For a binomial variate $X$ with parameters $n=5$ and $p=\frac{3}{4}$,if $\alpha=\frac{1}{9} P(X \geq 3)$ and $\beta=P(X \leq 2)$,then $256(\beta-\alpha)=$

Two radioactive materials $X_1$ and $X_2$ have decay constants $10 \lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei,then the ratio of the number of nuclei of $X_1$ to that of $X_2$ will be $1 / e$ after a time:

If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \dots$ is equal to:

Assertion $(A)$: The boiling points of noble gases increase from $He$ to $Xe$.
Reason $(R)$: The interatomic van der Waals' attractive forces increase from $He$ to $Xe$.
The correct answer is:

If $f(x) = \frac{\cos^2 x + \sin^4 x}{\sin^2 x + \cos^4 x}$, for $x \in R$, then $f(2002)$ is equal to

If the position vectors of the vertices of a triangle are $2 \hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3 \hat{j}-5 \hat{k}$, and $3 \hat{i}-4 \hat{j}-4 \hat{k}$, then the triangle is