Let f(x) = $$\begin{gathered} \left\{\mathrm{\alpha }\left(\mathrm{x}\right)\sin \left(\frac{\mathrm{\pi x}}{2}\right) \mathrm{for} \mathrm{x} \ne 2 \\ 1 \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$

where $\mathrm{\alpha }\left(\mathrm{x}\right)$ is such that $\underset{\mathrm{x}\to 0}{\lim }\left|\mathrm{\alpha }\left(\mathrm{x}\right)\right| = \infty$.

Then the function f(x) is contonuous at x = 0 if $\mathrm{\alpha }\left(\mathrm{x}\right)$ is chosen as

• $\frac{2}{\mathrm{\pi x}}$

• $\frac{1}{{\mathrm{x}}^2}$

• $\frac{2}{{\mathrm{\pi x}}^2}$

   

• $\frac{1}{\mathrm{x}}$

The points of the curve y = x³ + x - 2 at which its tangent are parallel to the straight line y = 4x - 1 are

• (2, 7), (- 2, - 11)

• (0, 2), (2^1/3, 2^1/3)

• (- 2^1/3, - 2^1/3), (0, - 4)

• (1, 0), (- 1, - 4)

The equation of the normal to the curve y = - $\sqrt{\mathrm{x}}$ + 2 at the point of its intersection with the bisector of the first quadrant is

• 4x - y + 16 = 0

• 4x - y = 16

• 2x - y - 1 = 0

• 2x - y + 1 = 0

Let the equation ofa curve is given in implicit form as y = $\tan \left(\mathrm{x} + \mathrm{y}\right)$. Then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ in terms of y is

• $\frac{2\left(1 + {\mathrm{y}}^2\right)}{{\mathrm{y}}^6}$

• $\frac{- 2\left(1 + {\mathrm{y}}^2\right)}{{\mathrm{y}}^6}$

• $\frac{- 2\left(1 + {\mathrm{y}}^2\right)}{{\mathrm{y}}^5}$

• $\frac{2{\left(1 + {\mathrm{y}}^2\right)}^2}{{\mathrm{y}}^5}$

The angle at which the curve y = x² and the curve x = $\frac{5}{3}\cos \left(\mathrm{t}\right)$, y = $\frac{5}{4}\sin \left(\mathrm{t}\right)$ intersect is

• ${\tan }^{-1}\left(\frac{2}{41}\right)$

• ${\tan }^{-1}\left(\frac{41}{2}\right)$

• $- {\tan }^{-1}\left(\frac{2}{41}\right)$

• $2{\tan }^{-1}\left(\frac{41}{2}\right)$

The maximum value of the function $\mathrm{y} = 2\tan \left(\mathrm{x}\right) - {\tan }^2\left(\mathrm{x}\right)$ over $\left[0, \frac{\mathrm{\pi }}{2}\right]$ is

• $\infty$

• 1

• 3

• 2

17.

The function f(x) = ${\mathrm{xtan}}^{-1}\left(\frac{1}{\mathrm{x}}\right) \mathrm{for} \mathrm{x} \ne 0$, f(0) = 0 is

• differentiable at x = 0

• neither continuous at x = 0 nor differentiable at x = 0

• not continuous at x = 0

• continuous at x = 0 but not differentiable at x = 0

The values of a and b for which the function y = alog_e(x ) + bx² + x, has extremum at the points x₁ = 1 and x₂ = 2 are

• $\mathrm{a} = \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = - \frac{1}{6}$

• $\mathrm{a} = - \frac{2}{3}, \mathrm{b} = \frac{1}{6}$

• $\mathrm{a} = - \frac{1}{3}, \mathrm{b} = - \frac{1}{6}$

The tangent to the graph of a continuous function y = f(x) at the point with abscissa x = a forms with the X-axis an angle of $\frac{\mathrm{\pi }}{3}$ and at the point with abscissa x = b an angle of $\frac{\mathrm{\pi }}{4}$, then what the value of integral ${\int }_{\mathrm{a}}^{\mathrm{b}}\left\{\mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)\right\}\mathrm{dx}$

(where f'(x) the derivative off w.r.t. xwhich is assumed to be continuous and similarly f"(x) the double derivative of f w.r.t. x)

• e^b + $\sqrt{3}{\mathrm{e}}^{\mathrm{a}}$

• ${\mathrm{e}}^{\mathrm{b}} - \sqrt{3}{\mathrm{e}}^{\mathrm{a}}$

• ${\mathrm{e}}^{\mathrm{b}} - \sqrt{3{\mathrm{e}}^{\mathrm{a}}}$

• $- {\mathrm{e}}^{\mathrm{b}} - \sqrt{3{\mathrm{e}}^{\mathrm{a}}}$

A closed figure S is bounded by the hyperbola x² - y² = a² and the straight line x = a + h; (h > 0, a > 0). This closed figure is rotated about the X-axis. Then, the volume of the solid ofrevolution is

• ${\mathrm{\pi h}}^2\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{6}\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{3}\left(3\mathrm{a} + \mathrm{h}\right)$

• $\frac{{\mathrm{\pi h}}^2}{2}\left(3\mathrm{a} + \mathrm{h}\right)$