$$\begin{gathered} \mathrm{Let} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{be} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation}, \mathrm{xy}\text{'} – \mathrm{y} = {\mathrm{x}}^2(\mathrm{xcosx} + \mathrm{sinx}), \mathrm{x} > 0. \\ \mathrm{If} \mathrm{y}\left(\mathrm{\pi }\right) = \mathrm{\pi }, \mathrm{then} \mathrm{y}\text{'}\text{'}\left(\frac{\mathrm{\pi }}{2}\right) + \mathrm{y}\left(\frac{\mathrm{\pi }}{2}\right) \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• $1 + \frac{\mathrm{\pi }}{2}$

• $1 + \frac{\mathrm{\pi }}{2} + \frac{{\mathrm{\pi }}^2}{4}$

• $2 + \frac{\mathrm{\pi }}{2} +\hspace{0.17em}\frac{{\mathrm{\pi }}^2}{4}$

• $2 + \frac{\mathrm{\pi }}{2}$

$\mathrm{If} \left(\mathrm{a} + \sqrt{2}\mathrm{bcos}\left(\mathrm{x}\right)\right)\left(\mathrm{a} - \sqrt{2}\mathrm{bcos}\left(\mathrm{y}\right)\right) = {\mathrm{a}}^2 - {\mathrm{b}}^2, \mathrm{where} \mathrm{a} > \mathrm{b}, \mathrm{then} \frac{\mathrm{dx}}{\mathrm{dy}} \mathrm{at} \left(\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{4}\right) \mathrm{is} :$

• $\frac{2\mathrm{a} + \mathrm{b}}{2\mathrm{a} - \mathrm{b}}$

• $\frac{\mathrm{a} + \mathrm{b}}{\mathrm{a} - \mathrm{b}}$

• $\frac{\mathrm{a} - \mathrm{b}}{\mathrm{a} + \mathrm{b}}$

• $\frac{\mathrm{a} - 2\mathrm{b}}{\mathrm{a} + 2\mathrm{b}}$

403.

Suppose a differentiable function f(x) satisfies the identity f(x + y) = f(x) + f(y) + xy² + x²y, for all real x and y. If

$\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x}} = 1$, then f'(3) is equal to :

$$\begin{gathered} \mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{y} + 3\mathrm{x}}{{\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right)} + 3 = 0 \mathrm{is} \\ (\mathrm{where} \mathrm{C} \mathrm{is} \mathrm{a} \mathrm{constant} \mathrm{of} \mathrm{integration}) \end{gathered}$$

• $\mathrm{x} - 2{\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right) = \mathrm{C}$

• $\mathrm{x} - {\log }_{\mathrm{e}}\left(\mathrm{y} = 3\mathrm{x}\right) = \mathrm{C}$

• $\mathrm{y} + 3\mathrm{x} - \frac{1}{2}{\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)}^2 = \mathrm{C}$

• $\mathrm{y} - \frac{1}{2}{\left({\log }_{\mathrm{e}}\left(\mathrm{y} + 3\mathrm{x}\right)\right)}^2 = \mathrm{C}$

$$\begin{gathered} \mathrm{If} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{is} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \frac{5 + {\mathrm{e}}^{\mathrm{x}}}{2 + \mathrm{y}}\frac{\mathrm{dy}}{\mathrm{dx}} {\mathrm{e}}^{\mathrm{x}} = 0 \mathrm{satisfying} \\ \mathrm{y}\left(0\right) = 1, \mathrm{then} \mathrm{a} \mathrm{value} \mathrm{of} {\mathrm{ylog}}_{\mathrm{e}}\left(13\right) \mathrm{is} :\hspace{0.17em} \end{gathered}$$

•  - 1

• 0

• 2

• 1

$$\begin{gathered} \mathrm{Let} \mathrm{y} = \mathrm{y}\left(\mathrm{x}\right) \mathrm{be} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \cos \left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{ysin}\left(\mathrm{x}\right) = \sin \left(2\mathrm{x}\right), \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right) \\ \mathrm{If} \mathrm{y}\left(\mathrm{\pi }/3\right) = 0, \mathrm{then} \mathrm{y}\left(\mathrm{\pi }/4\right) = ? \end{gathered}$$

• $\frac{1}{\sqrt{2}} - 1$

• 2 - $\sqrt{2}$

• $\sqrt{2} - 2$

• 2 + $\sqrt{2}$

$$\begin{gathered} \mathrm{If} \mathrm{y} = \left(\frac{2}{\mathrm{\pi }}\mathrm{x} - 1\right)\csc \left(\mathrm{x}\right) \mathrm{is} \mathrm{the} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation}, \\ \frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{p}\left(\mathrm{x}\right)\mathrm{y} = \frac{2}{\mathrm{\pi }}\csc \left(\mathrm{x}\right), 0 < \mathrm{x} < \frac{\mathrm{\pi }}{2} \mathrm{then} \mathrm{the} \mathrm{function} \mathrm{p}\left(\mathrm{x}\right) \mathrm{is} \mathrm{equal} \mathrm{to}: \end{gathered}$$

• $\tan \left(\mathrm{x}\right)$

• $\csc \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$

• $\sec \left(\mathrm{x}\right)$

Let z = x + iy be a non-zero complex number such that z² = i |z|², where i = $\sqrt{ - 1}$, then z lies on the:

• real axis

• line y = x

• line y = - x

• imaginary axis