AS June 2025 Paper 1 Q9

OCR ACurrent spec8 marksComplex Numbers

(a) Prove that if \(z\) is any complex number then \(z(z^*) = |z|^2\). [1]

(b) Prove that if \(w\) and \(z\) are any complex numbers then \((wz)^* = w^*z^*\). [2]

The complex number \(v\) is defined by \(v = (10 + 3\mathrm{i})(9 + 4\mathrm{i})\).

(c) In this question you must show detailed reasoning.
Express \(v\) in the form \(a + b\mathrm{i}\) where \(a\) and \(b\) are real. [2]

(d) In this question you must show detailed reasoning.
By considering \(v(v^*)\), write 10573 as the product of two prime factors. [3]

Mark scheme (a)

Scheme	Marks	AO
[\(z = a + b\mathrm{i}\), \(a\), \(b\) real] \([zz^*] = (a + b\mathrm{i})(a - b\mathrm{i}) = a^2 + b^2\) \(\|z\|^2 = a^2 + b^2\)	B1	2.1
	[1]

Notes

B1: AG
Must see either \((a + b\mathrm{i})(a - b\mathrm{i})\)
Or: \(z = a + b\mathrm{i}\)
\(z^* = a - b\mathrm{i}\)
\(zz^* = a^2 + b^2\)
Allow \(\left(\sqrt{(a^2 + b^2)}\right)^2\) for \(|z|^2\)
BOD lack of explicit equality shown
Must clearly attempt both sides
(Corrected from the printed mark scheme: the second line is printed as \(z = a - b\mathrm{i}\); it means \(z^* = a - b\mathrm{i}\).)

Mark scheme (b)

Scheme	Marks	AO
(\(z = a + b\mathrm{i}\), \(w = c + d\mathrm{i}\), \(a\), \(b\), \(c\), \(d\) real) \(\Rightarrow\) LHS \(= (ac - bd + (ad + bc)\mathrm{i})^*\) \(= ac - bd - (ad + bc)\mathrm{i}\)	M1	2.1
RHS \(= (a - b\mathrm{i})(c - d\mathrm{i}) = ac - bd - (ad + bc)\mathrm{i}\)	A1	1.1
	[2]

Notes

M1: AG. Setting up proof and finding one side.
\(w\) and \(z\) must be different.
For M1 BOD missing “big brackets”

A1: Finding other side and completing proof.
Must be complete and correct proof including insertion of all relevant brackets

Mark scheme (c)

Scheme	Marks	AO
\((10 + 3\mathrm{i})(9 + 4\mathrm{i}) = 90 - 12 + 27\mathrm{i} + 40\mathrm{i}\)	M1	1.1
\(= 78 + 67\mathrm{i}\)	A1	1.1
	[2]

Notes

M1: Expanding brackets using \(\mathrm{i}^2 = -1\)

A1: SCB1 for correct answer with no workings.

Mark scheme (d)

Scheme	Marks	AO
\(\therefore v(v^*) = \|v\|^2 = 78^2 + 67^2 = 10573\)	B1	1.1
\(= (10 + 3\mathrm{i})(10 - 3\mathrm{i})(9 + 4\mathrm{i})(9 - 4\mathrm{i})\)	M1	3.1a
\(= (100 + 9)(81 + 16) = 109 \times 97\)	A1	2.2a
	[3]

Notes

M1: Considering \(v(v^*)\) as the product of four complex numbers
Could be for awarded for finding \((10 + 3\mathrm{i})(10 - 3\mathrm{i}) = 109\) or \((9 + 4\mathrm{i})(9 - 4\mathrm{i}) = 97\)

A1: No need to assert primality of 109 and 97.
Need to see evidence that they have expressed \(v(v^*)\) as the product of four complex numbers.