$\int {\csc }^4\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\cot \left(\mathrm{x}\right) + \frac{{\cot }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $\tan \left(\mathrm{x}\right) + \frac{{\tan }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $- \cot \left(\mathrm{x}\right) - \frac{{\cot }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

• $- \tan \left(\mathrm{x}\right) - \frac{{\tan }^3\left(\mathrm{x}\right)}{3} +\hspace{0.17em}\mathrm{C}$

${\int }_0^{1000}{\mathrm{e}}^{\mathrm{x} - \left[\mathrm{x}\right]}\mathrm{dx}$ is

• e¹⁰⁰⁰ - 1

• $\frac{{\mathrm{e}}^{1000} - 1}{\mathrm{e} - 1}$

• 1000(e - 1)

• $\frac{\mathrm{e} - 1}{1000}$

23.

The area bounded by the curves y² = 4a²(x - 1) and lines x = 1 and y = 4a is

• 4a² sq unit

• $\frac{16\mathrm{a}}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{16{\mathrm{a}}^2}{3} \mathrm{sq} \mathrm{unit}$

• None of these

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

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

• (x + y)² = a²x + C

• (x + y)² = 2a²x + C

• None of the above

If A, B and C are the vertices of a triangle whose position vectors are $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ respectively and G is the centroid of the $∆$AABC , then $\overrightarrow{\mathrm{GA}} + \overrightarrow{\mathrm{GB}} + \overrightarrow{\mathrm{GC}}$ is

• $\overrightarrow{0}$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$

• $\frac{\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}}{3}$

• $\frac{\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}}}{3}$

A plane which passes through the point (3, 2, 0) and the line $\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - 6}{5} = \frac{3 - 4}{4}$ is

• x - y + z = 1

• x + y + z = 5

• x + 2y - z = 0

• 2x - y + z = 5

Two bodies of masses m and 4m are moving with equal momentum. The ratio of their KE is

• 1 : 4

• 4 : 1

• 1 : 1

• 1 : 2

A particle is projected from the top oftower 5 m high and at the same moment another particle is projected upward from the bottom of the tower with a speed of 10 m/s, meet at distance h' from the top oftower, then h is equal to

• 1.20 m

• 2.5 m

• 1.25 m

• None of these

$\int \frac{1 + \mathrm{x} + \sqrt{\mathrm{x} +\hspace{0.17em}{\mathrm{x}}^2}}{\sqrt{\mathrm{x}} + \sqrt{1 + \mathrm{x}}}\mathrm{dx}$ is equal to

• $\frac{1}{2}\sqrt{1 + \mathrm{x}} + \mathrm{C}$

• $\frac{2}{3}{\left(1 + \mathrm{x}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} + \mathrm{C}$

• $\sqrt{1 + \mathrm{x}} +\hspace{0.17em}\mathrm{C}$

• 2(1 + x)^3/2 + C