If the vectors $\overrightarrow{\mathrm{a}} +\hspace{0.17em}\mathrm{\lambda }\overrightarrow{\mathrm{b}} +\hspace{0.17em}3\overrightarrow{\mathrm{c}}$, $- 2\overrightarrow{\mathrm{a}} +\hspace{0.17em}3\overrightarrow{\mathrm{b}} -\hspace{0.17em}4\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} -\hspace{0.17em}3\overrightarrow{\mathrm{b}} +\hspace{0.17em}5\overrightarrow{\mathrm{c}}$ are coplanar, then the value of $\mathrm{\lambda }$ is

• 2

• - 1

• 1

• - 2

If $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, $\left|\overrightarrow{\mathrm{a}}\right| = 3, \left|\overrightarrow{\mathrm{b}}\right| = 5, \left|\overrightarrow{\mathrm{c}}\right| = 7$, then anle between $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}}$ is

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$5\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

If the vectors $\overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{a}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{a}}^2\hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}}= \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{b}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{b}}^2\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} +\hspace{0.17em}\mathrm{c}\hat{\mathrm{j}} +\hspace{0.17em}{\mathrm{c}}^2\hat{\mathrm{k}}$ are three non-coplanar vectors and $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2& 1 + {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& 1 + {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& 1 + {\mathrm{c}}^3\end{array}\right| = 0$, then the value of abc is

• 0

• 1

• 2

• - 1

Let $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$, $\overrightarrow{\mathrm{b}} = \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$  be three vectors. A vector in the plane of $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ whose projection on $\overrightarrow{\mathrm{a}}$ is magnitude $\sqrt{\frac{2}{3}}$, is

• $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 5\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 5\hat{\mathrm{k}}$

If the constant forces $\mathrm{}$ 2\hat{\mathrm{i}} - 5\hat{\mathrm{j}} + 6\hat{\mathrm{k}}$\stackrel{}{}$ and $$$- \hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$act on a particle due to which it is displaced from a point A (4,- 3, - 2) to a point B (6, 1,- 3), then the work done by the forces is

• 15 unit

• 9 unit

• - 15 unit

• - 9 unit

$\int {\cos }^3\left(\mathrm{x}\right) . {\mathrm{e}}^{\log \left(\sin \left(\mathrm{x}\right)\right)}\mathrm{dx}$ is equal to

• $- \frac{{\displaystyle {\sin }^4}\left(\mathrm{x}\right)}{4} + \mathrm{c}$

• $- \frac{{\displaystyle {\cos }^4\left(\mathrm{x}\right)}}{4} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\sin \left(\mathrm{x}\right)}}{4} +\hspace{0.17em}\mathrm{c}$

• None of the above

27.

The value of ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\cos \left(3\mathrm{x}\right) + 1}{2\cos \left(\mathrm{x}\right) - 1}\mathrm{dx}$ is

• 2

• 1

• $\frac{1}{2}$

• 0

The value of ${\int }_0^1{\tan }^{-1}\left(\frac{2\mathrm{x} - 1}{1 + \mathrm{x} - {\mathrm{x}}^2}\right)\mathrm{dx}$ is

• 1

• 0

• - 1

• None of the above

By the application of Simpson's one - third rule for numerical integration, with two subintervals, the value of ${\int }_0^1\frac{\mathrm{dx}}{1 +\hspace{0.17em}\mathrm{x}}$ is

• $\frac{17}{36}$

• $\frac{17}{25}$

• $\frac{25}{36}$

• $\frac{17}{24}$