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

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|.

Locate the points for which 3 < |z| < 4.

Find the value of 2x⁴ + 5x³ + 7x² – x + 41, when x = `-2 - sqrt(3)"i"`.

Find the value of k if for the complex numbers z₁ and z₂, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`

The real value of ‘a’ for which 3i³ – 2ai² + (1 – a)i + 5 is real is ______.

The value of `(- sqrt(-1))^(4"n" - 3)`, where n ∈ N, is ______.

If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n²) = ______.

State true or false for the following:

Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.