| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Square roots of complex numbers |
| Difficulty | Standard +0.8 This question requires finding square roots of complex numbers algebraically (setting up and solving simultaneous equations from $(a+bi)^2 = -1+4\sqrt{3}i$), then sketching a circle locus and finding the maximum argument using geometric reasoning with tangent lines from the origin. While methodical, it combines multiple non-trivial techniques and requires careful algebraic manipulation with surds, placing it moderately above average difficulty. |
| Spec | 4.02h Square roots: of complex numbers4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Square \(x + iy\) and equate real and imaginary parts to \(-1\) and \(4\sqrt{3}\) | M1 | |
| Obtain \(x^2 - y^2 = -1\) and \(2xy = 4\sqrt{3}\) | A1 | |
| Eliminate one unknown and find an equation in the other | M1 | |
| Obtain \(x^4 + x^2 - 12 = 0\) or \(y^4 - y^2 - 12 = 0\), or three term equivalent | A1 | |
| Obtain answers \(\pm(\sqrt{3} + 2i)\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Show a circle with centre \(-1 + 4\sqrt{3}\) in a relatively correct position | B1 | |
| Show a circle with radius 1 and centre not at the origin | B1 | |
| Carry out a complete method for calculating the greatest value of \(\arg z\) | M1 | |
| Obtain answer 1.86 or 106.4° | A1 | [4] |
**(i)** Square $x + iy$ and equate real and imaginary parts to $-1$ and $4\sqrt{3}$ | M1 |
Obtain $x^2 - y^2 = -1$ and $2xy = 4\sqrt{3}$ | A1 |
Eliminate one unknown and find an equation in the other | M1 |
Obtain $x^4 + x^2 - 12 = 0$ or $y^4 - y^2 - 12 = 0$, or three term equivalent | A1 |
Obtain answers $\pm(\sqrt{3} + 2i)$ | A1 | [5]
[If the equations are solved by inspection, give B2 for the answers and B1 for justifying them]
**(ii)** Show a circle with centre $-1 + 4\sqrt{3}$ in a relatively correct position | B1 |
Show a circle with radius 1 and centre not at the origin | B1 |
Carry out a complete method for calculating the greatest value of $\arg z$ | M1 |
Obtain answer 1.86 or 106.4° | A1 | [4]7 The complex number $u$ is given by $u = - 1 + ( 4 \sqrt { } 3 ) \mathrm { i }$.\\
(i) Without using a calculator and showing all your working, find the two square roots of $u$. Give your answers in the form $a + \mathrm { i } b$, where the real numbers $a$ and $b$ are exact.\\
(ii) On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying the relation $| z - u | = 1$. Determine the greatest value of $\arg z$ for points on this locus.\\
$8 \quad$ Let $f ( x ) = \frac { 5 x ^ { 2 } + x + 6 } { ( 3 - 2 x ) \left( x ^ { 2 } + 4 \right) }$.\\
\hfill \mbox{\textit{CAIE P3 2015 Q7 [9]}}