Arts (English Medium) Class 11 - CBSE Question Bank Solutions for Mathematics

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

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 ______.

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

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

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

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.

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.

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 

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 ______.