of $f(g(\frac{1}{n} \sum_{i=1}^{n} f(a_i)))$ is equal to.