CAIE P3 2023 November — Question 1 5 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2023
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProduct & Quotient Rules
TypePoints with specific gradient
DifficultyStandard +0.3 This is a straightforward application of the quotient rule followed by solving a quadratic equation. The question requires finding dy/dx using the quotient rule, setting it equal to 8, and solving for x-coordinates, then finding corresponding y-values. While it involves multiple steps, each step is routine and the problem type (finding points with specific gradient) is a standard textbook exercise with no novel insight required.
Spec1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
1 Find the exact coordinates of the points on the curve \(y = \frac { x ^ { 2 } } { 1 - 3 x }\) at which the gradient of the tangent is equal to 8 .
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
Use correct quotient or product rule\*M1
Obtain correct derivative in any formA1 e.g. \(\dfrac{(1-3x)2x - x^2(-3)}{(1-3x)^2} = \dfrac{2x-3x^2}{(1-3x)^2}\) or \(3x^2(1-3x)^{-2} + 2x(1-3x)^{-1}\)
Equate derivative to 8 and solve for \(x\)DM1 \(75x^2 - 50x + 8 = (15x-4)(5x-2)\)
Obtain answers \(x = \dfrac{2}{5}\) and \(\dfrac{4}{15}\)A1 Exact values required
Obtain answers \(y = -\dfrac{4}{5}\) and \(\dfrac{16}{45}\)A1 Allow A1 for one correct point
Total5 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct quotient or product rule | \*M1 | |
| Obtain correct derivative in any form | A1 | e.g. $\dfrac{(1-3x)2x - x^2(-3)}{(1-3x)^2} = \dfrac{2x-3x^2}{(1-3x)^2}$ or $3x^2(1-3x)^{-2} + 2x(1-3x)^{-1}$ |
| Equate derivative to 8 and solve for $x$ | DM1 | $75x^2 - 50x + 8 = (15x-4)(5x-2)$ |
| Obtain answers $x = \dfrac{2}{5}$ and $\dfrac{4}{15}$ | A1 | Exact values required |
| Obtain answers $y = -\dfrac{4}{5}$ and $\dfrac{16}{45}$ | A1 | Allow A1 for one correct point |
| **Total** | **5** | |
1 Find the exact coordinates of the points on the curve $y = \frac { x ^ { 2 } } { 1 - 3 x }$ at which the gradient of the tangent is equal to 8 .\\

\hfill \mbox{\textit{CAIE P3 2023 Q1 [5]}}
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