Let f : R $\to$ R be a function defined by f(x) = max {x, x²}. Let S denote the set of all points in R, where f is not differentiable. Then:

• $\mathrm{\varphi }(\mathrm{an} \mathrm{empty} \mathrm{set})$

• $\left\{1\right\}$

• $\left\{0\right\}$

• $\left\{0, 1\right\}$

For all twice differentiable functions f : R $\to$ R, with f(0) = f(1) = f'(0) = 0

• f''(x) = 0, for some x $\in$ (0, 1)

• $\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) = 0, \mathrm{at} \mathrm{every} \mathrm{point} \mathrm{x} \in (0, 1)$

• f''(0) = 0

• $\mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right) \ne 0, \mathrm{at} \mathrm{every} \mathrm{point} \mathrm{x} \in (0, 1)$

Suppose that a function f : R $\to$ R satisfies f(x + y) = f(x)f(y) for all x, y $\in$ R and f(1) = 3. If $\sum _{\mathrm{i} = 1}^{\mathrm{n}}\mathrm{f}\left(\mathrm{i}\right) = 363$, then n is equal to.......

$$\begin{gathered} \mathrm{If} \overrightarrow{\mathrm{x}} \mathrm{and} \overrightarrow{\mathrm{y}} \mathrm{be} \mathrm{two} \mathrm{non}-\mathrm{zero} \mathrm{vectors} \mathrm{such} \mathrm{that} \left|\overrightarrow{\mathrm{x}} + \overrightarrow{\mathrm{y}}\right| = \left|\overrightarrow{\mathrm{x}}\right|\mathrm{and} 2\overrightarrow{\mathrm{x}} + \mathrm{\lambda }\overrightarrow{\mathrm{y}} \mathrm{is} \mathrm{perpendicular} \\ \mathrm{to} \mathrm{y}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{is}...... \end{gathered}$$

For any integer n > 1, the number of positive divisors of n is denoted by d(n). Then, for a prime P, d (d (d(P)⁷)) is equal to

• 1

• 2

• 3

• p

$\sum _{\mathrm{k} = 1}^5\frac{1^3 + 2^3 + .... + {\mathrm{k}}^3}{1 + 3 + 5 + ... + \left(2\mathrm{k} - 1\right)} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 22.5

• 24.5

• 28.5

• 32.5

$$\begin{gathered} \mathrm{The} \mathrm{function} \mathrm{f}: \mathrm{C} \to \mathrm{C} \mathrm{defined}, \mathrm{by} \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{ax} + \mathrm{d}}{\mathrm{cx} + \mathrm{d}} \mathrm{for} \mathrm{x} \in \mathrm{C} \mathrm{where} \\ \mathrm{bd} \ne 0 \mathrm{reduces} \mathrm{to} \mathrm{a} \mathrm{constant} \mathrm{function}, \mathrm{if} \end{gathered}$$

• a = c

• b = d

• ad = bc

• ab = cd

The longest distance of the point (a, 0) from the curve 2x² + y² = 2x is

• 1 + a

• $\left|1 - \mathrm{a}\right|$

• $\sqrt{1 - 2\mathrm{a} +2{\mathrm{a}}^2}$

• $\sqrt{1 - 2\mathrm{a} + 3{\mathrm{a}}^2}$

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

• $\left\{- 14, - 13, . . . 0, . . . 13, 14\right\}$

• $\left\{- 14, - 13, . . . 0, . . . 14, 15\right\}$

• $\left\{- 15, - 14, . . . 0, . . . 14, 15\right\}$

• $\left\{- 15, - 14, . . . 0, . . . 13, 14\right\}$