A2 October 2020 Paper 1 Q4

OCR ACurrent spec4 marksComplex Numbers De Moivre's Theorem

4 In this question you must show detailed reasoning.

(a) Determine the square roots of \(25\mathrm{i}\) in the form \(r\mathrm{e}^{\mathrm{i}\theta}\), where \(0 \leqslant \theta \lt 2\pi\). [3]

(b) Illustrate the number \(25\mathrm{i}\) and its square roots on an Argand diagram. [1]

Mark scheme (a)

Scheme	Marks	AO
\(25\mathrm{i} = 25\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\) oe	B1	3.3
\(25\mathrm{i} = 25\mathrm{e}^{\frac{\pi}{2}\mathrm{i}}\) \(\Rightarrow \sqrt{25\mathrm{i}} = 5\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}\) and \(5\mathrm{e}^{\frac{5\pi}{4}\mathrm{i}}\)	M1	3.1a
So the square roots are \(5\mathrm{e}^{\frac{\pi}{4}\mathrm{i}}\) and \(5\mathrm{e}^{\frac{5\pi}{4}\mathrm{i}}\)	A1	3.2a
	[3]

Notes

B1: Conversion soi

M1: Attempt to find at least one root by rooting the modulus and halving the argument soi

A1: Both

Alternative method

Scheme	Marks
\(\sqrt{25\mathrm{i}} = a + b\mathrm{i}\)	B1
\(\Rightarrow a = b = \pm\dfrac{5}{2}\sqrt{2}\)	M1
\(\Rightarrow \sqrt{25\mathrm{i}} = \pm\dfrac{5}{2}\sqrt{2}(1 + \mathrm{i}) = \pm 5\left(\cos\dfrac{1}{4}\pi + \mathrm{i}\sin\dfrac{1}{4}\pi\right)\) \(\Rightarrow \sqrt{25\mathrm{i}} = 5\mathrm{e}^{\frac{1}{4}\pi\mathrm{i}}\) and \(5\mathrm{e}^{\frac{5}{4}\pi\mathrm{i}}\)	A1
	[3]

B1: Conversion and attempt to square

M1: \(a\) and \(b\). Ignore \(\pm\)

A1: Both

(corrected from the printed mark scheme: the last line of this method is printed as \(5\mathrm{e}^{\frac{1}{4}\pi}\) and \(5\mathrm{e}^{\frac{5}{4}\pi}\), without the \(\mathrm{i}\) in the exponents)

Argand diagram: a thick line from the origin straight up the positive imaginary axis representing 25i, and a shorter thick line through the origin at 45 degrees, from the third quadrant to the first quadrant, representing the two square roots — Mark scheme (b)

Scheme	Marks	AO
	B1	1.1
	[1]