Binomial Theorem

The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰ is

• 99⁵⁰ + 100⁵⁰

• both are equal

• 101⁵⁰

• None of the above

The coefficient of x⁵ in the expansion of (a + x²)⁵(1 + x)⁴ is

• 30

• 60

• 40

• None of these

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

• 15

• 21

• 14

• None of these

If in the expansion of (1 + x)^m (1 - x)ⁿ, the coefficient of x and x² are 3 and - 6 respectively, then m is

• 6

• 9

• 12

• 24

In the expansion of $\frac{\mathrm{a} + \mathrm{bx}}{{\mathrm{e}}^{\mathrm{x}}}$, the coefficient of x^r is

• $\frac{\mathrm{a} - \mathrm{b}}{\mathrm{r}!}$

• $\frac{\mathrm{a} -\mathrm{br}}{\mathrm{r}!}$

• ${\left(- 1\right)}^{\mathrm{r}}\frac{\mathrm{a} - \mathrm{br}}{\mathrm{r}!}$

• None of the above

The coefficient of xⁿ in the expansion of ${\log }_{\mathrm{a}}\left(1 + \mathrm{x}\right)$ is

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

• $\frac{{\left(- 1\right)}^{\mathrm{n} - 1}}{\mathrm{n}}{\log }_{\mathrm{a}}\left(\mathrm{e}\right)$

• $\frac{{\left(- 1\right)}^{\mathrm{n} - 1}}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\mathrm{a}\right)$

• $\frac{{\left(- 1\right)}^{\mathrm{n}}}{\mathrm{n}}{\log }_{\mathrm{a}}\left(\mathrm{e}\right)$

Find the value of (7.995)^1/3 correct to four decimal places.

• 1.9995

• 1.9996

• 1.9990

• 1.9991

The numerically greatest term in the expansion of (3 - 5x)¹¹ when x = $\frac{1}{5}$, is

• $55 \times 3^9$

• $55 \times 3^6$

• $45 \times 3^9$

• $45 \times 3^6$

The number of subsets of {1, 2, 3, ..., 9} containing atleast one odd number is

• 324

• 396

• 496

• 512

If X is a binomial variate with the range {0, 1, 2, 3, 4, 5, 6} and P(X = 2) = 4P(X = 4), then the parameter p of X is

• $\frac{1}{3}$

• $\frac{1}{2}$

• $\frac{2}{3}$

• $\frac{3}{4}$