Commerce (English Medium) Class 11 - CBSE Question Bank Solutions

If |z₁| = 1(z₁ ≠ –1) and z₂ = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z₂ is zero.

If |z₁| = |z₂| = ... = |z_n| = 1, then show that |z₁ + z₂ + z₃ + ... + z_n| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.

Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.

For any two complex numbers z₁, z₂ and any real numbers a, b, |az₁ – bz₂|² + |bz₁ + az₂|² = ______.

The value of `sqrt(-25) xx sqrt(-9)` is ______.

The number `(1 - i)^3/(1 - i^2)` is equal to ______.

The sum of the series i + i² + i³ + ... upto 1000 terms is ______.

Multiplicative inverse of 1 + i is ______.

If z₁ and z₂ are complex numbers such that z₁ + z₂ is a real number, then z₂ = ______.

If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.

State True or False for the following:

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

State True or False for the following:

For any complex number z the minimum value of |z| + |z – 1| is 1.

State True or False for the following:

The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).

State True or False for the following:

The inequality |z – 4| < |z – 2| represents the region given by x > 3.

Find `|(1 + i) ((2 + i))/((3 + i))|`.

Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.

The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.

The value of `(z + 3)(barz + 3)` is equivalent to ______.

If `((1 + i)/(1 - i))^x` = 1, then ______.

A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α² + β² = ______.