Let a, b, c and d be any four real numbers. Then, aⁿ +bⁿ = cⁿ + dⁿ holds for any natural number n, if

• a + b = c + d

• a - b = c - d

• a + b = c + d, a² + b² = c² + d²

• a - b = c - d, a² - b² = c² - d²

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} {\mathrm{x}}^2 - \mathrm{px} +\hspace{0.17em}1 = 0 \mathrm{and} \mathrm{\gamma } \mathrm{is} \mathrm{aroot} \mathrm{of} {\mathrm{x}}^2 + \mathrm{px} +\hspace{0.17em}1 = 0, \mathrm{then} \left(\mathrm{\alpha } + \mathrm{\gamma }\right)\left(\mathrm{\beta } + \mathrm{\gamma }\right)$ is

• 0

• 1

• - 1

• $\mathrm{\rho }$

The number of irrational terms in the binomial expansion of ${\left(3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} + 7^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}^{100}$

• 90

• 88

• 94

• 95

The quadratic expression ${\left(2\mathrm{x} +1\right)}^2 - \mathrm{px} + \mathrm{q} \ne 0$ for any real x, if

• p² - 16p - 8q < 0

• p² - 8p + 16q < 0

• p² - 8p - 16q < 0

• p² - 16p + 8q < 0

In a certain town, 60% of the families own a car, 30% own a house and 20% own both car and house. If a family is randomly chosen, then what is the probability that this family owns a car or a house but not both?

• 0.5

• 0.7

• 0.1

• 0.9

6.

The letters of the word 'COCHIN' are permuted and all the permutations are arranged in alphabetical order as in English dictionary. The number of words that appear before the word 'COCHIN', is

• 360

• 192

• 96

• 48

Let f: $\mathrm{R} \to \mathrm{R} \mathrm{be} \mathrm{defmed} \mathrm{as} \mathrm{f}\left(\mathrm{x}\right) = \frac{{\mathrm{x}}^2 - \mathrm{x} + 4}{{\mathrm{x}}^2 + \mathrm{x} + 4}$ Then, range of the function f(x) is

• $\left[\frac{3}{5}, \frac{5}{3}\right]$

• $\left(\frac{3}{5}, \frac{5}{3}\right)$

• $\left(- \infty , \frac{3}{5}\right) \cup \left(\frac{5}{3}, \infty \right)$

• $\left[- \frac{5}{3}, - \frac{3}{5}\right]$

The least value of 2x² + y² + 2xy + 2x -3y + 8 for real numbers x and y, is

• 2

• 8

• 3

• - 1/2

The minimum value of $\cos \left(\mathrm{\theta }\right) + \sin \left(\mathrm{\theta }\right) + \frac{2}{\sin \left(\mathrm{\theta }\right)} \mathrm{for} \mathrm{\theta } \in \left(0, \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

• $2 + \sqrt{2}$

• 2

• 1 + $\sqrt{2}$

• $2\sqrt{2}$

If $\cot \left(\frac{2\mathrm{x}}{3}\right) + \tan \left(\frac{\mathrm{x}}{3}\right) = \csc \left(\frac{\mathrm{kx}}{3}\right)$, then the value of k is

• 1

• 2

• 3

• - 1