| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2015 |
| Session | June |
| Marks | 7 |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of argument on loci |
| Difficulty | Standard +0.8 This question combines standard loci (shading a circle) with optimization requiring geometric insight. Part (i) is routine, but part (ii) requires visualizing that minimum arg(z) occurs where the ray from origin is tangent to the circle, then applying trigonometry to find the angle. This geometric optimization with complex numbers is moderately challenging but accessible with careful diagram work. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| (ii) | Circle with centre (20, –15) |
| Answer | Marks |
|---|---|
| for – 0.927 (or 2π – 0.927 = 5.356) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Page 5 | Mark Scheme | Syllabus |
| Cambridge Pre-U – May/June 2015 | 9795 | 01 |
Question 7:
--- 7 (i)
(ii) ---
7 (i)
(ii) | Circle with centre (20, –15)
Circle of radius = 7 stated or deducible from diagram
Allow B1 for a circle entirely in the 4th Quad.
Interior of circle shaded (Ignore boundary in/out)
for z1 in correct place
for (sufficient) distances
( )
for arg(z1) = − tan−13 +tan−1 7 or equivalent, using other inverse trig. functions
4 2 4
( )
−tan−14
[NB – this is using a result in Q13]
3
for – 0.927 (or 2π – 0.927 = 5.356) | B1
B1
B1
[3]
B1
B1
M1
A1
[4]
Page 5 | Mark Scheme | Syllabus | Paper
Cambridge Pre-U – May/June 2015 | 9795 | 01\begin{enumerate}[label=(\roman*)]
\item On an Argand diagram, shade the region whose points satisfy
$$|z - 20 + 15\text{i}| \leqslant 7.$$ [3]
\item The complex number $z_1$ represents that value of $z$ in the region described in part (i) for which $\arg(z)$ is least. Mark $z_1$ on your Argand diagram and determine $\arg(z_1)$ correct to 3 decimal places. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2015 Q7 [7]}}