A2 June 2022 Paper 1 Q8

AQACurrent spec11 marksComplex Numbers

(a) The complex number \(w\) is such that\[\arg(w + 2\mathrm{i}) = \tan^{-1}\frac{1}{2}\]

It is given that \(w = x + \mathrm{i}y\), where \(x\) and \(y\) are real and \(x \gt 0\)

Find an equation for \(y\) in terms of \(x\) [2 marks]

(b) The complex number \(z\) satisfies both\[-\frac{\pi}{2} \leqslant \arg(z + 2\mathrm{i}) \leqslant \tan^{-1}\frac{1}{2} \qquad \text{and} \qquad |z - 2 + 3\mathrm{i}| \leqslant 2\]

The region \(R\) is the locus of \(z\)

Sketch the region \(R\) on the Argand diagram below. [4 marks]

Blank Argand diagram on a square grid, with Re and Im axes through O, each marked from −6 to 6

(i) Calculate the exact value of \(|z_1|\) [3 marks]

(ii) Express \(z_1\) in the form \(a + \mathrm{i}b\), where \(a\) and \(b\) are real. [2 marks]

Mark scheme (a)

Scheme	Marks	AO
Deduces correct gradient or intercept PI	M1	2.2a
Obtains correct equation with \(y\) as the subject	A1	1.1b
	(2)

Typical solution

Gradient = \(\dfrac{1}{2}\)

Line passes through \((0, -2)\)

\[y = \frac{1}{2}x - 2\]

Argand diagram: the half-line from −2i through 4 on the real axis, and the circle centre 2 − 3i radius 2, which touches the half-line at 2 − i; the region inside the circle and below the half-line is shaded — Mark scheme (b)

Enlarged part of the diagram: the half-line from −2i to 4 making angle alpha with the real axis at 4, a dashed perpendicular from O to the half-line, and the shaded circular region — Mark scheme (c)

Scheme	Marks	AO
(i) Identifies the point in their region nearest to the origin. For example draws the perpendicular from the half-line to the origin or Finds \(y = -2x\)	B1F	3.1a
Finds the distance between their valid point and the origin For example uses \(\sin\left(\tan^{-1}\frac{1}{2}\right)\) or Finds the distance between the origin and the point of intersection of the lines \(y = \frac{1}{2}x - 2\) and \(y = -2x\)	M1	3.1a
Obtains correct exact value of \(\|z_1\|\)	A1	1.1b
	(3)
(ii) Uses their values from part (c)(i) to obtain value of \(a\) or \(b\)	M1	1.1a
Obtains correct value of \(z_1\) CSO	A1	1.1b
	(2)
	(11 marks)

Scheme	Marks	AO
Draws a half line from \(-2\mathrm{i}\) passing through 4. Condone full line ft their linear equation in part (a)	B1F	1.1b
Draws circle or arc of a circle, with centre at \(2 - 3\mathrm{i}\) or radius 2	M1	2.2a
Draws circle or arc of a circle, centre at \(2 - 3\mathrm{i}\) and radius 2	A1	1.1b
Correct region indicated	A1	2.2a
	(4)