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 and B are two points on the Argand plane such that the segment AB is bisected at the point (0, 0). If the point A, which is in the third quadrant has principal amplitude 0, then the principal amplitude of the point B is

• $- \mathrm{\theta }$

• $\mathrm{\pi } - \mathrm{\theta }$

• $\mathrm{\theta } - \mathrm{\pi }$

• $\mathrm{\pi } + \mathrm{\theta }$

20.

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