Let $h(x) = \min \{ x, x^2 \}$ for every real number $x$. Then:

Consider the function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$ defined by $f(x)=\frac{x^2-a x+1}{x^2+a x+1}, 0 < a < 2 .$
$1.$ Which of the following is true?
$(A)$ $(2+a)^2 f^{\prime \prime}(1)+(2-a)^2 f^{\prime \prime}(-1)=0$
$(B)$ $(2-a)^2 f^{\prime}(1)-(2+a)^2 f^{\prime \prime}(-1)=0$
$(C)$ $f^{\prime}(1) f^{\prime}(-1)=(2-a)^2$
$(D)$ $f^{\prime}(1) f^{\prime}(-1)=-(2+a)^2$
$2.$ Which of the following is true?
$(A)$ $f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$
$(B)$ $f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$
$(C)$ $f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$(D)$ $f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
$3.$ Let $g(x)=\int_0^{e^x} \frac{f^{\prime}(t)}{1+t^2} d t$. Which of the following is true?
$(A)$ $g^{\prime}(x)$ is positive on $(-\infty, 0)$ and negative on $(0, \infty)$
$(B)$ $g^{\prime}(x)$ is negative on $(-\infty, 0)$ and positive on $(0, \infty)$
$(C)$ $g^{\prime}(x)$ changes sign on both $(-\infty, 0)$ and $(0, \infty)$
$(D)$ $g^{\prime}(x)$ does not change sign on $(-\infty, \infty)$
Give the answer for questions $1, 2$ and $3.$

In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column $I$ with the properties in Column $II$.

Column $I$	Column $II$
$(A)$ $f(x) = x|x|$	$(p)$ continuous in $(-1, 1)$
$(B)$ $f(x) = \sqrt{|x|}$	$(q)$ differentiable in $(-1, 1)$
$(C)$ $f(x) = x + [x]$	$(r)$ strictly increasing in $(-1, 1)$
$(D)$ $f(x) = |x - 1| + |x + 1|$	$(s)$ not differentiable at least at one point in $(-1, 1)$

If $f(x) = \cos x \cos 2x \cos 4x \cos 8x \cos 16x$, then $f' \left( \frac{\pi}{4} \right)$ is equal to

Using the fact that $\sin (A+B)=\sin A \cos B+\cos A \sin B$ and the differentiation,obtain the sum formula for cosines.

Consider the function $f(x) = x \cos x - \sin x$. Identify the correct statement.