Let y = $\left(\frac{3^{\mathrm{x}} - 1}{3^{\mathrm{x}} + 1}\right)\sin \left(\mathrm{x}\right) + {\log }_{\mathrm{e}}\left(1 + \mathrm{x}\right), \mathrm{x} > - 1$. Then, at x = 0, $\frac{\mathrm{dy}}{\mathrm{dx}}$ equals

• 1

• 0

• - 1

• - 2

Maximum value of the function f(x) = $\frac{\mathrm{x}}{8} + \frac{2}{\mathrm{x}}$ on the interval [1, 6] is

• 1

• $\frac{9}{8}$

• $\frac{13}{12}$

• $\frac{17}{8}$

For  $- \frac{\mathrm{\pi }}{2} < \mathrm{x} < \frac{3\mathrm{\pi }}{2}$, the avlue of $\frac{\mathrm{d}}{\mathrm{dx}}\left({\tan }^{-1}\left(\frac{\cos \left(\mathrm{x}\right)}{1 + \sin \left(\mathrm{x}\right)}\right)\right)$ is equal to

• $\frac{1}{2}$

• - $\frac{1}{2}$

• 1

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

If f is a real-valued differentiable function such that f(x)f' (x) < 0 for all real x, then

• f(x) must be an increasing function

• f(x) must be a decreasing function

• $\left|\mathrm{f}\left(\mathrm{x}\right)\right|$ must be an increasing function

• $\left|\mathrm{f}\left(\mathrm{x}\right)\right|$ must be a decreasing function

Rolle's theorem is applicable in the interval [- 2, 2] for the function

• f(x) = x³

• f(x) = 4x⁴

• f(x) = 2x³ + 3

• f(x) = $\mathrm{\pi }\left|\mathrm{x}\right|$

The solution of $25\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} - 10\frac{d\mathrm{y}}{d\mathrm{x}} + \mathrm{y} = 0$, y(0) = 1, y(1) = $2{\mathrm{e}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ is

• $\mathrm{y} = {\mathrm{e}}^{5\mathrm{x}} + {\mathrm{e}}^{- 5\mathrm{x}}$

• $\mathrm{y} = \left(1 + \mathrm{x}\right){\mathrm{e}}^{5\mathrm{x}}$

• $\mathrm{y} = \left(1 + x\right){\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

• $\mathrm{y} = \left(1 + \mathrm{x}\right){\mathrm{e}}^{\raisebox{1ex}{$- \mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$

The system of linear equations $\mathrm{\lambda }$x + y + z = 3,  x - y - 2z = 6, - x + y + z = µ has

• infinite number of solutions for $\mathrm{\lambda } \ne$-1 and all µ

• infinite number of solutions for $\mathrm{\lambda }$ = - 1 and $\mathrm{\mu }$ = 3

• no solution for $\mathrm{\lambda } \ne - 1$

• unique solution for $\mathrm{\lambda }$ = - 1 and $\mathrm{\mu }$ = 3

68.

If f(x) and g(x) are twice differentiable functions on (0, 3) satisfying f"(x) = g''(c), f'(1) = 4g'(D) = 6, f(2) = 3, g(2) = 9, then f(1) - g(1) is

• 4

• - 4

• 0

• - 2

Two coins are available, one fair and the other two headed. Choose a coin and toss it once; assume that the unbiased coin is chosen with probability $\frac{3}{4}$. Given that the outcome is head, 4 the probability that the two headed coin was chosen, is

• $\frac{3}{5}$

• $\frac{2}{5}$

• $\frac{1}{5}$

• $\frac{2}{7}$

The general solution of the differential equation

$\frac{d\mathrm{y}}{d\mathrm{x}} = \frac{\mathrm{x} +\hspace{0.17em}\mathrm{y} +\hspace{0.17em}1}{2\mathrm{x} +\hspace{0.17em}2\mathrm{y} +\hspace{0.17em}1}$ is

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| + 3\mathrm{x} + 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| - 3\mathrm{x} + 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| - 3\mathrm{x} - 6\mathrm{y} = \mathrm{C}$

• ${\log }_{\mathrm{e}}\left|3\mathrm{x} + 3\mathrm{y} + 2\right| + 3\mathrm{x} - 6\mathrm{y} = \mathrm{C}$