NCERT Exemplar solutions for Mathematics [English] Class 11 chapter 8 - Binomial Theorem [Latest edition]

Find the r^th term in the expansion of `(x + 1/x)^(2r)`

Expand the following (1 – x + x²)⁴ 

Find the 4^th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`

Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`

Find the coefficient of x¹¹ in the expansion of `(x^3 - 2/x^2)^12`

Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x¹⁰?

Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.

Find the middle term in the expansion of `(2ax - b/x^2)^12`.

Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.

Show that `2^(4n + 4) - 15n - 16`, where n ∈ N is divisible by 225.

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

If n is a positive integer, find the coefficient of x^–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`

Which of the following is larger? 99⁵⁰ + 100⁵⁰  or 101⁵⁰

Find the coefficient of x⁵⁰ after simplifying and collecting the like terms in the expansion of (1 + x)¹⁰⁰⁰ + x(1 + x)⁹⁹⁹ + x²(1 + x)⁹⁹⁸ + ... + x¹⁰⁰⁰ .

If a₁, a₂, a₃ and a₄ are the coefficient of any four consecutive terms in the expansion of (1 + x)ⁿ, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`

The total number of terms in the expansion of (x + a)⁵¹ – (x – a)⁵¹ after simplification is ______.

If the coefficients of x⁷ and x⁸ in `2 + x^n/3` are equal, then n is ______.

If (1 – x + x²)ⁿ = a₀ + a₁ x + a₂ x² + ... + a_2n x²ⁿ , then a₀ + a₂ + a₄ + ... + a_2n equals ______.

The coefficient of x^p and x^q (p and q are positive integers) in the expansion of (1 + x)^{p + q} are ______.

The number of terms in the expansion of (a + b + c)ⁿ, where n ∈ N is ______.

The ratio of the coefficient of x¹⁵ to the term independent of x in `x^2 + 2^15/x` is ______.

If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.

Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`

If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.

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

Find the term independent of x in the expansion of `(3x - 2/x^2)^15`

Find the middle term (terms) in the expansion of `(x/a - a/x)^10`

Find the middle term (terms) in the expansion of `(3x - x^3/6)^9`

Find the coefficient of x¹⁵ in the expansion of (x – x²)¹⁰.

Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`

Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.

Find the value of r, if the coefficients of (2r + 4)^th and (r – 2)^th terms in the expansion of (1 + x)¹⁸ are equal.

If the coefficient of second, third and fourth terms in the expansion of (1 + x)²ⁿ are in A.P. Show that 2n² – 9n + 7 = 0.

Find the coefficient of x⁴ in the expansion of (1 + x + x² + x³)¹¹.

If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.

Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`

Find n in the binomial `(root(3)(2) + 1/(root(3)(3)))^n` if the ratio of 7^th term from the beginning to the 7^th term from the end is `1/6`

In the expansion of (x + a)ⁿ if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O² – E² = (x² – a²)ⁿ 

In the expansion of (x + a)ⁿ if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that 4OE = (x + a)²ⁿ – (x – a)²ⁿ 

If x^p occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`

Find the term independent of x in the expansion of (1 + x + 2x³) `(3/2 x^2 - 1/(3x))^9`

The total number of terms in the expansion of (x + a)¹⁰⁰ + (x – a)¹⁰⁰ after simplification is ______.

Given the integers r > 1, n > 2, and coefficients of (3r)^th and (r + 2)^nd terms in the binomial expansion of (1 + x)²ⁿ are equal, then ______.

The two successive terms in the expansion of (1 + x)²⁴ whose coefficients are in the ratio 1:4 are ______.

The coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^2n–1 are in the ratio ______.

If the coefficients of 2^nd, 3^rd and the 4^th terms in the expansion of (1 + x)ⁿ are in A.P., then value of n is ______.

If A and B are coefficient of x n in the expansions of (1 + x)²ⁿ and (1 + x)^2n–1 respectively, then `A/B` equals ______.

If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.

The largest coefficient in the expansion of (1 + x)³⁰ is ______.

The number of terms in the expansion of (x + y + z)ⁿ ______.

In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.

If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals ______.

The coefficient of a^–6b⁴ in the expansion of `(1/a - (2b)/3)^10` is ______.

Middle term in the expansion of (a³ + ba)²⁸ is ______.

The ratio of the coefficients of x^p and x^q in the expansion of (1 + x)^{p + q} is ______.

The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.

If 25¹⁵ is divided by 13, the reminder is ______.

The sum of the series `sum_(r = 0)^10  ""^20C_r` is `2^19 + (""^20C_10)/2`

The expression 7⁹ + 9⁷ is divisible by 64.

The number of terms in the expansion of [(2x + y³)⁴]⁷ is 8.

The sum of coefficients of the two middle terms in the expansion of (1 + x)^2n–1 is equal to ^2n–1C_n. 

The last two digits of the numbers 3⁴⁰⁰ are 01.

If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.

Number of terms in the expansion of (a + b)ⁿ where n ∈ N is one less than the power n.