| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Integration with differentiation context |
| Difficulty | Standard +0.3 Part (a) requires applying the product rule to find dy/dx, setting it to zero, and solving—a standard differentiation exercise. Part (b) involves integration by parts with a power function and ln x, which is a common textbook technique. Both parts are routine applications of core calculus methods with no novel insight required, making this slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct product rule or correct quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | \(y' = \frac{x^{-\frac{2}{3}}}{x} - \frac{2}{3}x^{-\frac{5}{3}}\ln x\) |
| Equate 2 term derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = e^{\frac{3}{2}}\) | A1 | Or exact equivalent |
| Obtain answer \(y = \frac{3}{2e}\) | A1 | Or exact equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Commence integration and reach \(ax^{\frac{1}{3}}\ln x + b\int x^{\frac{1}{3}} \cdot \frac{1}{x}\,dx\) | *M1 | |
| Obtain \(3x^{\frac{1}{3}}\ln x - 3\int x^{\frac{1}{3}} \cdot \frac{1}{x}\,dx\) | A1 | |
| Complete the integration and obtain \(3x^{\frac{1}{3}}\ln x - 9x^{\frac{1}{3}}\) | A1 | OE |
| Use limits correctly in an expression of the form \(px^{\frac{1}{3}}\ln x + qx^{\frac{1}{3}}\) \((pq \neq 0)\) | DM1 | \(6\ln 8 - 9\times 2 - 0 + 9\) |
| Obtain \(18\ln 2 - 9\) from full and correct working | A1 | AG. Need to see \(\ln 8 = 3\ln 2\) |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct product rule or correct quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | $y' = \frac{x^{-\frac{2}{3}}}{x} - \frac{2}{3}x^{-\frac{5}{3}}\ln x$ |
| Equate 2 term derivative to zero and solve for $x$ | M1 | |
| Obtain answer $x = e^{\frac{3}{2}}$ | A1 | Or exact equivalent |
| Obtain answer $y = \frac{3}{2e}$ | A1 | Or exact equivalent |
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## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Commence integration and reach $ax^{\frac{1}{3}}\ln x + b\int x^{\frac{1}{3}} \cdot \frac{1}{x}\,dx$ | *M1 | |
| Obtain $3x^{\frac{1}{3}}\ln x - 3\int x^{\frac{1}{3}} \cdot \frac{1}{x}\,dx$ | A1 | |
| Complete the integration and obtain $3x^{\frac{1}{3}}\ln x - 9x^{\frac{1}{3}}$ | A1 | OE |
| Use limits correctly in an expression of the form $px^{\frac{1}{3}}\ln x + qx^{\frac{1}{3}}$ $(pq \neq 0)$ | DM1 | $6\ln 8 - 9\times 2 - 0 + 9$ |
| Obtain $18\ln 2 - 9$ from full and correct working | A1 | AG. Need to see $\ln 8 = 3\ln 2$ |
---9 The equation of a curve is $y = x ^ { - \frac { 2 } { 3 } } \ln x$ for $x > 0$. The curve has one stationary point.
\begin{enumerate}[label=(\alph*)]
\item Find the exact coordinates of the stationary point.
\item Show that $\int _ { 1 } ^ { 8 } y \mathrm {~d} x = 18 \ln 2 - 9$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q9 [10]}}