$\mathrm{The} \mathrm{function} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^3 +\hspace{0.17em}{\mathrm{ax}}^2 + \mathrm{bx} +\hspace{0.17em}\mathrm{c}, {\mathrm{a}}^2 \le 3\mathrm{b} \mathrm{has}$

• one maximum value

• one minimum value

• no extreme value

• one maximum and one minimum value

$\mathrm{The} \mathrm{maximum} \mathrm{value} \mathrm{of} \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}, 0 < \mathrm{x} < \infty \mathrm{is}$

• $\infty$

• e

• 1

• e ^{- 1}

$\mathrm{z} = \tan \left(\mathrm{y} +\hspace{0.17em}\mathrm{ax}\right) +\hspace{0.17em}\sqrt{\mathrm{y} - \mathrm{ax}} \Rightarrow {\mathrm{z}}_{\mathrm{xx}} - {\mathrm{a}}^2{\mathrm{z}}_{\mathrm{yy}} =?$

• 0

• 2

• ${\mathrm{z}}_{\mathrm{x}} + {\mathrm{z}}_{\mathrm{y}} = 0$

• ${\mathrm{z}}_{\mathrm{x}}{\mathrm{z}}_{\mathrm{y}}$

The height of the cone of maximum volume inscribed in a sphere of radius R is

• $\frac{\mathrm{R}}{3}$

• $\frac{{\displaystyle 2\mathrm{R}}}{{\displaystyle 3}}$

• $\frac{4\mathrm{R}}{3}$

• $\frac{4\mathrm{R}}{\sqrt{3}}$

If the distance s travelled by a particle in time t is given by s = t² - 2t + 5, then its acceleration is

• 0

• 1

• 2

• 3

The length of the sub tangent at any point (x₁, y₁) on the curve y = 5^x is

• $5^{{\mathrm{x}}_1}$

• ${\mathrm{y}}_1{5}^{{\mathrm{x}}_1}$

• ${\log }_{\mathrm{e}}\left(5\right)$

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

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

• $\left(0, \infty \right)$

• $\left(1, \infty \right)$

• $\left(4, \infty \right)$

• $\left(5, \infty \right)$

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