$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

Let $f(x)$ be a polynomial of degree $6$ in $x$,in which the coefficient of $x^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If $\lim_{x \rightarrow 0} \frac{f(x)}{x^{3}}=1$,then $5 \cdot f(2)$ is equal to .............

The number of points, at which the function $f(x) = \max\{6x, 2+3x^2\} + |x-1| |\cos(x^2 - 1/4)|, x \in (-\pi, \pi)$, is not differentiable, is ————

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

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi / 4$ meets the curve again at the point $Q$,then $|PQ|$ is equal to

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow R$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow R$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$,where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in R$. Then
$(A)$ $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$
$(B)$ $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$
$(C)$ $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$
$(D)$ $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$