71.

If C is the reflection of A (2, 4) in x-axis and B is the reflection of C in y-axis, then | AB | is

• 20

• 2√5

• 4√5

• 4

The value of integral  is

• 1

• 2

• 0

• - 1

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} - \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{6}\right)\right| + \mathrm{c}$

• $\frac{1}{2}\log \left|\tan \left(\frac{\mathrm{x}}{4} + \frac{\mathrm{\pi }}{3}\right)\right| + \mathrm{c}$

If f(x) = f(a - x), then ${\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is equal to

• ${\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{{\mathrm{a}}^2}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $\frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

• $- \frac{\mathrm{a}}{2}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$

The value of ${\int }_0^{\infty }\frac{d\mathrm{x}}{\left({\mathrm{x}}^2 + 4\right)\left({\mathrm{x}}^2 + 9\right)}$ is

• $\frac{\mathrm{\pi }}{60}$

• $\frac{\mathrm{\pi }}{20}$

• $\frac{\mathrm{\pi }}{40}$

• $\frac{\mathrm{\pi }}{80}$

If I₁ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\sin }^2\left(\mathrm{x}\right)d\mathrm{x}$ and I₂ = ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}{\cos }^2\left(\mathrm{x}\right)d\mathrm{x}$, then

• I₁ = I₂

• I₁ < I₂

• I₁ > I₂

• I₂ = I₁ + $\frac{\mathrm{\pi }}{4}$

The general solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{e}}^{\mathrm{y} + \mathrm{x}} + {\mathrm{e}}^{\mathrm{y} - \mathrm{x}}$ is

• e^{- y} = e^x - e^{- x} + c

• e^{- y} = e^{- x} - e^x + c

• e^{- y} = e^x + e^- x + c

• e^y = e^x + e^{- x} + c

The integrating factor of the differential equation $\mathrm{xlog}\left(\mathrm{x}\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 2\log \left(\mathrm{x}\right)$ is given by

• e^x

• log(x)

• log(log(x))

• x

The order of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2}$ is

• 3

• 2

• 1

• 4

If ${\int }_{- 1}^4\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x} = 4 \mathrm{and} {\int }_2^4\left\{3 - \mathrm{f}\left(\mathrm{x}\right)\right\}d\mathrm{x} = 7$, then the value of

${\int }_{- 1}^2\mathrm{f}\left(\mathrm{x}\right)d\mathrm{x}$ is

• - 2

• 3

• 4

• 5