The equation of the plane passing through three non-collinear points $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ is :

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) = 0$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

The unit vector perpendicular to $\hat{\mathrm{i}} - \hat{\mathrm{j}}$ and coplanar with $\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$ and $2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ is :

• $\frac{2\hat{\mathrm{i}} - 5\hat{\mathrm{j}}}{\sqrt{29}}$

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

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}}}{\sqrt{2}}$

• $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

${\left(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right)}^2 + {\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right)}^2$ is equal to :

• ${\overrightarrow{\mathrm{a}}}^2 {\overrightarrow{\mathrm{b}}}^2$

• ${\overrightarrow{\mathrm{a}}}^2 + {\overrightarrow{\mathrm{b}}}^2$

• 1

• $2\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}$

$\int 32 {\mathrm{x}}^3 {\left(\log \left(\mathrm{x}\right)\right)}^2\mathrm{dx}$ is equal to

• 8x⁴(log(x))² + c

• ${\mathrm{x}}^4\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 4\left(\log \left(\mathrm{x}\right)\right) + 1\right\} + \mathrm{c}$

• $\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 4\left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{c}$

• $x^3\left\{8{\left(\log \left(\mathrm{x}\right)\right)}^2 - 2\left(\log \left(\mathrm{x}\right)\right)\right\} + \mathrm{c}$

$\int \frac{\cos \left(\mathrm{x}\right) - 1}{\sin \left(\mathrm{x}\right) + 1}{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to :

• $\frac{{\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)} + \mathrm{c}$

• $\mathrm{c} - \frac{{\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}$

• $\mathrm{c} - \frac{{\mathrm{e}}^{\mathrm{x}}}{1 + \sin \left(\mathrm{x}\right)}$

• $\mathrm{c} - \frac{{\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}$

If $\int \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \mathrm{g}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{c}, \mathrm{then} \int {\mathrm{f}}^{-1}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to :

• xf^-1(x) + c

• f(g^-1(x)) + c

• xf^-1(x) - g(f^-1(x)) + c

• g^-1(x) + c

27.

The value of ${\int }_1^2\frac{\mathrm{dx}}{\mathrm{x}\left(1 + {\mathrm{x}}^4\right)}$ is :

• $\frac{1}{4}\log \left(\frac{17}{32}\right)$

• $\frac{1}{4}\log \left(\frac{32}{17}\right)$

• $\log \left(\frac{17}{2}\right)$

• $\frac{1}{4}\log \left(\frac{17}{2}\right)$

The value of the integral ${\int }_{\mathrm{a}}^{\mathrm{b}}\frac{\sqrt{\mathrm{x}}\mathrm{dx}}{\sqrt{\mathrm{x}} + \sqrt{\mathrm{a} + \mathrm{b} - \mathrm{x}}}$ is :

• $\mathrm{\pi }$

• $\frac{1}{2}\left(\mathrm{b} - \mathrm{a}\right)$

• $\mathrm{\pi }/2$

• b - a

The area bounded by y = log(x), x-axis and ordinates x = 1, x = 2 is

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

• log(2/e)

• log(4/e)

• log(4)

The area of the segment of a circle of radius a subtending an angle of 2$\mathrm{\alpha }$ at the centre is :

• ${\mathrm{a}}^2\left(\mathrm{\alpha } + \frac{1}{2}\sin \left(2\mathrm{\alpha }\right)\right)$

• $\frac{1}{2}{\mathrm{a}}^2\sin \left(2\mathrm{\alpha }\right)$

• ${\mathrm{a}}^2\left(\mathrm{\alpha } - \frac{1}{2}\sin \left(2\mathrm{\alpha }\right)\right)$

• ${\mathrm{a}}^2\mathrm{\alpha }$