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

$A$ parallel plate capacitor of capacity $C_0$ is charged to a potential $V_0$. $(i)$ The energy stored in the capacitor when the battery is disconnected and the plate separation is doubled is $E_1$. (ii) The energy stored in the capacitor when the charging battery is kept connected and the separation between the capacitor plates is doubled is $E_2$. Then,the value of $E_1 / E_2$ is:

The maximum and minimum values of the function $f: R \rightarrow R$ defined by $f(x)=5 \cos x+3 \cos \left(x+\frac{\pi}{3}\right)+8$ for all $x \in R$,are respectively

Diborane reacts with ammonia under different conditions to give a variety of products. Which one among the following is not formed in these reactions?

Assertion : $HClO_4$ is a stronger acid than $HClO_3$.
Reason : Oxidation state of $Cl$ in $HClO_4$ is $+7$ and in $HClO_3$,it is $+5$.

For which of the following pairs of complexes is the crystal field splitting energy $\Delta_0$ for $II > \Delta_0$ for $I$?

$I$	$II$
$(a). [Cr(H_2O)_6]^{2+}$	$[Cr(H_2O)_6]^{3+}$
$(b). [Fe(H_2O)_6]^{3+}$	$[Fe(CN)_6]^{3-}$
$(c). [Fe(CN)_6]^{3-}$	$[Ru(CN)_6]^{3-}$
$(d). [NiF_6]^{4-}$	$[NiF_6]^{2-}$