41.

The position vectors of three non-collinear points A, Band C are a, b and c, respectively. The perpendicular distance of point C from the straight line AB is

• $\frac{\left|\mathrm{b} \times \mathrm{c}\right|}{\left|\mathrm{b} - \mathrm{c}\right|}$

• $\frac{\left|\mathrm{a} \times \mathrm{b}\right|}{\left|\mathrm{b} - \mathrm{a}\right|}$

• $\frac{\left|\mathrm{c} \times \mathrm{a}\right|}{\left|\mathrm{c} - \mathrm{a}\right|}$

• $\frac{\left|\mathrm{b} \times \mathrm{c} + \mathrm{c} \times \mathrm{a} + \mathrm{a} \times \mathrm{b}\right|}{\left|\mathrm{b} - \mathrm{a}\right|}$

If $\int \sin \left\{2{\tan }^{-1}\left(\sqrt{\frac{1 - \mathrm{x}}{1 + \mathrm{x}}}\right)\right\}\mathrm{dx} = {\mathrm{Asin}}^{-1}\left(\mathrm{x}\right) + \mathrm{Bx}\sqrt{1 - {\mathrm{x}}^2} + \mathrm{C}$, then A + B is equal to

• 10

• $\frac{1}{2}$

• 1

• $- \frac{1}{2}$

Three concurrent edges of a parallelopiped are given by

$$\begin{gathered} \mathrm{a} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}} \\ \mathrm{b} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}} \\ \mathrm{and} \mathrm{c} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}} \end{gathered}$$

The volume of the parallelopiped is

• 14 cu units

• 20 cu units

• 25 cu units

• 60 cu units

The curve, for which the area of the triangle formed by X-axis, the tangent line at any point P and line OP is equal to a, is given by

• $\mathrm{y} = \mathrm{x} - {\mathrm{Cx}}^2$

• $\mathrm{x} = \mathrm{Cy} \pm \frac{{\mathrm{a}}^2}{\mathrm{y}}$

• $\mathrm{y} = \mathrm{Cy} \pm \frac{{\mathrm{a}}^2}{\mathrm{x}}$

• None of these

Solution of the equation ${\cos }^2\left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \left(\tan \left(2\mathrm{x}\right)\right)\mathrm{y} = {\cos }^2\left(\mathrm{x}\right), \left|\mathrm{x}\right| < \frac{\mathrm{\pi }}{4}$, where $\mathrm{y}\left(\frac{\mathrm{\pi }}{6}\right) = \frac{3\sqrt{3}}{8}$, is given by 

• $\mathrm{y}\frac{\tan \left(2\mathrm{x}\right)}{1 - {\tan }^2\left(\mathrm{x}\right)} = 0$

• $\mathrm{y}\left(1 - {\tan }^2\left(\mathrm{x}\right)\right) = \mathrm{C}$

• $\mathrm{y} = \sin \left(2\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{y} = \frac{1}{2} \frac{\sin \left(2\mathrm{x}\right)}{1 - {\tan }^2\left(\mathrm{x}\right)}$

The equation of the curve through the point (1, 0), whose slope is $\frac{\mathrm{y} - 1}{{\mathrm{x}}^2 + \mathrm{x}}$, is

• 2x(y - 1) + x + 1 = 0

• (x + 1)(y - 1) + 2x = 0

• x(y - 1)(x + 1) + 2 = 0

• None of these

If A(- 1, 3, 2),B (2, 3, 5) and C(3, 5, - 2) are vertices of a $∆$ABC, then angles of $∆\mathrm{ABC}$ are

• $\angle \mathrm{A} = 90°, \angle \mathrm{B} = 30°, \angle \mathrm{C} = 60°$

• $\angle \mathrm{A} = \angle \mathrm{B} = \angle \mathrm{C} = 90°$

• $\angle \mathrm{A} = \angle \mathrm{B} = 45°, \angle \mathrm{C} = 90°$

• None of the above

${\int }_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{e}$}\right.}^1\left|\log \left(\mathrm{x}\right)\right|\mathrm{dx}$ is equal to

• $\frac{1}{\mathrm{e}}$

• e

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

• None of the above

$\underset{\mathrm{x}\to 0}{\lim }\frac{{\int }_0^{{\mathrm{x}}^2}\sin \left(\sqrt{\mathrm{t}}\right)\mathrm{dt}}{{\mathrm{x}}^3}$ is equal to

• $\frac{2}{3}$

• $\frac{1}{3}$

• 0

• $\infty$

If a, band care three non-coplanar vectors, then [a x b b x c c x a] is equal to

• [a b c]³

• [a b c]²

• 0

• None of these