Let f : [- 6, 6] $\to$ R be defined by f(x) = x² - 3.

Consider the following :

I. (f o f o f)( - 1) = (f o f o f)(1)

2. (f o f o f)( - 1) - 4(f o f o f)(1) = (f o f)(0)

Which of the above is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

Let f(x) = px + q and g(x) = mx + n. Then f(g(x)) = g(f(x)) is equivalent to

• f(p) = g(m)

• f(q) = g(n)

• f(n) = g(q)

• f(m) = g(p)

Let $\mathrm{f}\left(\mathrm{a}\right) = \frac{\mathrm{a} - 1}{\mathrm{a} + 1}$

Consider the following :

$$\begin{gathered} 1. \mathrm{f}\left(2\mathrm{a}\right) = \mathrm{f}\left(\mathrm{a}\right) + 1 \\ 2. \mathrm{f}\left(\frac{1}{\mathrm{a}}\right) = - \mathrm{f}\left(\mathrm{a}\right) \end{gathered}$$

Which of the above is/are correct ?

• 1 only

• 2 only

• both 1 and 2

• beither 1 nor 2

Which one of the following functions is neither even nor odd ?

• x² - 1

• x + $\frac{3}{\mathrm{x}}$

• $\left|\mathrm{x}\right|$

• x²(x - 3)

$\mathrm{The} \mathrm{remainder} \mathrm{and} \mathrm{the} \mathrm{quotient} \mathrm{of} \mathrm{the} \mathrm{binary} \mathrm{division} (101110{)}_2 : (110{)}_2, \mathrm{are} \mathrm{respectively} :$

• ${\left(111\right)}_2 \mathrm{and} {\left(100\right)}_2$

• ${\left(100\right)}_2 \mathrm{and} {\left(111\right)}_2$

• ${\left(101\right)}_2 \mathrm{and} {\left(101\right)}_2$

• ${\left(100\right)}_2 \mathrm{and} {\left(100\right)}_2$

If we define a relation R on the set N x N as (a, b) R(c, d) $\iff$ a + d = b + c for all (a, b), (c, d) $\in$ N x N, then the relation is :

• symmetric only

• symmetric and transitive only

• equivalence relation

• reflexive only

The inverse of the function y - 5^ln(x) is :

• $\mathrm{x} = {\mathrm{y}}^{\frac{1}{\ln \left(5\right)}}, \mathrm{y} > 0$

• $\mathrm{x} = {\mathrm{y}}^{\ln \left(5\right)}, \mathrm{y} > 0$

• $\mathrm{x} = {\mathrm{y}}^{\frac{1}{\ln \left(5\right)}}, \mathrm{y}< 0$

• $\mathrm{x} = 5\ln \left(\mathrm{y}\right), \mathrm{y} > 0$

A function is defined in (0, $\infty$) by : 

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{1 - {\mathrm{x}}^2 \mathrm{for} 0< \mathrm{x} \le 1 \\ \ln \left(\mathrm{x}\right) \mathrm{for} 1 < \mathrm{x} \le 2 \\ \ln \left(2\right) - 1 + 0.5\mathrm{x} \mathrm{for} 2 < \mathrm{x} < \infty \right\} \end{gathered}$$

Which one of the following is correct in respect of the derivative of the function, i.e., f'(x) ?

• f'(x) = 2x for $0 < \mathrm{x} \le 1$

• f'(x) = - 2x for $0 < \mathrm{x} \le 1$

• $\mathrm{f}\text{'}\left(\mathrm{x}\right) = - 2\mathrm{x} \mathrm{for} 0 < \mathrm{x} 1$

• $\mathrm{f}\text{'}\left(\mathrm{x}\right) = 0 \mathrm{for} 0 < \mathrm{x} < \infty$

39.

The function f (x) = $\left|\mathrm{x}\right| - {\mathrm{x}}^3$ is : 

• odd

• even

• both even and odd

• neither even nor odd

The maximum value of $\frac{\ln \left(\mathrm{x}\right)}{\mathrm{x}}$ is :

• e

• $\frac{1}{\mathrm{e}}$

• $\frac{2}{\mathrm{e}}$

• 1