51.

If f(x) =xⁿ, n being a non-negative integer, then the values of n for which $\mathrm{f}\text{'}(\mathrm{\alpha } + \mathrm{\beta }) = \mathrm{f}\text{'}\left(\mathrm{\alpha }\right) + \mathrm{f}\text{'}\left(\mathrm{\beta }\right)$ for all $\mathrm{\alpha }, \mathrm{\beta } > 0$ is

• 1

• 2

• 0

• 5

If the line ax + by + c = 0, ab $\ne$ 0, is a tangent to the curve xy = 1- 2x, then

• a> 0, b < 0

• a>0, b> 0

• a< 0, b > 0

• a< 0, b < 0

The equation of the plane through (1, 2,- 3) and (2,- 2, 1) and parallel to X-axis is

• y - z + 1 = 0

• y - z - 1 = 0

• y + z - 1 = 0

• y + z + 1 = 0

Three lines are drawn from the origin O with direction cosines proportional to (L,-1, 1), (2,-3, 0) and (1, 0, 3). The three lines are

• not coplanar

• coplanar

• perpendicular to each other

• coincident

Consider the non-constant differentiable function f of one variable which obeys the relation $\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{y}\right)}$ f(x - y ). If f'(0)= p and f(y) . f'(5) = q, then f'(- 5) is

• $\frac{{\mathrm{p}}^2}{\mathrm{q}}$

• $\frac{\mathrm{q}}{\mathrm{p}}$

• $\frac{\mathrm{p}}{\mathrm{q}}$

• q

If f(x)=log₅(log₃(x)), then f'(e) is equal to

• e log_e(5)

• e log_e(3)

• $\frac{1}{{\mathrm{elog}}_{\mathrm{e}}\left(5\right)}$

• $\frac{1}{{\mathrm{elog}}_{\mathrm{e}}\left(3\right)}$

Let F(x)=e^x, G(x)=e^{- x} and H(x) = G(F(x)), where x is a real variable. Then, $\frac{\mathrm{dH}}{\mathrm{dx}}$ at x= 0 is

• 1

• - 1

• $- \frac{1}{\mathrm{e}}$

• $\text{- e}$

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

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

$\int \cos \left(\log \left(\mathrm{x}\right)\right)\mathrm{dx} = \mathrm{F}\left(\mathrm{x}\right) +\hspace{0.17em}\mathrm{C}$, where C is an arbitrary constant. Here, F(x) is equal to 

• $\mathrm{x}\left[\cos \left(\log \left(\mathrm{x}\right)\right) +\hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\mathrm{x}\left[\cos \left(\log \left(\mathrm{x}\right)\right) - \hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\frac{\mathrm{x}}{2}\left[\cos \left(\log \left(\mathrm{x}\right)\right) +\hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$

• $\frac{\mathrm{x}}{2}\left[\cos \left(\log \left(\mathrm{x}\right)\right) - \hspace{0.17em}\sin \left(\log \left(\mathrm{x}\right)\right)\right]$