Using binomial theorem, the value of (0.999)³  correct to 3 decimal places is

• 0.999

• 0.998

• 0.997

• 0.995

Show that, for a positive integer n, the coefficient of x^k, ($0 \le \mathrm{k} \le \mathrm{n}$) in the expansion of 1 + (1 + x) + (1 + x) + ... + (1 + x)ⁿ is ${}^{\mathrm{n} + 1}\mathrm{C}_{\mathrm{n} - \mathrm{k}}$

213.

The coefficient of x^{- 10} in ${\left({\mathrm{x}}^2 - \frac{1}{{\mathrm{x}}^3}\right)}^{10}$ is

• - 252

• - 210

• - (5!)

• 210

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