On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z + 2 - 3i| ≤ 2 and arg z < ¾π.

Verify that −1 + √2i is a root of the equation z⁴ + 3z² + 2z + 12 = 0.

The complex number u is defined by u = `(√2 – a√2i) / (1 + 2i)`, where a is a positive integer.

The polynomial x³ + 5x² + 31x + 75 is denoted by p(x).

Find the quotient and remainder when x⁴ - 3x³ + 9x² - 12x + 27 is divided by x² + 5.

Find the coefficient of x² in the expansion of (2x - 5)√4-x.

The positive numbers p and q are such that ln(p/q) = a and ln(q²p) = b. Express ln(p⁷q) in terms of a and b.