If f(x + 2y, x - 2y) = xy, then f(x, y) is equal to

• $\frac{1}{4}\mathrm{xy}$

• $\frac{1}{4}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)$

• $\frac{1}{8}\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)$

• $\frac{1}{2}\left({\mathrm{x}}^2 + {\mathrm{y}}^2\right)$

The value of ${\int }_{- 2}^2\left(\mathrm{xcos}\left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right) + 1\right)d\mathrm{x}$

• 2

• 0

• - 2

• 4

The general solution of the differential equation

$\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} +\hspace{0.17em}8\frac{d\mathrm{y}}{d\mathrm{x}} + 16\mathrm{y} = 0$ is

• (A + B)e^5x

• (A + Bx)e^{- 4x}

• (A + Bx²)e^4x

• (A + Bx⁴)e^4x

If x² + y² = 4, then $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{x}$ is equal to

• 4

• 0

• 1

• - 1

$\int \frac{{\mathrm{x}}^3\mathrm{dx}}{1 + {\mathrm{x}}^8}$ is equal to

• $4{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

• $\frac{1}{4}{\tan }^{-1}\left({\mathrm{x}}^3\right) + \mathrm{C}$

• $\mathrm{x} +\hspace{0.17em}4{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

• ${\mathrm{x}}^2 +\hspace{0.17em}\frac{1}{4}{\tan }^{-1}\left({\mathrm{x}}^4\right) + \mathrm{C}$

${\int }_{\mathrm{\pi }}^{16\mathrm{\pi }}\left|\sin \left(\mathrm{x}\right)\right|d\mathrm{x}$ is equal to

• 0

• 32

• 30

• 28

67.

The degree and order of the differential equation

y = $\mathrm{x}{\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2 + {\left(\frac{d\mathrm{x}}{d\mathrm{y}}\right)}^2$ are respectively

• 1, 1

• 2, 1

• 4, 1

• 1, 4

$\int \frac{\cos \left(2\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $2\sin \left(\mathrm{x}\right) + \log \left|\sec \left(\mathrm{x}\right) +\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) - \log \left|\sec \left(\mathrm{x}\right) -\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) - \log \left|\sec \left(\mathrm{x}\right) +\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

• $2\sin \left(\mathrm{x}\right) + \log \left|\sec \left(\mathrm{x}\right) -\hspace{0.17em}\tan \left(\mathrm{x}\right)\right| + \mathrm{C}$

$\int \frac{{\sin }^8\left(\mathrm{x}\right) - {\cos }^8\left(\mathrm{x}\right)}{1 - 2{\sin }^2\left(\mathrm{x}\right){\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$

• $- \frac{1}{2}\sin \left(2\mathrm{x}\right) + \mathrm{C}$

• $\frac{1}{2}\sin \left(2\mathrm{x}\right) + \mathrm{C}$

• $\frac{1}{2}\sin \left(\mathrm{x}\right) + \mathrm{C}$

• $- \frac{1}{2}\sin \left(\mathrm{x}\right) + \mathrm{C}$

The general solution of the differential equation ${\log }_{\mathrm{e}}\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right) = \mathrm{x} + \mathrm{y}$ is

• e^x + e^{- y} = C

• e^x + e^y = C

• e^y + e^{- x} = C

• e^{- x} + e^{- y} = C