The charge on $As_2S_3$ sol is due to the adsorbed

Let $\overrightarrow{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$.
Assertion $(A)$: The identity $|\overrightarrow{a} \times \hat{i}|^2+|\overrightarrow{a} \times \hat{j}|^2+|\overrightarrow{a} \times \hat{k}|^2=2|\overrightarrow{a}|^2$ holds for $\overrightarrow{a}$.
Reason $(R)$: $\overrightarrow{a} \times \hat{i}=a_3 \hat{j}-a_2 \hat{k}$, $\overrightarrow{a} \times \hat{j}=a_1 \hat{k}-a_3 \hat{i}$, and $\overrightarrow{a} \times \hat{k}=a_2 \hat{i}-a_1 \hat{j}$.
Which of the following is correct?

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

The image of the point $(4, -13)$ with respect to the line $5x + y + 6 = 0$ is

$A$ car travels from point $A$ to point $B$ with a speed of $40 \ km/h$ and returns from $B$ to $A$ with a speed of $60 \ km/h$. What is the average speed of the car for the entire journey (in $km/h$)?

If $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$,then $x \in$