41.

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

• 1

• 1 + sin(1)

• 1 - sin(1)

• (sin(1) - 1

Distance of the point (2, 3, 4) from the plane 3x - 6 y + 2z + 11 = 0 is

• 0

• 1

• 2

• 3

The three lines of a triangle are given by (x² - y²)(2x + 3y - 6) = 0. If the point (- 2, $\mathrm{\lambda }$) lies inside and ($\mathrm{\mu }$, 1) lies outside the triangle, then

• $\mathrm{\lambda } \in \left(1, \frac{10}{3}\right), \mathrm{\mu } \in \left(- 3, 5\right)$

• $\mathrm{\lambda } \in \left(2, \frac{10}{3}\right), \mathrm{\mu } \in \left(- 1, 1\right)$

• $\mathrm{\lambda } \in \left(- 1, \frac{9}{2}\right), \mathrm{\mu } \in \left(- 2, \frac{10}{3}\right)$

• None of the above

If f(x) = ${\int }_2^{\mathrm{x}}\frac{\mathrm{dt}}{\sqrt{1 + {\mathrm{t}}^4}}$ and g is the inverse of f. Then, the value of g'(0) is

• 1

• 17

• $\sqrt{17}$

• None of the above

The value of $\hat{\mathrm{i}} . \left(\hat{\mathrm{j}} \times \hat{\mathrm{k}}\right) + \hat{\mathrm{j}} . \left(\hat{\mathrm{k}} \times \hat{\mathrm{i}}\right) + \hat{\mathrm{k}} . \left(\hat{\mathrm{i}} \times \hat{\mathrm{j}}\right)$ is

• 0

• 1

• 3

• - 3

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \mathrm{ytan}\left(\mathrm{x}\right) - {\mathrm{y}}^2\sec \left(\mathrm{x}\right)$ is

• $\tan \left(\mathrm{x}\right) = \left(\mathrm{C} + \sec \left(\mathrm{x}\right)\right)\mathrm{y}$

• $\sec \left(\mathrm{y}\right) = \left(\mathrm{C} + \tan \left(\mathrm{y}\right)\right)\mathrm{x}$

• $\sec \left(\mathrm{x}\right) = \left(\mathrm{C} + \tan \left(\mathrm{x}\right)\right)\mathrm{y}$

• $\tan \left(\mathrm{y}\right) = \left(\mathrm{C} + \sec \left(\mathrm{x}\right)\right)\mathrm{x}$

${\int }_0^{100}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}\mathrm{dx}$ is equal to

• 50(e - 1)

• 75(e - 1)

• 90(e - 1)

• 100(e - 1)

The values of $\mathrm{\lambda }$, such that (x, y, z) if (0, 0, 0) and $\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$x + $\left(3\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$y + $\left(- 4\hat{\mathrm{i}} + 5\hat{\mathrm{j}}\right)$ are

• 0, 1

• - 1, 1

• - 1, 0

• - 2, 0

By Simpson's $\frac{1}{3}$rd rule, the approximate value of the integral ${\int }_1^2{\mathrm{e}}^{\raisebox{1ex}{$- \mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\mathrm{dx}$ using four intervals, is

• 0.377

• 0.487

• 0.477

• 0.387

A variable plane is at a constant distance p from the origin O and meets the axes at A, B and C. The locus of the centroid of the tetrahedron OABC is

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{1}{{\mathrm{p}}^2}$

• $\frac{1}{{\mathrm{x}}^2} + \frac{1}{{\mathrm{y}}^2} + \frac{1}{{\mathrm{z}}^2} = \frac{16}{{\mathrm{p}}^2}$

• x² + y² + z² = 16p²

• x² + y² + z² = p²