If $y=\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3 \cos^2 x-5) \cos^3 x$,then $96 y'(\frac{\pi}{6})$ is equal to :

Let $f(x) = x|x|$,$g(x) = \sin x$ and $h(x) = (g \circ f)(x)$. Then

Two curves $C_1 : y = x^2 - 3$ and $C_2 : y = kx^2, k \in R$,intersect each other at two different points. The tangent drawn to $C_2$ at one of the points of intersection $A \equiv (a, y_1), (a > 0)$ meets $C_1$ again at $B(1, y_2), (y_1 \neq y_2)$. The value of '$a$' is

If a function $f(x)$ defined by $f(x)=\begin{cases} a e^{x}+b e^{-x}, & -1 \leq x<1 \\ c x^{2}, & 1 \leq x \leq 3 \\ a x^{2}+2 c x, & 3 < x \leq 4 \end{cases}$ is continuous for some $a, b, c \in R$ and $f'(0)+f'(2)=e$,then the value of $a$ is:

If $y = \log(\tan(x/2)) + \sin^{-1}(\cos x)$,then $dy/dx$ is

Which of the following is not true?