Binomial Theorem MATHS - Binomial Theorem

If in the expansion of $\Big(\text{x}-\frac{1}{3\text{x}^{3}}\Big)^{9},$ the term independent of x is:

1. $\text{T}_{3}$

2. $\text{T}_{4}$

3. $\text{T}_{5}$

4. None of these.

The number of terms with integral coefficients in the expansion of $\Big(17^{\frac{1}{3}}+35^{\frac{1}{2}}\text{x}\Big)^{600}$ is:

1. 100
2. 50
3. 150
4. 101

If in the expansion of $(1+\text{x})^{20},$ the coefficients of rth and (r + 4) terms are equal, then r is equal to:

1. 7
2. 8
3. 9
4. 10

If the sum of the binomial coefficients of the expansion $\Big(2\text{x}+\frac{1}{\text{x}}\Big)^{\text{n}}$ is equal to 256, then the term independent of x is:

1. 1120
2. 1020
3. 512
4. None of these.

If the coefficients of 2^nd, 3^rd and 4^th terms in the expansion of $(1+\text{x})^{\text{n}}, \text{n}\in\text{N}$ are in A.P. then n = 

1. 7
2. 14
3. 2
4. None of these.

Write the coefficient of the middle term in the expansion of $(1+\text{x})^{2\text{n}}.$

Write the total number of terms in the expansion of $(\text{x}+\text{a})^{100}+(\text{x}-\text{a})^{100}.$

Find the ratio of the coefficients of x^p and x^q in the expansion of $(1+\text{x})^{\text{p}+\text{q}}.$

If $(1-\text{x}+\text{x}^{2})^{\text{n}}=\text{a}^{0}+\text{a}_{1}\text{x}+\text{a}_{2}\text{x}^{2}+...+\text{a}_{2\text{n}}\text{x}^{2\text{n}},$find the value of $\text{a}_{0}+\text{a}_{2}+\text{a}_{4}+...+\text{a}_{2\text{n}}.$

Find the coefficient of:

x in the expansion of $(1-2\text{x}^3+3\text{x}^5)\Big(1+\frac{1}{\text{x}}\Big)^8.$

Find the middle term in the expansion of:

$\Big(\frac{\text{x}}{\text{a}}-\frac{\text{a}}{\text{x}}\Big)^{10}$

Find the 11th term from the beginning and the 11th term from the end in the expansion of $\Big(2\text{x}-\frac{1}{\text{x}^2}\Big)^{25}$

Find the 4th term from the beginning and 4th term from the end in the expansion of $\Big(\text{x}+\frac{2}{\text{x}}\Big)^9.$

Find the 7th term in the expansion of $\Big(3\text{x}^2-\frac{1}{\text{x}^3}\Big)^{10}$

Using binomial theorem, write down the expansions of the following:

$\Big(\text{x}-\frac{1}{\text{x}}\Big)^6$

Find the coefficient of:

x in the expansion of $(1+3\text{x}+7\text{x}^2)(1-\text{x})^{16}.$

If in the expansion of $(1+\text{x})^{\text{n}}$ the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.

Find the middle terms(s) in the expansion of:

$\Big(\frac{\text{p}}{\text{x}}+\frac{\text{x}}{\text{p}}\Big)^{9}$

Evaluate the following:

$(2+\sqrt3)^7+(2-\sqrt3)^7$

If the 2^nd, 3^rd and 4^th terms in the expansion of $(\text{x}+\text{a})^{\text{n}}$ are 240, 729 and 1080 respectively find x, a, n.

Evaluate the following:

$\Big\{\text{a}^2+\sqrt{\text{a}^2-1}\Big\}^4+\Big\{\text{a}^2-\sqrt{\text{a}^2-1}\Big\}^4$

If n is a positive integer, prove that $3^{3\text{n}}-26\text{n}-1$ is divisible by 676.

Find the term independent of x in the expansion of the following expressions:

$(1+\text{x}+2\text{x}^{3})\Big(\frac{3}{2}\text{x}^{2}-\frac{1}{3\text{x}}\Big)^{9}$

Find the sixth term in the expansion $\Big(\text{y}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}\Big)^{\text{n}},$ if the binomial coefficient of the term from the end is 45.