The function $f(x) = x^{3} - 6x^{2} + ax + b$ is such that $f(2) = f(4) = 0$. Consider two statements.
$(S_1)$ There exists $x_{1}, x_{2} \in (2, 4)$,$x_{1} < x_{2}$,such that $f^{\prime}(x_{1}) = -1$ and $f^{\prime}(x_{2}) = 0$.
$(S_2)$ There exists $x_{3}, x_{4} \in (2, 4)$,$x_{3} < x_{4}$,such that $f$ is decreasing in $(2, x_{4})$,increasing in $(x_{4}, 4)$ and $2f^{\prime}(x_{3}) = \sqrt{3}f(x_{4})$.
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 $b.$

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

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 $f(x) = x \cos^{-1}(-\sin |x|)$,$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Then which of the following is true?

Let $f(x) = \sin x + (x^3 - 3x^2 + 4x - 2) \cos x$ for $x \in (0, 1)$. Consider the following statements:
$I.$ $f$ has a zero in $(0, 1)$.
$II.$ $f$ is monotone in $(0, 1)$.
Then,