If f(x) = $$\begin{gathered} \left\{\frac{{\mathrm{x}}^2 - 9}{\mathrm{x} - 3}, \mathrm{if} \mathrm{x} \ne 3 \\ 2\mathrm{x} + \mathrm{k}, \mathrm{otherwise}\right\} \end{gathered}$$ is continuous at x = 3, then k is equal to :

• 3

• 0

• - 6

• $\frac{1}{6}$

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{x}}^{\mathrm{x}}\right)$ is equal to :

• $\log \left(\mathrm{x}\right)$

• $\log \left({\mathrm{e}}^{\mathrm{x}}\right)$

• ${\mathrm{x}}^{\mathrm{x}}\log \left(\mathrm{x}\right)$

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

If the displacements of a particle at time t is given by s² = at² + 2bt + c, then acceleration varies as :

• $\frac{1}{{\mathrm{s}}^2}$

• $\frac{1}{\mathrm{s}}$

• $\frac{1}{{\mathrm{s}}^3}$

• s³

Let f(x) be twice differentiable such that f''(x) = - f(x), f'(x) = g(x), where f'(x) and f''(x) represent the first and second derivatives of f(x) respectively. Also, if h(x) = [f(x)]² + [g(x)]² and h(S) = 5, then h(10) is equal to :

• 3

• 10

• 13

• 5

775.

If a particle is moving such that the velocity acquired is proportional to the square root of the distance covered, then its acceleration is :

• a constant

• $\propto {\mathrm{s}}^2$

• $\propto \frac{1}{{\mathrm{s}}^2}$

• $\propto \mathrm{s}$

If f(x) = $$\begin{gathered} \left\{\frac{2^{\mathrm{x}} - 1}{\sqrt{1 + \mathrm{x}} - 1}, - 1 \le \mathrm{x} < \infty , \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous everywhere, then k is equal to:

• $\frac{1}{2}\log \left(2\right)$

• $\log \left(4\right)$

• $\log \left(8\right)$

• $\log \left(2\right)$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to :

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $- \frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $- \frac{\mathrm{y}}{\mathrm{x}}$

If g(x) = min (x, x) where x is a real number, then :

• g(x) is an increasing function

• g(x) is a decreasing function

• g(x) is a constant function

• g(x) is a continuous function except at x = 0

If y = 2^x . 3^{2x - 1}, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to :

• $\left(\log \left(2\right)\right)\left(\log \left(3\right)\right)$

• $\left(\log \left(18\right)\right)$

• $\left(\log {\left(18\right)}^2\right){\mathrm{y}}^2$

• $\left(\log \left(18\right)\right)\mathrm{y}$

If y = x + x² + x³ + ... to $\infty$ where $\left|\mathrm{x}\right| < 1$, then for $\left|\mathrm{y}\right| < 1$, $\frac{\mathrm{dx}}{\mathrm{dy}}$ is equal to :

• y + y² + y³ + ... to $\infty$

• 1 - y + y² - y³ + ... to $\infty$

• 1 - 2y + 3y² - ... to $\infty$

• 1 + 2y + 3y² + ... to $\infty$