$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left\{\mathrm{a}, \mathrm{b}, \mathrm{c}\right\} \mathrm{and} \mathrm{B} = \left\{1, 2, 3, 4\right\}.\mathrm{Then} \mathrm{the} \mathrm{number} \mathrm{of} \mathrm{elements} \mathrm{in} \mathrm{the} \mathrm{set} \\ \mathrm{C}=\left\{\mathrm{f} : \mathrm{A} \to \mathrm{B}|2 \in \mathrm{f}(\mathrm{A}) \mathrm{and} \mathrm{f} \mathrm{is} \mathrm{not} \mathrm{one}–\mathrm{one}\right\}\mathrm{is}... \end{gathered}$$

The position of moving car at time tis given by f(t) = at² + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average speed of the car over the time interval [t₁, t₂] is attained at the point

• $\frac{{\mathrm{t}}_1 + {\mathrm{t}}_2}{2}$

• $\frac{{\mathrm{t}}_2 - {\mathrm{t}}_1}{2}$

• $2\mathrm{a}\left({\mathrm{t}}_1 +\hspace{0.17em} {\mathrm{t}}_2\right) \hspace{0.17em}+ \mathrm{b}$

• $\mathrm{a}\left({\mathrm{t}}_2 - {\mathrm{t}}_1\right) + \mathrm{b}$

$$\begin{gathered} \mathrm{For} \mathrm{a} \mathrm{suitably} \mathrm{chosen} \mathrm{real} \mathrm{constant} \mathrm{a}, \mathrm{let} \mathrm{a} \mathrm{function}, \mathrm{f} :\mathbf{ }\mathbf{R}\mathbf{ }\mathbf{-}\mathbf{ }\left\{\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{a}\right\}\mathbf{ }\mathbf{\to }\mathbf{ }\mathbf{R}\mathbf{ }\mathrm{be} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \frac{\mathrm{a} - \mathrm{x}}{\mathrm{a} + \mathrm{x}}. \mathrm{Further} \mathrm{suppose} \mathrm{that} \mathrm{for} \mathrm{any} \mathrm{real} \mathrm{number} \mathrm{x} \ne 0 \mathrm{and} \\ \mathrm{f}\left(\mathrm{x}\right) \ne - \mathrm{a}, \left(\mathrm{fof}\right)\left(\mathrm{x}\right) = \mathrm{x}. \mathrm{Then} \mathrm{f}\left( - \frac{1}{2}\right) = ? \end{gathered}$$

• 3

•  - 3

• $\frac{1}{3}$

• $- \frac{1}{3}$

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