${\int }_0^{\mathrm{\pi }}\frac{\mathrm{\theta sin}\left(\mathrm{\theta }\right)}{1 + {\cos }^2\left(\mathrm{\theta }\right)}\mathrm{d\theta }$ is equal to

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{{\mathrm{\pi }}^2}{3}$

• ${\mathrm{\pi }}^2$

• $\frac{{\mathrm{\pi }}^2}{4}$

If $\int {\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)\mathrm{dx} = \mathrm{f}\left(\mathrm{x}\right) - \log \left(1 + {\mathrm{x}}^2\right)$ then f(x) is equal to

• 2xtan^-1(x)

• - 2xtan^-1(x)

• xtan^-1(x)

• - xtan^-1(x)

$\mathrm{If} \mathrm{dx} + \mathrm{dy} = (\mathrm{x} + \mathrm{y})\left(\mathrm{dx} - \mathrm{dy}\right), \mathrm{then} \log \left(\mathrm{x} + \mathrm{y}\right) = ?$

• x + y + c

• x + 2y + c

• x - y + c

• 2x + y + c

$\mathrm{If} {\mathrm{x}}^2\mathrm{y} - {\mathrm{x}}^3\frac{\mathrm{dy}}{\mathrm{dx}} = {\mathrm{y}}^4\mathrm{cosx}, \mathrm{then} {\mathrm{x}}^3{\mathrm{y}}^{- 3} \mathrm{is} \mathrm{equal} \mathrm{to}$

• sin(x)

• 2sin(x) + c

• - 3sin(x) + c

• 3cos(x) + c

Observe the following statements

$$\begin{gathered} \mathrm{I}. \mathrm{If} \mathrm{dy} +\hspace{0.17em}2\mathrm{xydx} = 2{\mathrm{e}}^{- {\mathrm{x}}^2}\mathrm{dx}, \mathrm{then} {\mathrm{ye}}^{{\mathrm{x}}^2} = 2\mathrm{x} + \mathrm{c}, \\ \mathrm{II}. \mathrm{If} {\mathrm{ye}}^{- {\mathrm{x}}^2} - 2\mathrm{x} = \mathrm{c}, \mathrm{then} \\ \mathrm{dx} = \left(2{\mathrm{e}}^{- {\mathrm{x}}^2} - 2\mathrm{xy}\right)\mathrm{dy} \\ \mathrm{which} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} \mathrm{statement} \end{gathered}$$

• Both I and II are true

• Neither I nor II is true

• I is false, But II is true

• I is true, But II is false

$\mathrm{If} \frac{\mathrm{dy}}{\mathrm{dx}} =\frac{ \mathrm{y} +\hspace{0.17em}\mathrm{xtan}\left({\displaystyle \frac{\mathrm{y}}{\mathrm{x}}}\right)}{\mathrm{x}}, \mathrm{then} \sin \left(\frac{\mathrm{y}}{\mathrm{x}}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• cx²

• cx

• cx³

• cx⁴

77.

The area (in square units) bounded by the curves y² = 4x and x² = 4y in the plane is

• $\frac{8}{3}$

• $\frac{16}{3}$

• $\frac{32}{3}$

• $\frac{64}{3}$

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{f}: \mathrm{C} \to \mathrm{C} \mathrm{defined}, \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{ax} + \mathrm{d}}{\mathrm{cx} + \mathrm{d}} \mathrm{for} \mathrm{x} \in \mathrm{C} \mathrm{where} \\ \mathrm{bd} \ne 0 \mathrm{reduces} \mathrm{to} \mathrm{a} \mathrm{constant} \mathrm{function}, \mathrm{if} \end{gathered}$$

• a = c

• b = d

• ad = bc

• ab = cd

$\mathrm{If} \frac{\tan \left(3\mathrm{A}\right)}{\tan \left(\mathrm{A}\right)} = \mathrm{a}, \mathrm{then} \frac{\sin \left(3\mathrm{A}\right)}{\sin \left(\mathrm{A}\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{2\mathrm{a}}{\mathrm{a} + 1}$

• $\frac{2\mathrm{a}}{\mathrm{a} - 1}$

• $\frac{\mathrm{a}}{\mathrm{a} + 1}$

• $\frac{\mathrm{a}}{\mathrm{a} - 1}$

The parabola with directrix x + 2y - 1 = 0 and focus (1, 0) is

• 4x² - 4xy + y² - 8x + 4y + 4 = 0
  
  

• 4x² + 4xy + y² - 8x + 4y + 4 = 0
  
  

• 4x² + 5xy + y² + 8x - 4y + 4 = 0
  
  

• 4x - 4xy + y - 8x - 4y + 4 = 0