The focal length of a mirror is given by . In finding the values of u and v, the errors are equal to ' 'p'. Then, the relative error in f is

• $\mathrm{p}\left(\frac{1}{\mathrm{u}} + \frac{1}{\mathrm{v}}\right)$

• $\frac{\mathrm{p}}{2}\left(\frac{{\displaystyle 1}}{{\displaystyle \mathrm{u}}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{v}}}\right)$

• $\mathrm{p}\left(\frac{{\displaystyle 1}}{{\displaystyle \mathrm{u}}} - \frac{{\displaystyle 1}}{{\displaystyle \mathrm{v}}}\right)$

$\mathrm{If} \mathrm{u} = \log \left({\mathrm{x}}^3 +\hspace{0.17em}{\mathrm{y}}^3 + {\mathrm{z}}^3 - 3\mathrm{xyz}\right), \mathrm{then} \left(\mathrm{x} + \mathrm{y} + \mathrm{z}\right)\left({\mathrm{u}}_{\mathrm{x}} + {\mathrm{u}}_{\mathrm{y}} \hspace{0.17em}+{\mathrm{u}}_{\mathrm{z}}\right) = ?$

• 0

• x - y + z

• 2

• 3

$\int {\mathrm{e}}^{\mathrm{x}}\left(\frac{2 + \sin \left(2\mathrm{x}\right)}{1 + \cos \left(2\mathrm{x}\right)}\right)\mathrm{dx} = ?$

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

• $2{\mathrm{e}}^{\mathrm{x}}{\sec }^2\left(\mathrm{x}\right) +\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\cos \left(2\mathrm{x}\right) + \mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\tan \left(\mathrm{x}\right) + \mathrm{C}$

$\mathrm{If} \int \frac{\mathrm{x} - \sin \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx} = \mathrm{xtan}\left(\frac{\mathrm{x}}{2}\right) + \mathrm{plog}\left|\sec \left(\frac{\mathrm{x}}{2}\right)\right| + \mathrm{C}, \mathrm{then} \mathrm{p} = ?$

• - 4

• 4

• 2

• - 2

75.

$\mathrm{If} \int \frac{\mathrm{dx}}{\mathrm{xlog}\left(\mathrm{x} - 2\right)\log \left(\mathrm{x} - 3\right)} = \mathrm{I} + \mathrm{C}, \mathrm{then} \mathrm{I} =?$

• $\frac{1}{\mathrm{x}}\log \left|\frac{\log \left(\mathrm{x} - 3\right)}{\log \left(\mathrm{x} - 2\right)}\right|$

• $\log \left|\frac{\log \left(\mathrm{x} - 3\right)}{\log \left(\mathrm{x} - 2\right)}\right|$

• $\log \left|\frac{\log \left(\mathrm{x} - 2\right)}{\log \left(\mathrm{x} - 3\right)}\right|$

• $\log \left|\log \left(\mathrm{x} - 3\right)\log \left(\mathrm{x} - 2\right)\right|$

$\mathrm{If} {\int }_0^{\mathrm{b}}\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2} = {\int }_{\mathrm{b}}^{\infty }\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}, \mathrm{then} \mathrm{b} = ?$

• ${\tan }^{-1}\left(\frac{1}{3}\right)$

• $\frac{\sqrt{3}}{2}$

• $\sqrt{2}$

• 1

The approximate value of ${\int }_1^3\frac{\mathrm{dx}}{2 + 3\mathrm{x}}$ using Simpson's rule and dividing the interval [1, 3] into two equal parts is

• $\frac{1}{3}\log \left(\frac{11}{5}\right)$

• $\frac{107}{110}$

• $\frac{22}{110}$

• $\frac{119}{440}$

An integrating factor of the equation $\left(1 + \mathrm{y} +\hspace{0.17em}{\mathrm{x}}^2\mathrm{y}\right)\mathrm{dx} + \left(\mathrm{x} + {\mathrm{x}}^3\right)\mathrm{dy} = 0$ is

• e^x

• x²

• $\frac{1}{\mathrm{x}}$

• x

$$\begin{gathered} \mathrm{The} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{differential} \mathrm{equation} \\ \frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{ytan}\left(2\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}}\sec \left(2\mathrm{x}\right) \mathrm{is} \end{gathered}$$

• $\mathrm{ysin}\left(2\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}} + \mathrm{C}$

• $\mathrm{ycos}\left(2\mathrm{x}\right) = \int {\mathrm{e}}^{\mathrm{x}} . 1\mathrm{dx} + \mathrm{C}$

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

• $\mathrm{ycos}\left(2\mathrm{x}\right) + e^x = \mathrm{C}$

$\mathrm{If} \frac{\left(1 + \mathrm{i}\right)\mathrm{x} - \mathrm{i}}{2 + \mathrm{i}} + \frac{\left(1 + 2\mathrm{i}\right)\mathrm{y} + \mathrm{i}}{2 - \mathrm{i}} = 1, \mathrm{then} \left(\mathrm{x}, \mathrm{y}\right) =?$

• $\left(\frac{7}{3}, \frac{- 7}{15}\right)$

• $\left(\frac{7}{3}, \frac{7}{15}\right)$

• $\left(\frac{7}{5}, \frac{- 7}{15}\right)$

• $\left(\frac{7}{5}, \frac{7}{15}\right)$