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 $[-\frac{3}{2}, 2]$

Two differentiable functions $f(x)$ and $g(x)$ are such that $f''(x) > 0$ and $g''(x) < 0$ for all $x \in (a,b)$ and $\int_{a}^{b} f(x) dx = \int_{a}^{b} g(x) dx$. If $f(x) = g(x)$ for $x = \alpha, \beta \in (a,b)$ $(\alpha < \beta)$,then:

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 $b.$

For the function $f(x) = x^4 (12 \ln x - 7)$,which of the following statements is true?

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 $f(x)=2^x-x^2, x \in R$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f^{\prime}(x)$ intersect the $x$-axis,then the value of $m+n$ is