$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \left[\frac{4}{5}\right] \mathrm{for} \\ \mathrm{x} \in \mathrm{R}, \mathrm{where} \left[\mathrm{y}\right] \mathrm{denotes} \mathrm{the} \mathrm{greatest} \mathrm{integer} \\ \mathrm{not} \mathrm{exceeding} \mathrm{y}, \mathrm{then} \left\{\mathrm{f}\left(\mathrm{x}\right) : \left|\mathrm{x}\right| < 71\right\} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• $\left(0, \infty \right)$

• $\left(1, \infty \right)$

• $\left(4, \infty \right)$

• $\left(5, \infty \right)$

The condition that f(x) = a + bx² + cx + d has no extreme value, is

• ${\mathrm{b}}^2 > 3\mathrm{ac}$

• ${\mathrm{b}}^2 = 4\mathrm{ac}$

• ${\mathrm{b}}^2 = 3\mathrm{ac}$

• ${\mathrm{b}}^2 < 3\mathrm{ac}$

If there is an error of ± 0.04cm in the measurement of the diameter of a sphere, then the approximate percentage error in its volume, when the radius is 10cm, is

• $\pm 1.2$

• $\pm 0 . 06$

• $\pm 0 . 006$

• $\pm 0 . 6$

If x² + y² = 25, then log₅[max(3x + 4y)] is

• 2

• 3

• 4

• 5

If the tangent to the curve y = x + siny at a point (a, b) is parallel to the line joining $\left(0, \frac{3}{2}\right) \mathrm{and} \left(\frac{1}{2}, 2\right)$ then

• $\left|\mathrm{b} - \mathrm{a}\right| = 1$

• $\mathrm{b} = \frac{\mathrm{\pi }}{2} + \mathrm{a}$

• $\left|\mathrm{a} +\mathrm{b}\right| = 1$

• b = a

$\mathrm{The} \mathrm{derivative} \mathrm{of} {\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} - 1}{\mathrm{x}}\right) \mathrm{with} \mathrm{respect} \mathrm{to} {\tan }^{-1}\left(\frac{2\mathrm{x}\sqrt{1 - {\mathrm{x}}^2}}{1 - 2{\mathrm{x}}^2}\right) \mathrm{at} \mathrm{x} = \frac{1}{2} \mathrm{is} :$

• $\frac{\sqrt{3}}{10}$

• $\frac{\sqrt{3}}{12}$

• $\frac{2\sqrt{3}}{5}$

• $\frac{2\sqrt{3}}{3}$