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

If (2 + i) and $\left(\sqrt{5} - 2\mathrm{i}\right)$ are the roots of the equation (x² + ax + b )(x² + ex + d) = 0, where a, b, c and d are real constants, then product of all the roots of the equation is

• 40

• 9$\sqrt{5}$

• 45

• 35

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 the point $\left(2\cos \left(\mathrm{\theta }\right), 2\sin \left(\mathrm{\theta }\right)\right) \mathrm{for} (0, 2\mathrm{\pi })$ lies in the region between the lines x + y = 2 and x - y = 2 containing the origin, then 0 lies in

• $\left(0, \frac{\mathrm{\pi }}{2}\right) \cup \left(\frac{3\mathrm{\pi }}{2}, 2\mathrm{\pi }\right)$

• $\left[0, \mathrm{\pi }\right]$

• $\left(\frac{\mathrm{\pi }}{2}, \frac{3\mathrm{\pi }}{2}\right)$

• $\left[\frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{2}\right]$

35.

Number of points having distance $\sqrt{5}$  from the straight line x - 2y + 1 = 0 and a distance $\sqrt{13}$ is from the line 2x + 3y - 1 = 0, is

• 1

• 2

• 4

• 5

Let P(x) be a polynomial, which when divided by (x - 3) and (x - 5) leaves remainders 10 and 6, respectively. If the polynomial is divided by (x - 3) (x - 5), then the remainder is

• - 2x + 16

• 16

• 2x - 16

• 60

Let f: $\mathrm{R} \to \mathrm{R}$ be differentiable at x = 0. If f(0) = 0 and f'(0) = 2, then the value of

$\underset{\mathrm{x}\to 0}{\lim }\frac{1}{\mathrm{x}}\left[\mathrm{f}\left(\mathrm{x}\right) + \mathrm{f}\left(2\mathrm{x}\right) + \mathrm{f}\left(3\mathrm{x}\right) + ... + \mathrm{f}\left(2015\mathrm{x}\right)\right]$ is

• 2015

• 0

• $2015 \times 2016$

• $2015 \times 2014$

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

• 15

• 6

• 12

• 13

A person goes to office by car, scooter, bus and train, probability of which are 1/7, 3/7, 2/7 and 1/7, respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 2/9, 1/9, 4/9, and 1/9, respectively. Given that he reached office in time, the probability that he travelled by a car, is

• 1/7

• 2/7

• 3/7

• 4/7

Let f : N $\to$ R be such that f(1) = 1 and f(1) + 2f(2) +  3f(3) + ... + nf(n) = n(n+ 1) f(n), for all n  $\in$ N, n $\ge$ 2, where N is the set of natural numbers and R is the set of real numbers. Then, the value of f(500) is

• 1000

• 500

• 1/500

• 1/1000