Complex Numbers and Quadratic Equations

If (1 - i) is a root of the equation x² + ax + b = 0, where a, b ∈ ℝ, then find the values of a and b.

If α and β are different complex numbers with |β| = 1, then find |(^{β - α}/_{1 - ᾱβ})|

If z₁ = 2 - i, z₂ = 1 + i, find |(^{z₁ + z₂ + 1}/_{z1 - z2 + 1})|

If (a + ib)(c + id)(e + if)(g + ih) = A + iB, then show that

(a² + b²)(c² + d²)(e² + f²)(g² + h²) = A² + B²

Find the modulus and the argument of the complex number z = -√3 + i

Find the square-root of 3 + 4i

Express the following in the form a + ib:

^{5 + √2i}/_{1 - √2i}

Find the number of non-zero integral solutions of the equation |1 - i|^x = 2^x.

Solve the quadratic equation: x² - x + (1 + i) = 0

Reduce (¹/_{1 - 4i} - ²/_{1 + i})(^{3 - 4i}/_{5 + i}) to the standard form.

Convert the following in the polar form:

(i) ^{(1 + 7i)}/_{(2 - i)²}

(ii) ^{(1 + 3i)}/_{(1 - 2i)}

If (^{1 + i}/_{1 - i})^m = 1, then find the least positive integral value of m.

Find the condition that roots of the equation ax² + bx + c = 0 are in the ratio m : n.

If x - iy = √(^{a - ib}/_{c - id}), prove that (x² + y²)² = ^{a2 + b2}/_{c² + d²}.

Find the multiplicative inverse of 2 – 3i.

If 4x + i(3x - y) = 3 + i(-6), where x and y are real numbers, then find the value of x and y.