Further Maths full proofs and worked examples

Prove by mathematical induction that 1(3 × 1 + 1) + 2(3 × 2 + 1) + ⋯ + n(3n + 1) = n(n + 1)² for every positive integer n.

6 marks

Your answer and working

The polynomial p(z) = z³ − 7z² + 25z − 39 has real coefficients. One root is 2 + 3i.

Solve p(z) = 0, showing your working.

6 marks

Your answer and working

Let $z=x+iy$ .

(a) On an Argand diagram, sketch the locus $|z-(2+i)|=3$ , stating its centre and radius.

3 marks

(b) Find the complex numbers on this locus for which $\operatorname{Re}(z)=2$ . Show your working.

3 marks

Your answer and working

The matrix $A=\begin{pmatrix}2&1\\1&3\end{pmatrix}$ represents a linear transformation of the plane, acting on column coordinate vectors. A triangle has vertices $O=(0,0)$ , $P=(2,0)$ and $Q=(0,3)$ .

(a) Find the image of each vertex and sketch the image triangle.

3 marks

(b) Find the area of the image triangle. Show how the matrix determines your answer.

3 marks

Your answer and working

Further Maths full proofs and worked examples

Write the whole induction proof

Find every root

Sketch a complex locus

Transform a triangle