If f''(0) = k, $\mathrm{k} \ne 0$, then the value of $\underset{\mathrm{x}\to 0}{\lim } \frac{2\mathrm{f}\left(\mathrm{x}\right) - 3\mathrm{f}\left(2\mathrm{x}\right) +\hspace{0.17em}\mathrm{f}\left(4\mathrm{x}\right)}{{\mathrm{x}}^2}$ is

• k

• 2k

• 3k

• 4k

12.

If  y = ${\mathrm{e}}^{{\mathrm{msin}}^{- 1}\left(\mathrm{x}\right)}$ then $\left(1 - {\mathrm{x}}^2\right)\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - \mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} - \mathrm{ky} = 0$, where k is equal to

• m²

• 2

• - 1

• - m²

Solution of ${\left(\mathrm{x} + \mathrm{y}\right)}^2\frac{d\mathrm{y}}{d\mathrm{x}} = {\mathrm{a}}^2$ ( 'a' being a constant) is

• $\frac{\left(\mathrm{x} + \mathrm{y}\right)}{\mathrm{a}} = \tan \left(\frac{\mathrm{y} + \mathrm{C}}{\mathrm{a}}\right)$, C is an arbitrary

• $\mathrm{xy} = \mathrm{atan}\left(\mathrm{Cx}\right)$, C is an arbitrary

• $\frac{\mathrm{x}}{\mathrm{a}} = \tan \left(\frac{\mathrm{y}}{\mathrm{C}}\right)$, C is an arbitrary

• $\mathrm{xy} = \tan (\mathrm{x} + \mathrm{C})$, C is an arbitrary

The integrating factor of the first order differential equation

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

• e^x

• $\mathrm{x} - \frac{1}{\mathrm{x}}$

• $\mathrm{x} + \frac{1}{\mathrm{x}}$

• $\frac{1}{{\mathrm{x}}^2}$

Let a and B be the roots of x² + x + 1 = 0. If n be a positive integer, then ${\mathrm{\alpha }}^{\mathrm{n}} + {\mathrm{\beta }}^{\mathrm{n}}$ is

• $2\cos \left(\frac{2\mathrm{n\pi }}{3}\right)$

• $2\sin \left(\frac{2\mathrm{n\pi }}{3}\right)$

• $2\cos \left(\frac{\mathrm{n\pi }}{3}\right)$

• $2\sin \left(\frac{\mathrm{n\pi }}{3}\right)$

For real x, the greatest value of $\frac{{\mathrm{x}}^2 + 2\mathrm{x} + 4}{2{\mathrm{x}}^2 + 4\mathrm{x} + 9}$ is

• 1

• - 1

• $\frac{1}{2}$

• $\frac{1}{4}$

If the solution of the differential equation $\mathrm{x}\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = {\mathrm{xe}}^{\mathrm{x}} \mathrm{be} \mathrm{xy} = {\mathrm{e}}^{\mathrm{x}}\mathrm{\varphi }\left(\mathrm{x}\right) + \mathrm{C}$, then $\mathrm{\varphi }\left(\mathrm{x}\right)$ is equal to

• x + 1

• x - 1

• 1 - x

• x

The order of the differential equation of all parabolas whose axis of symmetry along X-axis is

• 2

• 3

• 1

• None of these

General solution of $\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}} + {\mathrm{by}}^2 = \mathrm{acos}\left(\mathrm{x}\right), 0 < \mathrm{x} < 1$ is

• ${\mathrm{y}}^2 = 2\mathrm{a}\left(2\mathrm{bsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

• $\left(4{\mathrm{b}}^2 + 1\right){\mathrm{y}}^2 = 2\mathrm{a}\left(\sin \left(\mathrm{x}\right) + 2\mathrm{bcos}\left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

• $\left(4{\mathrm{b}}^2 + 1\right){\mathrm{y}}^2 = 2\mathrm{a}\left(\sin \left(\mathrm{x}\right) + 2\mathrm{bcos}\left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{2\mathrm{bx}}$

• ${\mathrm{y}}^2 = 2\mathrm{a}\left(2\mathrm{bsin}\left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) + {\mathrm{ce}}^{- 2\mathrm{bx}}$

The integrating factor of the differential equation

$\left(1 + {\mathrm{x}}^2\right)\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}$ is

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

• 1 + x²

• ${\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{x}\right)}$

• ${\log }_{\mathrm{e}}\left(1 + {\mathrm{x}}^2\right)$