AS June 2025 Paper 1 Q1
1
Express \(z\) in cartesian form. [3]
| Scheme | Marks | AO |
|---|---|---|
| \(x = 7\cos 2.2\) or \(y = 7\sin 2.2\) | M1 | 1.1 |
| \(z = -4.12\ldots\) | A1 | 1.1 |
| \(\ldots + 5.66\mathrm{i}\) | A1 | 1.1 |
| [3] |
Notes
M1: Allow for \(7\cos\theta\) where \(\theta\) from an attempt to convert to degrees
Allow for related angle
Note: \(\cos 2.2 = -0.5885\ldots\)
\(\sin 2.2 = 0.8084964\ldots\)
A1 A1: Allow answers rounding to \(-4.12 + 5.66\mathrm{i}\)
Must be in Cartesian form \((-4.12 + 5.66\mathrm{i})\) for full marks
\(-4.11950782\ldots + 5.6594748\ldots\mathrm{i}\)
If written as \((\pm 4.12, \pm 5.66)\) then allow SC B1 as long as one sign correct
Alternative method (ALT)
| Scheme | Marks |
|---|---|
| \(x^2 + y^2 = 49\) \(\dfrac{y}{x} = \tan 2.2 = -1.3738\ldots\) \(x^2 + (-1.3738x)^2 = 49\) \(x^2 = 16.97\ldots\) | M1 |
| \(z = -4.12\ldots\) | A1 |
| \(\ldots + 5.66\mathrm{i}\) | A1 |
M1: Setting up simultaneous equations and substituting to find an equation in \(x\) or \(y\) only.
Allow errors in substitution for M1
Allow \(x^2 + y^2 = 7\) or \(\tan 2.2 = \dfrac{x}{y}\)
Do not allow \(\tan\left(\dfrac{y}{x}\right) = 2.2\)
A1 A1: A0 if any other values given as well unless clearly rejected
| Scheme | Marks | AO |
|---|---|---|
| \((a + b\mathrm{i})^2 = a^2 - b^2 + 2ab\mathrm{i}\) | B1 | 1.1 |
| \(a^2 - b^2 = 1\) and \(2ab = 4\sqrt{3}\) (where \(a\) and \(b\) are real) | M1 | 1.1 |
| \(b = \dfrac{2\sqrt{3}}{a} \Rightarrow a^2 - \left(\dfrac{2\sqrt{3}}{a}\right)^2 = 1\) \(\therefore \left(a^2\right)^2 - a^2 - 12 = 0\) | M1 | 1.1 |
| \(a\) real \(\Rightarrow a^2 = 4\) only | A1 | 1.1 |
| so \(\pm\left(2 + \sqrt{3}\mathrm{i}\right)\) oe only | A1 | 1.1 |
| [5] |
Notes
B1: soi in solution
M1: Condone loss of exact form for M1M1A1
This correct line implies B1
Allow M1 for sign mistake when expanding
M1: Eliminating \(b\) or \(a\) to obtain 3 term quadratic in \(a^2\) or \(b^2\). Non-zero terms on same side. “\(= 0\)” seen or can be implied by solution.
\(\left(a^2\right)^2\) or \(a^4\).
Eliminating \(a\) leads to \(b^4 + b^2 - 12 = 0\)
Factorised forms: \(\left(a^2 - 4\right)\left(a^2 + 3\right)\) or \(\left(b^2 + 4\right)\left(b^2 - 3\right)\)
A1: Or equivalent in \(b\)
Some indication of root rejection
Evidence could be just the positive value of \(a^2\) appearing, but if both roots appear both must be correct and one must be rejected
A1: Allow \(2 + \sqrt{3}\mathrm{i}\), \(-2 - \sqrt{3}\mathrm{i}\)
Not \(\pm 2 \pm \sqrt{3}\mathrm{i}\). Not \(\pm 2 + \sqrt{3}\mathrm{i}\).
Not after loss of exact form unless recovery clear.
Independent of previous A mark