Using binomial theorem, expand each of the following:

(1 – 2x)⁵

Using binomial theorem, expand each of the following:

(2x – 3)⁶

Using binomial theorem, expand each of the following:

(3x + 2y)⁵

Using binomial theorem, expand each of the following:

(2x – 3y)⁴

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

(1 + 2x – 3x²)⁴

Using binomial theorem, expand each of the following:

Using binomial theorem, expand each of the following:

(3x² – 2ax + 3a²)³

Evaluate :

Evaluate :

Evaluate :

Evaluate :

Prove that

Using binominal theorem, evaluate each of the following :

(i) (101)⁴ (ii) (98)⁴

(iii)(1.2)⁴

Using binomial theorem, prove that (2³ⁿ - 7n -1) is divisible by 49, where n N.

Prove that

Find the 7^th term in the expansion of.

Find the 9^th term in the expansion of .

Find the 16^th term in the expansion of.

Find the 13^th term in the expansion of .

Find the coefficients of x⁷ and x⁸ in the expansion of .

Find the ratio of the coefficient of x¹⁵ to the term independent of x in the expansion of .

Show that the ratio of the coefficient of x¹⁰ in the expansion of (1 – x2)¹⁰ and the term independent of x in the expansion of is 1 : 32.

Find the term independent of x in the expansion of (91 + x + 2x³) .

Find the coefficient of x in the expansion of (1 – 3x + 7x²) (1 – x)¹⁶.

Find the coefficient of

(i)x⁵ in the expansion of (x + 3)⁸

(ii) x⁶ in the expansion of .

(iii) x^-15 in the expansion of .

(iv) a⁷b⁵ in the expansion of (a – 2b)¹².

Show that the term containing x³ does not exist in the expansion of .

Show that the expansion of does not contain any term involving x⁹.

Show that the expansion of does not contain any term involving x^-1.

Write the general term in the expansion of

(x² – y)⁶

Find the 5^th term from the end in the expansion of .

Find the 4^th term from the end in the expansion of .

Find the 4^th term from the beginning and end in the expansion of .

Find the middle term in the expansion of :

(i) (3 + x)⁶

(ii)

(iii)

(iv)

Find the two middle terms in the expansion of :

(x² + a²)⁵

Find the two middle terms in the expansion of :

Find the two middle terms in the expansion of :

Find the two middle terms in the expansion of :

Find the term independent of x in the expansion of :

Find the term independent of x in the expansion of :

Find the term independent of x in the expansion of :

Find the term independent of x in the expansion of :

Find the coefficient of x⁵ in the expansion of (1 + x)³ (1 – x)⁶.

Find numerically the greatest term in the expansion of (2 + 3x)⁹, where .

If the coefficients of 2^nd, 3^rd and 4^th terms in the expansion of (1 + x)²ⁿ are in AP, show that 2n² – 9n + 7 = 0.

Find the 6^th term of the expansion (y^1/2 + x^1/3)ⁿ, if the binomial coefficient of the 3^rd term from the end is 45.

If the 17^th and 18^th terms in the expansion of (2 + a)⁵⁰ are equal, find the value of a.

Find the coefficient of x⁴ in the expansion of (1 + x)ⁿ (1 – x)ⁿ. Deduce that C₂ = C₀C₄ – C₁C₃ + C₂C₂ – C₃C₁ + C₄C₀, where C_r stands for ⁿC_r.

Prove that the coefficient of xn in the binomial expansion of (1 + x)²ⁿ is twice the coefficient of xⁿ in the binomial expansion of (1 + x)^2n-1.

Find the middle term in the expansion of

Show that the term independent of x in the expansion of is -252.

If the coefficients of x² and x³ in the expansion of (3 + px)⁹ are the same then prove that .

Show that the coefficient of x^-3 in the expansion of is -330.

Show that the middle term in the expansion of is 252.

Show that the coefficient of x⁴ in the expansion of is .

Prove that there is no term involving x⁶ in the expansion of.

Show that the coefficient of x⁴ in the expansion of (1 + 2x + x²)⁵ is 212.

Write the number of terms in the expansion of

Which term is independent of x in the expansion of?

Write the coefficient of the middle term in the expansion of (1 + x)²ⁿ.

Write the coefficient of x⁷y² in the expansion of (x + 2y)⁹

If the coefficients of (r – 5)th and (2r – 1)th terms in the expansion of (1 + x)³⁴ are equal, find the value of r.

Write the 4^th term from the end in the expansion of

Find the coefficient of xⁿ in the expansion of (1 + x) (1 – x)ⁿ.

In the binomial expansion of (a + b)ⁿ, the coefficients of the 4^th and 13^th terms are equal to each other. Find the value of n.

Find the positive value of m for which the coefficient of x² in the expansion of (1 + x)^m is 6.