If the position vectors of the vertices A, B and C are 6i, 6j and k respectively w.r.t. origin 0, then the volume of the tetrahedron OABC is

• 6

• 3

• $\frac{1}{6}$

• $\frac{1}{3}$

If three vectors 2i - j - k, i + 2j - 3k and 3i + $\mathrm{\lambda }$j + 5k are coplanar, then the value of $\mathrm{\lambda }$ is

• - 4

• - 2

• - 1

• - 8

13.

The vector perpendicular to the vectors 4i - j + 3k and - 2i + j - 2k whose magnitude is 9

• 3i + 6j - 6k

• 3i - 6j + 6k

• - 3i + 6j + 6k

• None of the above

The area of the region bounded by the curves x² + y² = 8 and y² = 2x is

• $2\mathrm{\pi } + \frac{1}{3}$

• $\mathrm{\pi } + \frac{1}{3}$

• $2\mathrm{\pi } + \frac{4}{3}$

• $\mathrm{\pi } + \frac{4}{3}$

The value of ${\int }_0^{\mathrm{\pi }}\log \left(1 +\hspace{0.17em}\cos \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $- \frac{\mathrm{\pi }}{2}\log \left(2\right)$

• $\mathrm{\pi log}\left(\frac{1}{2}\right)$

• $\mathrm{\pi log}\left(2\right)$

• $\frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 2}}\log \left(2\right)$

The value of ${\int }_3^4\sqrt{\left(4 - \mathrm{x}\right)\left(\mathrm{x} - 3\right)}\mathrm{dx}$ is

• $\frac{\mathrm{\pi }}{16}$

• $\frac{\mathrm{\pi }}{8}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

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

• $\frac{1}{\mathrm{n}}\log \left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• $\log \left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{n}}\log \left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) + \mathrm{C}$

• $\log \left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

The value of $\int \cos \left(\log \left(\mathrm{x}\right)\right)\mathrm{dx}$ is

• $\frac{1}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) + \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

• $\frac{\mathrm{x}}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) + \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

• $\frac{\mathrm{x}}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) - \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

• $\frac{1}{2}\left\{\sin \left(\log \left(\mathrm{x}\right)\right) - \cos \left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{C}$

The value of $\int {\mathrm{e}}^{\mathrm{x}}\left[\frac{1 + \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\right]\mathrm{dx}$ is

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{x}}\sec \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\sec \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{x}}\tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

The value of $\int \frac{1}{3\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + 3}\mathrm{dx}$ is

• $\log \left(\frac{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $\frac{1}{2}\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C

• $2\log \left(\frac{2\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}{\tan \left({\displaystyle \frac{\mathrm{x}}{2}}\right) + 1}\right)$ + C