The function f(x) = 1 - x³ - x⁵ is decreasing for :

• $1 \le \mathrm{x} \le 5$

• $\mathrm{x} \le 1$

• $\mathrm{x} \ge 1$

• all values of x

12.

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^{\mathrm{x}}\right)$ is equal to :

• $\log \left(\mathrm{x}\right)$

• $\log \left({\mathrm{e}}^{\mathrm{x}}\right)$

• ${\mathrm{x}}^{\mathrm{x}}\log \left(\mathrm{x}\right)$

• ${\mathrm{x}}^{\mathrm{x}}{\log }_{\mathrm{e}}\left(\mathrm{x}\right)$

If the displacements of a particle at time t is given by s² = at² + 2bt + c, then acceleration varies as :

• $\frac{1}{{\mathrm{s}}^2}$

• $\frac{1}{\mathrm{s}}$

• $\frac{1}{{\mathrm{s}}^3}$

• s³

If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle, is :

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

The function y = a(1 - cos(x)) is maximum when x is equal to

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $- \frac{\mathrm{\pi }}{2}$

• - $\frac{\mathrm{\pi }}{6}$

$If $ {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }$\mathrm{}$ then x + y + z is :

• xyz

• 0

• 1

• 2xyz

If $\mathrm{\theta } = {\sin }^{-1}\left[\sin \left(- 600°\right)\right]$, then one of the possible values of $\mathrm{\theta }$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2\mathrm{\pi }}{3}$

• $- \frac{2\mathrm{\pi }}{3}$

A force of magnitude 5 unit acting along the vector $2\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$ displaces the point of application from (1, 2, 3) to (5, 3, 7). Then the work done is :

• 50/7 unit

• 50/3 unit

• 25/3

• 25/4

An unit vector perpendicular to both $\hat{\mathrm{i}} + \hat{\mathrm{j}} \mathrm{and} \hat{\mathrm{j}} + \hat{\mathrm{k}}$ is :

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

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

• $\frac{\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}}{\sqrt{3}}$

• $\frac{\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}}{\sqrt{3}}$

The area of the triangle whose vertices are (1, 2, 3), (2, 5, -1) and (-1, 1, 2) is :

• 150 sq unit

• 145 sq unit

• $\frac{\sqrt{155}}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{155}{2} \mathrm{sq} \mathrm{unit}$