June 2025 Paper 2 Q2
2
In this question you must show detailed reasoning.
Solve the following equations.
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(x^2 - 5 = 8^{\frac{2}{3}}\) | M1 | 2.1 |
| \(x^2 - 5 = 4\) or \(x^2 = 9\) | A1 | 1.1 |
| \(x = \pm 3\) | A1 | 1.1 |
| [3] |
Notes
M1: Attempt to raise both sides to power \(\frac{2}{3}\) (May see \(\sqrt[\frac{3}{2}]{8}\))
May be implied by \(x^2 - 5 = 4\) or \(x^2 = 9\)
A1: www (e.g. A0 if \(8^{\frac{3}{2}} = 4\) seen).
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(3y = \ln 2\) | M1 | 2.1 |
| \(y = \frac{1}{3}\ln 2\) or 0.231 (3 sf) | A1 | 1.1 |
| [2] |
Notes
M1: Attempt to take logs of both sides.
A1: SCB1 for correct answer without working (max 1/2).
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(\left(x^2\right)^2 - 3\left(x^2\right) - 4 = 0\) \(\Rightarrow \left(x^2 - 4\right)\left(x^2 + 1\right) = 0\) or \(u = x^2 \Rightarrow u^2 - 3u - 4 = 0\) \(\Rightarrow (u - 4)(u + 1) = 0\) | M1 | 2.1 |
| \(x^2 = 4\) (or \(u = 4\)) | A1 | 1.1 |
| \(x = \pm 2\) | A1 | 1.1 |
| \(x^2 = -1\) has no roots | B1 | 1.1 |
| [4] |
Notes
M1: For recognising the given quartic as a quadratic in \(x^2\) or \(u = x^2\) and attempting to solve (allow a sign error in factorisation).
One of these four lines of working must be seen (oe), but the other may be implied by later correct work. Condone missing \(= 0\).
A1: Or \((x + 2)(x - 2)\) seen (NB this mark not implied by correct answers)
A1: Or \(x = 2\) and \(x = -2\) (and no other solutions e.g. A0 if \(x = 4\) seen)
B1: Accept equivalents but there must be an indication that there are no (real) solutions from \(x^2 = -1\) (e.g. ‘N/A’ or ‘X’ etc.) Or \(x = \pm i\)
If M0 then allow SCB2 for \((x - 2)(x + 2) \Rightarrow x = \pm 2\) (max. 2/4)
Answers only without working scores 0/4