| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Roots of polynomial equations |
| Difficulty | Standard +0.3 Part (a) requires substituting a complex root into a cubic equation and using the conjugate root theorem - standard techniques for A-level. Part (b) involves expressing w in exponential form and algebraic manipulation with trigonometric identities, which is routine for P3/Further Maths complex numbers. Both parts are straightforward applications of well-practiced methods with no novel insight required, making this slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks |
|---|---|
| (a) Either: Substitute and expand \((-1 + \sqrt{5}i)^3\) completely | M1 |
| Use \(i^2 = -1\) correctly at least once | M1 |
| Obtain \(a = -12\) | A1 |
| State that the other complex root is \(-1 - \sqrt{5}i\) | B1 |
| Or1: State that the other complex root is \(-1 - \sqrt{5}i\) | B1 |
| State the quadratic factor \(z^2 + 2z + 6\) | B1 |
| Divide the cubic by a 3-term quadratic, equate remainder to zero and solve for \(a\) or, using a 3-term quadratic, factorise the cubic and determine \(a\) | M1 |
| Obtain \(a = -12\) | A1 |
| Or2: State that the other complex root is \(-1 - \sqrt{5}i\) | B1 |
| State or show the third root is 2 | B1 |
| Use a valid method to determine \(a\) | M1 |
| Obtain \(a = -12\) | A1 |
| Or3: Substitute and use De Moivre to cube \(\sqrt{6}\text{cis}(114.1°)\), or equivalent | M1 |
| Find the real and imaginary parts of the expression | M1 |
| Obtain \(a = -12\) | A1 |
| State that the other complex root is \(-1 - \sqrt{5}i\) | B1 |
| Total: 4 marks | |
| (b) Either: Substitute \(w = \cos 2\theta + i\sin 2\theta\) in the given expression | B1 |
| Use double angle formulae throughout | M1 |
| Express numerator and denominator in terms of \(\cos\theta\) and \(\sin\theta\) only | A1 |
| Obtain given answer correctly | A1 |
| Or: Substitute \(w = e^{2i\theta}\) in the given expression | B1 |
| Divide numerator and denominator by \(e^{i\theta}\), or equivalent | M1 |
| Express numerator and denominator in terms of \(\cos\theta\) and \(\sin\theta\) only | A1 |
| Obtain the given answer correctly | A1 |
| Total: 4 marks |
**(a)** Either: Substitute and expand $(-1 + \sqrt{5}i)^3$ completely | M1 |
Use $i^2 = -1$ correctly at least once | M1 |
Obtain $a = -12$ | A1 |
State that the other complex root is $-1 - \sqrt{5}i$ | B1 |
Or1: State that the other complex root is $-1 - \sqrt{5}i$ | B1 |
State the quadratic factor $z^2 + 2z + 6$ | B1 |
Divide the cubic by a 3-term quadratic, equate remainder to zero and solve for $a$ or, using a 3-term quadratic, factorise the cubic and determine $a$ | M1 |
Obtain $a = -12$ | A1 |
Or2: State that the other complex root is $-1 - \sqrt{5}i$ | B1 |
State or show the third root is 2 | B1 |
Use a valid method to determine $a$ | M1 |
Obtain $a = -12$ | A1 |
Or3: Substitute and use De Moivre to cube $\sqrt{6}\text{cis}(114.1°)$, or equivalent | M1 |
Find the real and imaginary parts of the expression | M1 |
Obtain $a = -12$ | A1 |
State that the other complex root is $-1 - \sqrt{5}i$ | B1 |
| Total: 4 marks |
**(b)** Either: Substitute $w = \cos 2\theta + i\sin 2\theta$ in the given expression | B1 |
Use double angle formulae throughout | M1 |
Express numerator and denominator in terms of $\cos\theta$ and $\sin\theta$ only | A1 |
Obtain given answer correctly | A1 |
Or: Substitute $w = e^{2i\theta}$ in the given expression | B1 |
Divide numerator and denominator by $e^{i\theta}$, or equivalent | M1 |
Express numerator and denominator in terms of $\cos\theta$ and $\sin\theta$ only | A1 |
Obtain the given answer correctly | A1 |
| Total: 4 marks |
---\begin{enumerate}[label=(\alph*)]
\item It is given that $-1 + (\sqrt{5})i$ is a root of the equation $z^3 + 2z + a = 0$, where $a$ is real. Showing your working, find the value of $a$, and write down the other complex root of this equation. [4]
\item The complex number $w$ has modulus 1 and argument $2\theta$ radians. Show that $\frac{w - 1}{w + 1} = i\tan\theta$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2014 Q7 [8]}}