${\int }_1^3\frac{\cos \left(\log \left(\mathrm{x}\right)\right)}{\mathrm{x}}\mathrm{dx}$ is equal to

• 1

• $\cos \left(\log \left(3\right)\right)$

• $\sin \left(\log \left(3\right)\right)$

• $\frac{\mathrm{\pi }}{4}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\sqrt{1 + \sin \left(2\mathrm{x}\right)}}\mathrm{dx}$ is equal to

• $- \frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

Probability P(A) = 0. 7, P(B) = 0.4, P(A $\cap$ B) = 0.3, then P(A $\cap$ B') is equal to

• 0.1

• 0.3

• 0.2

• 0.4

${\int }_1^2{\mathrm{e}}^{\mathrm{x}}\left(\frac{1}{\mathrm{x}} - \frac{1}{{\mathrm{x}}^2}\right)\mathrm{dx}$ is equal to

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2}$

• $\frac{{\mathrm{e}}^2}{2} - \mathrm{e}$

• $\frac{{\mathrm{e}}^2}{2} + \mathrm{e}$

• $\frac{{\mathrm{e}}^2}{2} - 2$

Solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = 3^{\mathrm{x} +\hspace{0.17em}\mathrm{y}}$ is

• 3^{x + y} = c

• 3^x + 3^y = c

• 3^{x - y}

• 3^x + 3^{- y} = c

Area of rhombus is ..., where diagonals are $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 5\hat{\mathrm{k}} \mathrm{and} \overrightarrow{\mathrm{b}} = - \hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}$

• $\sqrt{21.5}$

• $\sqrt{31.5}$

• $\sqrt{28.5}$

• $\sqrt{38.5}$

27.

Degree and order of the differential equation $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$ are respectively

• 1, 2

• 2, 1

• 2, 2

• 1, 1

Let ABCD be a parallelogram whose diagonals intersect at P and O be the origin, then $\overrightarrow{\mathrm{OA}} + \overrightarrow{\mathrm{OB}} + \overrightarrow{\mathrm{OC}} + \overrightarrow{\mathrm{OD}}$

• $\overrightarrow{\mathrm{OP}}$

• $2\overrightarrow{\mathrm{OP}}$

• $3\overrightarrow{\mathrm{OP}}$

• $4\overrightarrow{\mathrm{OP}}$