NCERT Exemplar Solutions
Class 11 - Mathematics - Chapter 5: COMPLEX NUMBERS AND QUADRATIC EQUATIONS

Long Answer Questions

If |z + 1| = z + 2(1 + i), then find z.

If arg(z − 1) = arg(z + 3i), then find x − 1 : y, where z = x + iy.

Show that \(\left|\dfrac{z - 2}{z - 3}\right| = 2\) represents a circle. Find its centre and radius.

If \(\dfrac{z - 1}{z + 1}\) is a purely imaginary number (z ≠ −1), then find |z|.

z₁ and z₂ are two complex numbers such that |z₁| = |z₂| and arg(z₁) + arg(z₂) = π, then show that z₁ = − z̄₂.

If |z₁| = 1 (z₁ ≠ −1) and z₂ = \(\dfrac{z₁ − 1}{z₁ + 1}\), show that the real part of z₂ is zero.

If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, then find \(\arg\left(\dfrac{z₁}{z₄}\right) + \arg\left(\dfrac{z₂}{z₃}\right).\)

If |z₁| = |z₂| = … = |zₙ| = 1, then show that

\[|z₁ + z₂ + ⋯ + zₙ| = \left|\dfrac{1}{z₁} + \dfrac{1}{z₂} + ⋯ + \dfrac{1}{zₙ}\right|.\]

For complex numbers z₁ and z₂, if arg(z₁) − arg(z₂) = 0, then show that |z₁ − z₂| = ||z₁| − |z₂||.

Solve the system of equations Re(z²) = 0, |z| = 2.

Find the complex number satisfying the equation z + √2 |z + 1| + i = 0.

Write the complex number \(z = \dfrac{1 - i}{\cos \dfrac{\pi}{3} + i \sin \dfrac{\pi}{3}}\) in polar form.

If z and w are two complex numbers such that |zw| = 1 and arg(z) − arg(w) = \(\dfrac{\pi}{2}\), then show that \(\overline{z} w = −i\).