If $\int e^x(1+x) \cdot \sec ^2(x e^x) \, dx = f(x) + \text{constant}$,then $f(x)$ is equal to

Electrons ejected from the surface of a metal,when light of certain frequency is incident on it,are stopped fully by a retarding potential of $3 \ V$. Photoelectric effect in this metallic surface begins at a frequency $6 \times 10^{14} \ s^{-1}$. The frequency of the incident light in $s^{-1}$ is : [Planck's constant $= 6.4 \times 10^{-34} \ J \ s$,charge on the electron $= 1.6 \times 10^{-19} \ C$]

Match the following columns:

Column-$I$	Column-$II$
$(A)$ Linnaeus	$(i)$ First to discover viruses
$(B)$ Ivanowsky	$(ii)$ First to discover lichens
$(C)$ Iyengar	$(iii)$ Father of Indian Bryology
$(D)$ Tulasne	$(iv)$ Father of Taxonomy
$(E)$ Shiv Ram Kashyap	$(v)$ Father of modern Phycology in India

Let $f$ be a non-zero real-valued continuous function satisfying $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in R$. If $f(2)=9$,then $f(6)$ is equal to

$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^\beta\end{array}\right] \\ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

If $\alpha, \beta, \gamma$ are the roots of $x^3+2x^2-3x-1=0$,then $\alpha^{-2}+\beta^{-2}+\gamma^{-2}=$