Arts (English Medium) कक्षा ११ - CBSE Question Bank Solutions

Prove the statement by using the Principle of Mathematical Induction:

2 + 4 + 6 + ... + 2n = n² + n for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ – 1 for all natural numbers n.

Prove the statement by using the Principle of Mathematical Induction:

1 + 5 + 9 + ... + (4n – 3) = n(2n – 1) for all natural numbers n.

A sequence a₁, a₂, a₃ ... is defined by letting a₁ = 3 and a_k = 7a_{k – 1} for all natural numbers k ≥ 2. Show that a_n = 3.7^n–1 for all natural numbers.

A sequence b₀, b₁, b₂ ... is defined by letting b₀ = 5 and b_k = 4 + b_{k – 1} for all natural numbers k. Show that b_n = 5 + 4n for all natural number n using mathematical induction.

A sequence d₁, d₂, d₃ ... is defined by letting d₁ = 2 and d_k = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that d_n = `2/(n!)` for all n ∈ N.

Prove that for all n ∈ N.
cos α + cos(α + β) + cos(α + 2β) + ... + cos(α + (n – 1)β) = `(cos(alpha + ((n - 1)/2)beta)sin((nbeta)/2))/(sin  beta/2)`.

Prove that, cosθ cos2θ cos2²θ ... cos2^n–1θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.

Prove that, sinθ + sin2θ + sin3θ + ... + sinnθ = `((sin ntheta)/2 sin  ((n + 1))/2 theta)/(sin  theta/2)`, for all n ∈ N.

Show that `n^5/5 + n^3/3 + (7n)/15` is a natural number for all n ∈ N.

Prove that `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24`, for all natural numbers n > 1.

Prove that number of subsets of a set containing n distinct elements is 2ⁿ, for all n ∈ N.

If 10ⁿ + 3.4ⁿ⁺² + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.

For all n ∈ N, 3.5²ⁿ⁺¹ + 2³ⁿ⁺¹ is divisible by ______.

If xⁿ – 1 is divisible by x – k, then the least positive integral value of k is ______.

If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.

State whether the following statement is true or false. Justify.

Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.

Evaluate: (1 + i)⁶ + (1 – i)³ 

If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a² + b²)

If z₁, z₂, z₃ are complex numbers such that `|z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3|` = 1, then find the value of |z₁ + z₂ + z₃|.