Binomial Theorem

If in the expansion of (a- 2b)ⁿ, the sum of the 5th and 6th term is zero, then the value of $\frac{\mathrm{a}}{\mathrm{b}}$ is

• $\frac{\mathrm{n} - 4}{5}$

• $\frac{2\left(\mathrm{n} - 4\right)}{5}$

• $\frac{5}{\mathrm{n} - 4}$

• $\frac{5}{2\left(\mathrm{n} - 4\right)}$

(2³ⁿ - 1) will be divisible by $\left(\forall \mathrm{n} \in \mathrm{N}\right)$

• 25

• 8

• 7

• 3

Sum of the last 30 coefficients in the expansion of (1 + x)⁵⁹ , when expanded in ascending power of x is

• 2⁵⁹

• 2⁵⁸

• 2³⁰

• 2²⁹

If (1 - x + x²)ⁿ = a₀ + a₁x + ... + a_2nx²ⁿ, then the value of a₀ + a₂ + a₄ + ... + a_2n is

• $3^{\mathrm{n}} + \frac{1}{2}$

• $3^{\mathrm{n}} - \frac{1}{2}$

• $\frac{3^{\mathrm{n}} - 1}{2}$

• $\frac{3^{\mathrm{n}} + 1}{2}$

The coefficient of xⁿ, where n is any positive integer, in the expansion of ${\left(1 + 2\mathrm{x} + 3{\mathrm{x}}^2 + ... \infty \right)}^{\frac{1}{2}}$

• 1

• $\frac{\mathrm{n} + 1}{2}$

• 2n + 1

• n + 1

If C₀, C₁, C₂, ..., C_n denote the coefficients in the expansion of (1 + x)ⁿ, then the value of C₁ + 2C₂ + 3C₃ + ... + nC_n is

• n . 2^{n - 1}

• (n + 1)2^{n - 1}

• (n + 1)2ⁿ

• (n + 2)2^{n - 1}

If the coefficients of x² and x³ in the expansion of (3 + ax)⁹ be same, then the value of 'a' is

• 3/7

• 7/3

• 7/9

• 9/7

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 ${}^{\mathrm{n} + 1}\mathrm{C}_{\mathrm{n} - \mathrm{k}}$

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

• - 252

• - 210

• - (5!)

• 210