| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under curve with fractional/negative powers or roots |
| Difficulty | Moderate -0.3 This is a straightforward two-part question testing standard calculus techniques: differentiation using chain rule to find a tangent equation, and integration using substitution. Both parts follow routine procedures with no conceptual challenges—slightly easier than average due to the predictable structure and standard methods, though the algebraic manipulation (especially the substitution in part b) requires care. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks |
|---|---|
| 7(a) | 4 |
| Answer | Marks |
|---|---|
| 1 | M1 |
| Answer | Marks |
|---|---|
| Obtain correct −8(2x+1) 3 or unsimplified equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | M1 | Gradient must come from a differentiated |
| Answer | Marks |
|---|---|
| 2 4 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 7(b) | 2 |
| Answer | Marks |
|---|---|
| 2 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain correct 9(2x+1)3 or unsimplified equivalent | A1 | |
| Use correct limits correctly to find area | M1 | Substitute correct limits into an integrated |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain 27 | A1 | SC B1 if M1 A1 M0 scored. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 7:
--- 7(a) ---
7(a) | 4
−
Differentiate to obtain form k (2x+1) 3
1 | M1
4
−
Obtain correct −8(2x+1) 3 or unsimplified equivalent | A1
7
Attempt equation of tangent at ,6 with numerical gradient
2 | M1 | Gradient must come from a differentiated
expression.
1 31
Obtain y=− x+ or equivalent of requested form
2 4 | A1
4
Question | Answer | Marks | Guidance
--- 7(b) ---
7(b) | 2
Integrate to obtain form k (2x+1)3
2 | M1
2
Obtain correct 9(2x+1)3 or unsimplified equivalent | A1
Use correct limits correctly to find area | M1 | Substitute correct limits into an integrated
expression.
36 – 9 minimum working required.
Obtain 27 | A1 | SC B1 if M1 A1 M0 scored.
4
Question | Answer | Marks | Guidance\includegraphics{figure_7}
The diagram shows part of the curve with equation $y = \frac{12}{\sqrt{2x+1}}$. The point $A$ on the curve has coordinates $\left(\frac{7}{2}, 6\right)$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the tangent to the curve at $A$. Give your answer in the form $y = mx + c$. [4]
\item Find the area of the region bounded by the curve and the lines $x = 0$, $x = \frac{7}{2}$ and $y = 0$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q7 [8]}}