If the normal to the curve y = f(x) at (3, 4) makes an angle $\frac{3\mathrm{\pi }}{4}$ with the positive x-axis, then f'(3) is equal to :

• - 1

• $\frac{3}{4}$

• 1

• - $\frac{3}{4}$

2.

The number of integral solutions of x₁ + x₂ + x₃ = 0, with x_i $\ge$ - 5, is :

• ${}^{15}\mathrm{C}_2$

• ${}^{16}\mathrm{C}_2$

• ${}^{17}\mathrm{C}_2$

• ${}^{18}\mathrm{C}_2$

Let A = {1, 2, 3, ... , n} and B = {a, b, c}, then the number of functions from A to B that are onto is:

• 3ⁿ - 2ⁿ

• 3ⁿ - 2ⁿ - 1

• 3(2ⁿ - 1)

• 3ⁿ - 3(2ⁿ - 1)

Everybody in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is:

• 9

• 12

• 10

• 14

In a group G = {1, 3, 7, 9} under multiplication modulo 10, the inverse of 7 is :

• 7

• 3

• 9

• 1

A box contains 9 tickets numbered 1 to 9 inclusive. If 3 tickets are drawn from the box one at a time, the probability that they are alternatively either {odd, even, odd} or {even, odd, even}is:

• $\frac{5}{17}$

• $\frac{4}{17}$

• $\frac{5}{16}$

• $\frac{5}{18}$

If $\mathrm{P}\left(\mathrm{A}\right) = \frac{1}{12}, \mathrm{P}\left(\mathrm{B}\right) = \frac{5}{12} \mathrm{and} \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right) = \frac{1}{15}$, then P(A $\cup$ B) is equal to :

• $\frac{89}{180}$

• $\frac{90}{180}$

• $\frac{91}{180}$

• $\frac{92}{180}$

If b² $\ge$ 4ac for the equation ax⁴ + bx² + c = 0, then all the roots of the equation will be real if:

• b > 0, a < 0, c > 0

• b < 0, a > 0, c > 0

• b > 0, a > 0, c > 0

• b > 0, a > 0, c < 0

If x > 0 and log₃(x) + ${\log }_3\left(\sqrt{\mathrm{x}}\right) + {\log }_3\left(\sqrt[4]{\mathrm{x}}\right) + {\log }_3\left(\sqrt[8]{\mathrm{x}}\right) + {\log }_3\left(\sqrt[16]{\mathrm{x}}\right)$ = 4, then x equals:

• 9

• 81

• 1

• 27

The number of real roots of the equation ${\left(\mathrm{x} + \frac{1}{\mathrm{x}}\right)}^3 + \mathrm{x} + \frac{1}{\mathrm{x}} = 0$ is :

• 0

• 2

• 4

• 6