21.

For a positive integer $\mathrm{n}, {\left(1 + \frac{1}{\mathrm{x}}\right)}^{\mathrm{n}}$ is expanded in increasing powers of x. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12 , then n is equal to _____

If the variance of the terms in an increasing A.P. b₁, b₂, b₃, ... , b₁₁ is 90, then the common difference of this A.P. is

Let the position vectors of points 'A' and 'B' be $\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}} \mathrm{and} 2\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$, respectively. A point 'P' divides the line segment AB internally in the ratio $\mathrm{\lambda }$ : 1 ($\mathrm{\lambda }$ > 0). If O is the origin and $\overrightarrow{\mathrm{OB}} . \overrightarrow{\mathrm{OP}} - 3\overrightarrow{\mathrm{OA}} \times {\overrightarrow{\mathrm{OP}}}^2 = 6$ then $\mathrm{\lambda }$ is equal to

Let [t] denote the greatest integer less than or equal to t. Then the value of ${\int }_1^2\left|2\mathrm{x} - \left[3\mathrm{x}\right]\right|d\mathrm{x}$ is

$\mathrm{If} \mathrm{y} =\sum _{\mathrm{k} = 1}^6 {\cos }^{-1}\left\{\frac{3}{5}\cos \left(\mathrm{kx}\right) - \frac{4}{5}\sin \left(\mathrm{kx}\right)\right\} \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{at} \mathrm{x} = 0 \mathrm{is}$