June 2025 Paper 1 Q7
7
In this question you must show detailed reasoning.
A curve has parametric equations \(x = t^3 + t^2\), \(y = t^2 + 2t\) for all real values of \(t\).
The curve passes through the point \(P\) with coordinates \((2, 3)\).
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = 3t^2 + 2t\), \(\dfrac{\mathrm{d}y}{\mathrm{d}t} = 2t + 2\), \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\;\frac{\mathrm{d}y}{\mathrm{d}t}\;}{\frac{\mathrm{d}x}{\mathrm{d}t}}\) | M1 | 1.1a |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{2t + 2}{3t^2 + 2t}\) | A1 | 2.1 |
| \(t = 1\) | B1 | 3.1a |
| \(y - 3 = \frac{4}{5}(x - 2)\) | M1 | 1.1 |
| \(5y = 4x + 7\) AG | A1 | 2.1 |
| [5] |
Notes
M1: Attempt to find \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\). Correctly combine attempts at algebraic or numerical derivatives
A1: Correct \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\). Algebraic or numerical (ie \(\frac{4}{5}\)) www
B1: Correct \(t\) stated or explicitly used. Ignore any additional \(t\) values
M1: Attempt equation of tangent, using \((2, 3)\) and their numerical gradient (from clearly using their \(t\) value in their attempt at parametric differentiation)
If using \(y = mx + c\) then M1 is awarded when value for \(c\) is obtained
M0 if using \(x = 3\), \(y = 2\)
A1: Obtain given answer www
If trying to start with a Cartesian equation there are a number of different, correct, solutions. To gain any credit it would have to be a valid attempt in obtaining the initial equation, condoning only a slip. Please consult with your TL if seen.
| Scheme | Marks | AO |
|---|---|---|
| DR | ||
| \(5(t^2 + 2t) = 4(t^3 + t^2) + 7\) | M1* | 3.1a |
| \(4t^3 - t^2 - 10t + 7\ (= 0)\) | A1 | 2.1 |
| \((t - 1)(t - 1)(4t + 7)\) \(t = -\frac{7}{4}\) | M1dep* | 3.1a |
| \(\left(-\frac{147}{64}, -\frac{7}{16}\right)\) oe | A1 | 2.1 |
| [4] |
Notes
M1*: Attempt to solve equations of tangent and curve simultaneously. Or valid attempt to obtain an equation in \(x\) or \(y\) only
A1: Obtain correct cubic. Brackets expanded and like terms collected
Some (usually BC) methods may not explicitly state simplified cubic; if M1 awarded and \(t = -\frac{7}{4}\) seen www then allow first A1 BOD, with then possibly SCB1
M1dep*: Attempt to solve cubic to obtain \(t\) value other than \(t = 1\). DR so method needed
Factorise to 3 linear factors and solve oe, eg using division
If using \((t - 1)^2\) as repeated factor, M1 for \((t - 1)^2(4t \pm 7)\)
If using \((t - 1)\) as linear factor then need both attempt at quadratic factor (factorisation or division) and also attempt to solve quadratic – BOD for slips
A1: Obtain correct coordinates, in exact form, www. Ignore \((2, 3)\) if also given, but A0 if other point(s) as well
If M1 A1 awarded for correct cubic, but no method shown to solve either cubic, or quadratic from \((t - 1)\)(quadratic factor), then SC B1 for correct coordinates www ie max of 3 marks
See comment in part (a) with regard to using Cartesian equation