If (1 + x)¹⁵ = a₀ + a₁x + ... + a₁₅x¹⁵, then $\sum _{\mathrm{r} = 1}^{15}\mathrm{r}\frac{{\mathrm{a}}_{\mathrm{r}}}{{\mathrm{a}}_{\mathrm{r}} - 1}$ is equal to

• 110

• 115

• 120

• 135

The coefficient of x³y⁴z⁵ in the expansion of (xy + yz + xz)⁶ is

• 70

• 60

• 50

• None of these

If $\left|\mathrm{x}\right| < \frac{1}{2}$, then the coefficient of x^r in the expansion of $\frac{1 +\hspace{0.17em}2\mathrm{x}}{{\left(1 - 2\mathrm{x}\right)}^2}$, is

• r2^r

• (2r - 1)2^r

• r2^{2r + 1}

• (2r + 1)2^r

$\mathrm{The} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{n}} \mathrm{in} \frac{1 - 2\mathrm{x}}{{\mathrm{e}}^{\mathrm{x}}} \mathrm{is} :$

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

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

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

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

For $\left|\mathrm{x}\right| < 1$, the constant term in the expansion of $\frac{1}{{\left(\mathrm{x} - 1\right)}^2\left(\mathrm{x} - 2\right)} \mathrm{is}$

• 2

• 1

• 0

• $- \frac{1}{2}$

The numbers a_n = 6ⁿ - 5n for n = 1, 2, 3, . . . when divided by 25 leave the remainder

• 9

• 7

• 3

• 1

$\mathrm{For} \left|\mathrm{x}\right| < \frac{1}{5}, \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^3 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} \frac{1}{\left(1 - 5\mathrm{x}\right)\left(1 - 4\mathrm{x}\right)} \mathrm{is}$

• 369

• 370

• 371

• 372

If the coefficients of rth and (r + 1)th terms in the expansion of (3 + 7x)²⁹ are equal, then r is equal to

• 14

• 15

• 18

• 21

Suppose X follows a binomial distribution with parameters n and p, where 0 < p < 1. If $\frac{\mathrm{P}\left(\mathrm{X} = \mathrm{r}\right)}{\mathrm{P}\left(\mathrm{X} = \mathrm{n} - \mathrm{r}\right)}$ is independent of n for every r, then p is equal to

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$

80.

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{3}{8}$