Which of the following statements regarding growth is incorrect?

If the area of the auxiliary circle of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b)$ is twice the area of the ellipse,then the eccentricity of the ellipse is

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?

The work functions for metals $A, B$ and $C$ are respectively $1.92\, eV$,$2.0\, eV$ and $5\, eV$. According to Einstein's photoelectric equation,the metals which will emit photoelectrons for a radiation of wavelength $4100\, \mathring{A}$ is/are:

$\lim _{x \rightarrow 0} \frac{\tan x - \sin x}{x^3}$ is equal to

If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to