A-Level Further Maths: Pure Mathematics Practice

What is the modulus of the complex number z = 3 - 4i?

Express z = 1 + √3i in polar form re^(iθ) with θ in radians.

Using De Moivre's theorem, find (1 + i)^8.

Find the sum of the four fourth roots of 16 in Cartesian form.

Given M = [[1,0,2],[3,1,-1],[0,2,1]], find the entry in row 2, column 3 of M².

A 3×3 matrix M has determinant 5. What is the determinant of 2M?

Matrix A = [[2,1],[0,3]] and B = [[1,-1],[4,2]]. Find (AB)⁻¹.

A transformation is represented by matrix T = [[0,-1,0],[1,0,0],[0,0,1]]. What 3D transformation does this represent?

What is the first step in any proof by mathematical induction?

Prove by induction: 1 + 2 + 3 + ... + n = n(n+1)/2. For the inductive step, assuming the formula holds for n = k, what expression represents the sum to k+1?

Prove by induction that 7^n - 1 is divisible by 6 for all n ∈ ℕ. In the inductive step, 7^(k+1) - 1 is rewritten as:

Prove by induction that the n×n matrix M^n = M for all n ≥ 1, given M = [[1,1],[0,1]]. What property of M must be used in the inductive step?

Which of the following is the correct definition of cosh x?

Evaluate cosh(ln 2) without a calculator.

Find arsinh(1) in exact logarithmic form.

Find d/dx [cosh(2x)] and ∫ sinh(3x) dx.