Relations and Functions

The number of real roots of equation log_e(x) + ex = 0 is

• 0

• 1

• 2

• 3

Let the number of elements of the sets A and B be p and q, respectively. Then, the number of relations from the set A to the set B is

• 2^{p + q}

• 2^pq

• p + q

• pq

Eleven apples are distributed among a girl and a boy, Then, which one of the following statements is true ?

• atleast one of them will receive 7 apples

• the girl receives atleast 4 apples or the boy receives atleast 9 apples

• the girl receives atleast 5 apples or the boy receives atleast 8 apples

• the girl receives atleast 4 apples or the boy receives atleast 8 apples

The domain of definition of the function

$\mathrm{f}\left(\mathrm{x}\right) = \sqrt{1 + {\log }_{\mathrm{e}}\left(1 - \mathrm{x}\right)}$

• $- \infty < \mathrm{x} \le 0$

• $- \infty < \mathrm{x} \le \frac{\mathrm{e} - 1}{\mathrm{e}}$

• $- \infty < \mathrm{x} \le 1$

• $\mathrm{x} \ge 1 - \mathrm{e}$

The function f(x) = $\log \left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right)$ satisfies the equation

• f(x + 2) - 2f(x + 1) + f(x) = 0

• f(x) + f(x + 1) = f{x(x + 1)}

• f(x) + f(y) = $\mathrm{f}\left(\frac{\mathrm{x} + \mathrm{y}}{1 + \mathrm{xy}}\right)$

• f(x + y) = f(x)f(y)

The range of the function f (x) = ${\log }_{\mathrm{e}}\left(\sqrt{4 - {\mathrm{x}}^2}\right)$ is given by

• $\left(0, \infty \right)$

• $\left(- \infty , \infty \right)$

• $\left(- \infty , {\log }_{\mathrm{e}}\left(2\right)\right)$

• $\left({\log }_{\mathrm{e}}\left(2\right), \infty \right)$

If ${\log }_5\left({\log }_5\left({\log }_2\left(\mathrm{x}\right)\right)\right) = 0$, then the value of x is

• 32

• 125

• 625

• 25

A mapping f : N ➔ N, where N is the set of natural numbers is defined as

f(n) = n² for n odd

f(n) = 2n + 1, for n even

for n ∈ N. Then f is

• surjective but not injective

• injective but not surjective

• bijective

• neither injective nor surjective

A function f: A $\to$ B, where A = {x: $- 1 \le \mathrm{x} \le 1$} and B = {y: $1 \le \mathrm{y} \le 2$} is defined by the rule y = f(x) = 1 + x² Which of the following statement is true ?

• f is injective but not surjective

• f is surjective but not injective

• f is both injective and surjective

• f is neither injective nor surjective

The equation e^x - x - 1 = 0 has, apart from x = 0

• one other real root

• two real roots

• no other real root

• infinite number of real roots