| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Integration with differentiation context |
| Difficulty | Standard +0.8 Part (a) requires quotient rule differentiation and solving a quartic equation numerically. Part (b) involves a non-obvious substitution (u = √3x²) requiring careful manipulation of differentials and recognizing the resulting arctangent integral form. The combination of techniques and the substitution's complexity elevate this above standard exercises. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form, e.g. \(\dfrac{(1+3x^4) - x\times 12x^3}{(1+3x^4)^2}\) | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(0.577\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(du = 2\sqrt{3x}\,dx\), or equivalent | B1 | |
| Substitute for \(x\) and \(dx\) | M1 | |
| Obtain integrand \(\dfrac{1}{2\sqrt{3}(1+u^2)}\), or equivalent | A1 | |
| State integral of the form \(a\tan^{-1}u\) and use limits \(u=0\) and \(u=\sqrt{3}\) (or \(x=0\) and \(x=1\)) correctly | M1 | |
| Obtain answer \(\dfrac{\sqrt{3}}{18}\pi\), or exact equivalent | A1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use quotient or product rule | M1 | |
| Obtain correct derivative in any form, e.g. $\dfrac{(1+3x^4) - x\times 12x^3}{(1+3x^4)^2}$ | A1 | |
| Equate derivative to zero and solve for $x$ | M1 | |
| Obtain answer $0.577$ | A1 | |
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## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $du = 2\sqrt{3x}\,dx$, or equivalent | B1 | |
| Substitute for $x$ and $dx$ | M1 | |
| Obtain integrand $\dfrac{1}{2\sqrt{3}(1+u^2)}$, or equivalent | A1 | |
| State integral of the form $a\tan^{-1}u$ and use limits $u=0$ and $u=\sqrt{3}$ (or $x=0$ and $x=1$) correctly | M1 | |
| Obtain answer $\dfrac{\sqrt{3}}{18}\pi$, or exact equivalent | A1 | |
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\includegraphics[max width=\textwidth, alt={}, center]{3149080d-ad1a-4d2e-8e20-eb9977ced619-08_318_750_260_699}
The diagram shows the curve $y = \frac { x } { 1 + 3 x ^ { 4 } }$, for $x \geqslant 0$, and its maximum point $M$.
\begin{enumerate}[label=(\alph*)]
\item Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.
\item Using the substitution $u = \sqrt { 3 } x ^ { 2 }$, find by integration the exact area of the shaded region bounded by the curve, the $x$-axis and the line $x = 1$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q6 [9]}}