$\int \frac{{\mathrm{x}}^{\mathrm{e} - 1} + {\mathrm{e}}^{\mathrm{x} - 1}}{{\mathrm{x}}^{\mathrm{e}} + {\mathrm{e}}^{\mathrm{x}}}\mathrm{dx}$

• $\frac{- 1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $- \mathrm{elog}\left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\frac{1}{\mathrm{e}}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

• $\mathrm{e}\log \left|{\mathrm{x}}^{\mathrm{e}} +\hspace{0.17em}{\mathrm{e}}^{\mathrm{x}}\right| + \mathrm{C}$

If a, b and c are non-zero vectors such that a and b are not perpendicular to each other, then the vector r which is perpendicular to a and satisfying r x b = c x b  is

• $\frac{\left(\mathrm{a} \times \mathrm{b}\right) \times \mathrm{c}}{\mathrm{c} \mathrm{a}}$

• $\frac{\mathrm{b} \times \left(\mathrm{a} \times \mathrm{c}\right)}{\mathrm{b} \mathrm{c}}$

• $\frac{\left(\mathrm{b} \times \mathrm{c}\right) \times \mathrm{a}}{\mathrm{a} \mathrm{b}}$

• $\frac{\left(\mathrm{c} \times \mathrm{b}\right) \times \mathrm{a}}{\mathrm{ac}}$

$\int \frac{\mathrm{x}+ \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx} = ?$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{C}$

• $\mathrm{xsin}\left(\frac{\mathrm{x}}{2}\right) + \cos \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \sec \left(\frac{\mathrm{x}}{2}\right)\hspace{0.17em}+ \mathrm{C}$

• $\mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \sec \left(\frac{\mathrm{x}}{2}\right)\hspace{0.17em}+ \mathrm{C}$

$\mathrm{If} {\mathrm{I}}_{\mathrm{n}} = \int \frac{\sin \left(\mathrm{nx}\right)}{\cos \left(\mathrm{x}\right)}\mathrm{dx}, \mathrm{then} {\mathrm{I}}_{\mathrm{n}} = ?$

• $\frac{ - 2}{\mathrm{n} - 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{2}{\mathrm{n} - 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} + {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{ - 2}{\mathrm{n} + 1}\sin \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

• $\frac{ - 2}{\mathrm{n} + 1}\cos \left(\mathrm{n} - 1\right)\mathrm{x} - {\mathrm{I}}_{\mathrm{n} - 2}$

The differential equation corresponding to the family of circles inthe plane touching the Y-axis at the origin, is

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2 - {\mathrm{x}}^2}{2\mathrm{xy}}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2\mathrm{xy}}{{\mathrm{x}}^2 + {\mathrm{y}}^2}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{2\mathrm{xy}}$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2 + {\mathrm{x}}^2}{2\mathrm{xy}}$

$$\begin{gathered} \mathrm{The} \mathrm{triad} (\mathrm{x}, \mathrm{y}, \mathrm{z}) \mathrm{of} \mathrm{real} \mathrm{number} \mathrm{such} \mathrm{that} \\ \left(3\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right) = \left(2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}\right)\mathrm{x} + \left(\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)\mathrm{y} + \left( - 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}\right)\mathrm{z} \mathrm{is} \end{gathered}$$

• (- 2, 5, 3)

• (2, - 5, 3)

• (2, 5, 3)

• (2, 5, - 3)

If the volume of the tetrahedron formed by the coterminous edges a, b and c is 4, then the volume of the parallelopiped formed by the coterminous edges a x b, b x c and c x a is

• 576

• 48

• 16

• 144

The incentre of the triangle formed by the straight lines $\mathrm{y} = \sqrt{3}\mathrm{x}, \mathrm{y} = - \sqrt{3}\mathrm{x} \mathrm{and} \mathrm{y} = 3 \mathrm{is}$  and y = 3 is

• (0, 2)

• (1, 2)

• (2, 0)

• (2, 1)

The tangents to the parabola y = 4ax from an external point P make angles ${\mathrm{\theta }}_1 \mathrm{and} {\mathrm{\theta }}_2$, with the axis of the parabola. Such that $\tan \left({\mathrm{\theta }}_1\right) + \tan \left({\mathrm{\theta }}_2\right) = \mathrm{b}$ where b is constant. Then P lies on

• y = x + b

• y + x = b

• $\mathrm{y} = \frac{\mathrm{x}}{\mathrm{b}}$

• y = bx

80.

The points on the straight line 3x - 4y + 1 = 0 which are at a distance of 5 units from the point (3, 2) are

• $\left( - 2, - \frac{7}{4}\right)\left( - 3, \frac{ - 5}{2}\right)$

• $\left(4, \frac{11}{4}\right)\left( - 1, - 1\right)$

• $\left(1, \frac{1}{2}\right)\left(2, \frac{5}{4}\right)$

• $\left(7, 5\right),\left( - 1, - 1\right)$