If the displacement, velocity and acceleration of a particle at time t be x, v and f respectively, then which one is true ?

• $\mathrm{f} = {\mathrm{v}}^3\frac{d^2\mathrm{t}}{d{\mathrm{x}}^2}$

• $\mathrm{f} = - {\mathrm{v}}^3\frac{d^2\mathrm{t}}{d{\mathrm{x}}^2}$

• $\mathrm{f} = {\mathrm{v}}^2\frac{d^2\mathrm{t}}{d{\mathrm{x}}^2}$

• $\mathrm{f} = - {\mathrm{v}}^2\frac{d^2\mathrm{t}}{d{\mathrm{x}}^2}$

The displacement x of a particle at time t is given by x = At² + Bt + C where A, B, C are constants and v is velocity of a particle, then the value of 4Ax - v² is

• 4AC + B²

• 4AC - B²

• 2AC - B²

• 2AC + B²

The displacement of a particle at time t is x,  where x = t⁴ - kt³. If the velocity of the particle at time t = 2 is minimum, then 

• k = 4

• k = - 4

• k = 8

• k = - 8

$\int \frac{\log \left(\sqrt{\mathrm{x}}\right)}{3\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{c}$

• $\frac{2}{3}{\left(\log \left(\sqrt{\mathrm{x}}\right)\right)}^2 + \mathrm{c}$

• $\frac{2}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

• $\frac{1}{3}{\left(\log \left(\mathrm{x}\right)\right)}^2 + \mathrm{c}$

$\int {\mathrm{e}}^{\mathrm{x}}\left(\frac{2}{\mathrm{x}} - \frac{2}{{\mathrm{x}}^2}\right)\mathrm{dx}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{2{\mathrm{x}}^2} + \mathrm{c}$

• $\frac{2{\mathrm{e}}^{\mathrm{x}}}{\mathrm{x}} + \mathrm{c}$

• $\frac{2{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^2} + \mathrm{c}$

The value of the integral $\int \frac{\mathrm{dx}}{{\left({\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}}\right)}^2}$

• $\frac{1}{2}\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $\frac{1}{2}\left({\mathrm{e}}^{- 2\mathrm{x}} +\hspace{0.17em}1\right) + \mathrm{c}$

• $- \frac{1}{2}{\left({\mathrm{e}}^{2\mathrm{x}} +\hspace{0.17em}1\right)}^{- 1} + \mathrm{c}$

• $\frac{1}{4}\left({\mathrm{e}}^{2\mathrm{x}} -\hspace{0.17em}1\right) + \mathrm{c}$

67.

$\int \sqrt{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

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

• $2\sqrt{2}\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em} \mathrm{c}$

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

• $\sqrt{2}\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em} \mathrm{c}$

The general solution of the differential equation

$100\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - 20\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 0$ is

• y = (c₁ + c₂x)e^x

• y = (c₁ + c₂x)e^- x

• $\mathrm{y} = \left({\mathrm{c}}_1 + {\mathrm{c}}_2\mathrm{x}\right){\mathrm{e}}^{\frac{\mathrm{x}}{10}}$

• y = c₁e^x + c₂e^{- x}

If y'' - 3y' + 2y = 0 where y(0) = 1, y'(0) = 0, then the value of y at x = log(2) is

• 1

• - 1

• 2

• 0

The degree of the differential equation $\mathrm{x} = 1 + \left(\frac{d\mathrm{y}}{d\mathrm{x}}\right) + \frac{1}{2!}{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + \frac{1}{3!}{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^3 + ...$

• 3

• 2

• 1

• not defined