The sum of n terms of the following series 1³ + 3³ + 5³ + 7³ + ... is

• n²(2n² - 1)

• n³(n - 1)

• n³ + 8n + 4

• 2n⁴ + 3n²

If w is an imginary cube root of unity, then the value of (2 - w)(2 - w²) + 2(3 - w)(3 - w²) + ... + (n - 1)(n - w)(n - w²) is

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} +\hspace{0.17em}1\right)}^2 - \mathrm{n}$

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} + 1\right)}^2 + \mathrm{n}$

• $\frac{{\mathrm{n}}^2}{4}{\left(\mathrm{n} + 1\right)}^2$

• $\frac{{\mathrm{n}}^2}{4}\left(\mathrm{n} + 1\right) - \mathrm{n}$

If the first and (2n - 1)th terms of an AP, GP and HP are equal and their nth terms are respectively a, b, c, then always

• a = b = c

• $\mathrm{a} \ge \mathrm{b} \ge \mathrm{c}$

• a + c = b

• ac - b² = 0

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

The value of ${\left(\frac{1 + \sqrt{3}\mathrm{i}}{1 - \sqrt{3}\mathrm{i}}\right)}^{64} + {\left(\frac{1 - \sqrt{3}\mathrm{i}}{1 + \sqrt{3}\mathrm{i}}\right)}^{64}$

• 0

• - 1

• 1

• i

46.

Let x₁, x₂, ..., x₁₅ be 15 distinct numbers chosen from 1, 2, 3, ..., 15. Then, the value of (x₁ - 1)(x₂ - 1)(x₃ - 1)...(x₁₅ - 1) is

• always $\le$ 0

• 0

• always even

• always odd

Let d(n) denotes the number of divisors of n including 1 and itself. Then, d (225), d (1125) and d(640) are

• in AP

• in HP

• in GP

• consecutive integers

Let S = $\left\{\left(\mathrm{a}, \mathrm{b}, \mathrm{c}\right) \in \mathrm{N} \times \mathrm{N} \times \mathrm{N}: \mathrm{a} + \mathrm{b} + \mathrm{c} = 21, \mathrm{a} \le \mathrm{b} \le \mathrm{c}\right\} \mathrm{and} \mathrm{T} = \left\{\left(\mathrm{a}, \mathrm{b}, \mathrm{c}\right) \in \mathrm{N} \times \mathrm{N} \times \mathrm{N}: \mathrm{a}, \mathrm{b}, \mathrm{c} \mathrm{are} \mathrm{in} \mathrm{AP}\right\}$, where N is the set of all natural numbers. Then, the number of elements in the set $\mathrm{S} \cap \mathrm{T}$

• 6

• 7

• 13

• 14

If x and y are digits such that 17! = 3556xy428096000, then x + y equals

• 15

• 6

• 12

• 13

Let f(x) = x + 1/2. Then, the number of real values of x for which the three unequal terms f(x), f(2x), f(4x) are in HP is

• 1

• 0

• 3

• 2