The number of solutions of the equation $2e^{|x|} \tan^{-1}|x| = 1$ is -

Let $f: R \rightarrow R$ be defined by $f(x) = \frac{x^2-3x-6}{x^2+2x+4}$. Then which of the following statements is (are) $TRUE$?
$(A)$ $f$ is decreasing in the interval $(-2, -1)$
$(B)$ $f$ is increasing in the interval $(1, 2)$
$(C)$ $f$ is onto
$(D)$ Range of $f$ is $[-\frac{3}{2}, 2]$

Consider the polynomial $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f(x)$ and let $t=|s|$.
$1.$ The real number $s$ lies in the interval
$(A)$ $\left(-\frac{1}{4}, 0\right)$ $(B)$ $\left(-1,-\frac{3}{4}\right)$
$(C)$ $\left(-\frac{3}{4},-\frac{1}{2}\right)$ $(D)$ $\left(0, \frac{1}{4}\right)$
$2.$ The area bounded by the curve $y=f(x)$ and the lines $x=0, y=0$ and $x=t$,lies in the interval
$(A)$ $\left(\frac{3}{4}, 3\right)$ $(B)$ $\left(\frac{21}{64}, \frac{11}{16}\right)$
$(C)$ $(9,10)$ $(D)$ $\left(0, \frac{21}{64}\right)$
$3.$ The function $f^{\prime}(x)$ is
$(A)$ increasing in $\left(-t,-\frac{1}{4}\right)$ and decreasing in $\left(-\frac{1}{4}, t\right)$
$(B)$ decreasing in $\left(-t,-\frac{1}{4}\right)$ and increasing in $\left(-\frac{1}{4}, t\right)$
$(C)$ increasing in $(-t, t)$ $(D)$ decreasing in $(-t, t)$
Give the answer for questions $1, 2$ and $3.$

Let $f(x) = \min (\{x\}, \{e^{-x}\})$ for $x \in [0, 10]$. If $C$ and $D$ are the number of points where $f(x)$ is discontinuous and non-differentiable respectively,then $(C + D)$ is equal to (where $\{.\}$ denotes the fractional part function).

If $(x-a)^{2}+(y-b)^{2}=c^{2},$ for some $c > 0,$ prove that $\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2} y}{d x^{2}}}$ is a constant independent of $a$ and $b.$

$f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1$,which one of the following is $NOT$ correct?