What are the order and degree, respectively, of the differential equation ${\left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right)}^2 = {\mathrm{y}}^4 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^5 ?$

• 4, 5

• 2, 3

• 3, 2

• 5, 4

The differential equation of the family of curves y = pcos (ax) + qsin(ax), where p, q are arbitrary constants, is

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - {\mathrm{a}}^2\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \mathrm{ay} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{ay} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\mathrm{a}}^2\mathrm{y} = 0$

$\mathrm{The} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{curve} \mathrm{passing} \mathrm{through} \mathrm{the} \mathrm{point} ( - 1, - 2) \mathrm{which} \mathrm{satisfies} \frac{\mathrm{dy}}{\mathrm{dx}} = - {\mathrm{x}}^2 - \frac{1}{{\mathrm{x}}^3}, \mathrm{is}$

• $17{\mathrm{x}}^2\mathrm{y} - 6{\mathrm{x}}^2 + 3{\mathrm{x}}^5 - 2 = 0$

• $6{\mathrm{x}}^2\mathrm{y} + 17{\mathrm{x}}^2 + 2{\mathrm{x}}^5 - 3 = 0$

• $6\mathrm{xy} - 2{\mathrm{x}}^2 + 17{\mathrm{x}}^5 + 3 = 0$

• $17{\mathrm{x}}^2\mathrm{y} + 6\mathrm{xy} - 3{\mathrm{x}}^5 +\hspace{0.17em}5 = 0$

What is the order of the differential equation whose solution is 

$\mathrm{y} = \mathrm{acos}\left(\mathrm{x}\right) + \mathrm{bsin}\left(\mathrm{x}\right) +{\mathrm{ce}}^{ - \mathrm{x}} =\mathrm{d}, \mathrm{where} \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{and} \mathrm{d} \mathrm{are} \mathrm{arbitrary} \mathrm{constants} ?$

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \ln \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \mathrm{ax} + \mathrm{by} ? \end{gathered}$$

• ${\mathrm{ax}}^{\mathrm{ax} } + {\mathrm{be}}^{\mathrm{by}} = \mathrm{c}$

• $\frac{1}{\mathrm{a}}{\mathrm{e}}^{\mathrm{ax}} + \frac{1}{\mathrm{b}}{\mathrm{e}}^{\mathrm{by}} = 0$

• ${\mathrm{ae}}^{\mathrm{ax}} + {\mathrm{be}}^{ - \mathrm{bx}} = \mathrm{c}$

• $\frac{1}{\mathrm{b}}{\mathrm{e}}^{\mathrm{ax}} +\hspace{0.17em}\frac{1}{\mathrm{b}}{\mathrm{e}}^{ - \mathrm{by}} = \mathrm{c}$

$\mathrm{If} \mathrm{\mu } = {\mathrm{e}}^{\mathrm{ax}}\sin \left(\mathrm{bx}\right) \mathrm{and} \mathrm{\nu } = {\mathrm{e}}^{\mathrm{ax}}\cos \left(\mathrm{bx}\right), \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{\mu }\frac{\mathrm{du}}{\mathrm{dx}} + \mathrm{v}\frac{\mathrm{dv}}{\mathrm{dx}} = ?$

• ae^2ax

• $\left({\mathrm{a}}^2 + {\mathrm{b}}^2\right){\mathrm{e}}^{\mathrm{ax}}$

• abe^2ax

• $\left(\mathrm{a} + \mathrm{b}\right){\mathrm{e}}^{\mathrm{ax}}$

$\mathrm{If} \mathrm{y} = \sin \left(\ln \left(\mathrm{x}\right)\right), \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} = 0$

• ${\mathrm{x}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

• ${\mathrm{x}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = 0$

$$\begin{gathered} \mathrm{What} \mathrm{is} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \frac{\mathrm{dx}}{\mathrm{dy}} = \frac{\mathrm{x} + \mathrm{y} + 1}{\mathrm{x} + \mathrm{y} - 1} ? \end{gathered}$$

• y - x + ln(x + y) = c

• y + x + 2ln(x + y) = c

• y - x + 2ln(x + y) = c

• y + x + 2ln(x + y) = c

19.

$\mathrm{What} \mathrm{is} \underset{\mathrm{x}\to \frac{\mathrm{\pi }}{6}}{\lim }\frac{2{\sin }^2\left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) - 1}{2{\sin }^2\left(\mathrm{x}\right) - 3\sin \left(\mathrm{x}\right) + 1} = ?$

• $- \frac{1}{2}$

• $- \frac{1}{3}$

•  - 2

•  - 3

$\mathrm{What} \mathrm{is} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dy}}^2} = ?$

• $- {\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{ - 1}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{ - 3}$

• ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{ - 1}{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{ - 2}$

• $- \left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{ - 3}$

• ${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}^{ - 1}$