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

State true or false for the following:

The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.

State true or false for the following:

The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.

State true or false for the following:

If three complex numbers z₁, z₂ and z₃ are in A.P., then they lie on a circle in the complex plane.

State true or false for the following:

If n is a positive integer, then the value of iⁿ + (i)ⁿ⁺¹ + (i)ⁿ⁺² + (i)ⁿ⁺³ is 0.

What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?

What is the smallest positive integer n, for which (1 + i)²ⁿ = (1 – i)²ⁿ?

What is the reciprocal of `3 + sqrt(7)i`.

What is the principal value of amplitude of 1 – i?

What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?

1 + i² + i⁴ + i⁶ + ... + i²ⁿ is ______.

If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.

The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.

The equation |z + 1 – i| = |z – 1 + i| represents a ______.

Number of solutions of the equation z² + |z|² = 0 is ______.

For a positive integer n, find the value of `(1 - i)^n (1 - 1/i)^"n"`

Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.

If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).

If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.

If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.

If z = x + iy, then show that `z  barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.