$\mathrm{If} {\mathrm{a}}_{\mathrm{k}} \mathrm{is} \mathrm{the} \mathrm{coefficient} \mathrm{of} {\mathrm{x}}^{\mathrm{k}} \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} {\left(1 + \mathrm{x} + {\mathrm{x}}^2\right)}^{\mathrm{n}} \mathrm{for} \mathrm{k} = 0, 1, 2, . . , 2\mathrm{n}, \mathrm{then}$

• - a₀

• 3ⁿ

• n 3^{n + 1}

• n 3ⁿ

The coefficient of x^k in the expansion of $\frac{1 - 2\mathrm{x} - {\mathrm{x}}^2}{{\mathrm{e}}^{- \mathrm{x}}}$ is

• $\frac{1 - \mathrm{k} - {\mathrm{k}}^2}{\mathrm{k}!}$

• $\frac{{\mathrm{k}}^2 + 1}{\mathrm{k}!}$

• $\frac{1 - \mathrm{k}}{\mathrm{k}!}$

• $\frac{1}{\mathrm{k}!}$

The coefficient of x²⁴ in the expansion of (1 + x²)¹²(1 + x¹²)(1 + x²⁴) is

• ${}_{12}\mathrm{C}_6$

• ${}_{12}\mathrm{C}_6 + 2$

• ${}_{12}\mathrm{C}_6 + 4$

• ${}_{12}\mathrm{C}_6 +\hspace{0.17em}6$

254.

If x is numerically so small so that x and higher powers of x can be neglected, then ${\left(1 + \frac{2\mathrm{x}}{3}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\left(32 +\hspace{0.17em}5\mathrm{x}\right)}^{\raisebox{1ex}{$- 1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.} \mathrm{is} \mathrm{approximately} \mathrm{equal} \mathrm{to}$

• $\frac{32 + 31\mathrm{x}}{64}$

• $\frac{31 + 32\mathrm{x}}{64}$

• $\frac{31 - 32\mathrm{x}}{64}$

• $\frac{1 - 2 x}{64}$

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

$$\begin{gathered} \mathrm{If} \mathrm{ab} \ne 0 \mathrm{and} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{the} \mathrm{coefficients} \mathrm{of} {\mathrm{x}}^7 \\ \mathrm{and} {\mathrm{x}}^4 \mathrm{in} \mathrm{the} \mathrm{expansion} \mathrm{of} {\left(\frac{{\mathrm{x}}^2}{\mathrm{a}} - \frac{\mathrm{b}}{\mathrm{x}}\right)}^{11} \mathrm{is} 0, \mathrm{then} \end{gathered}$$

• a = b

• a + b = 0

• ab = - 1

• ab = 1

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

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$