Solution of Chapter 8. Binomial Theorem (NCERT - Maths Book)

Expand each of the expressions in Exercises 1 to 5.

(1 – 2x)⁵

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

(2x – 3)⁶

Expand each of the expressions in Exercises 1 to 5.

Expand each of the expressions in Exercises 1 to 5.

Using binomial theorem, evaluate each of the following:

(96)³

Using binomial theorem, evaluate each of the following:

(102)⁵

Using binomial theorem, evaluate each of the following:

(101)⁴

Using binomial theorem, evaluate each of the following:

(99)⁵

Using Binomial Theorem, indicate which number is larger (1.1)¹⁰⁰⁰⁰ or 1000.

Find (a + b)⁴ – (a – b)⁴. Hence, evaluate

Find (x + 1)⁶ + (x – 1)⁶. Hence or otherwise evaluate .

Show that 9ⁿ⁺¹ – 8n – 9 is divisible by 64, whenever n is a positive integer.

Prove that

Find the coefficient of

x⁵ in (x + 3)⁸

Find the coefficient of

a⁵b⁷ in (a – 2b)¹² .

Write the general term in the expansion of

(x² – y)⁶

Write the general term in the expansion of

(x² – yx)¹², x ≠ 0.

Find the 4th term in the expansion of (x – 2y)¹².

Find the 13^th term in the expansion of

Find the middle terms in the expansions of

Find the middle terms in the expansions of

In the expansion of (1 + a)^m+n, prove that coefficients of a^m and aⁿ are equal.

The coefficients of the (r – 1)^th, r^th and (r + 1)^th terms in the expansion of (x + 1)ⁿ are in the ratio 1 : 3 : 5. Find n and r.

Prove that the coefficient of xⁿ in the expansion of (1 + x)²ⁿ is twice the coefficient of xⁿ in the expansion of (1 + x)^{2n – 1}.

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

Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375, respectively.

Find a if the coefficients of x² and x³ in the expansion of (3 + ax)⁹ are equal.

Find the coefficient of x⁵ in the product (1 + 2x)⁶ (1 – x)⁷ using binomial theorem.

If a and b are distinct integers, prove that a – b is a factor of aⁿ – bⁿ, whenever n is a positive integer. [Hint write aⁿ = (a – b + b)ⁿ and expand]

Evaluate

Find the value of

Find an approximation of (0.99)⁵ using the first three terms of its expansion.

Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of is √6 : 1

Find the expansion of (3x² – 2ax + 3a²)³ using binomial theorem.