For the reaction $CH_{4(g)} + 2O_{2(g)} \rightleftharpoons CO_{2(g)} + 2H_2O_{(l)} \, \Delta H_r = -170.8 \, kJ \, mol^{-1}$,which of the following statements is not true?

If the mean and variance of a binomial variate $X$ are $8$ and $4$ respectively, then $P(X < 3)$ equals to

An object with a mass $10 \ kg$ moves at a constant velocity of $10 \ m/s$. $A$ constant force then acts for $4 \ s$ on the object and gives it a speed of $2 \ m/s$ in the opposite direction. The impulse acting on the object is (in $N \cdot s$):

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?

If $\int\limits_0^{\frac{\pi }{3}} {\frac{{\tan \theta }}{{\sqrt {2k \sec \theta } }}} \,d\theta = 1 - \frac{1}{\sqrt{2}}, (k > 0),$ then the value of $k$ is

The number of rational terms in the expansion of $(3^{1/4} + 7^{1/6})^{144}$ is