| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Finding maximum/minimum on curve |
| Difficulty | Standard +0.8 This question requires finding a maximum using quotient rule differentiation and solving a cubic equation, then integration by substitution with u = 1 + x³ to find area and solve ln(1+p³)/3 = 1 for p. The substitution is non-trivial but recognizable, and solving for p requires algebraic manipulation of logarithms. More challenging than routine calculus but within standard P3 scope. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use the quotient rule | M1 | |
| Obtain correct derivative in any form | A1 | |
| Equate derivative to zero and solve for \(x\) | M1 | |
| Obtain answer \(x = \sqrt[3]{2}\), or exact equivalent | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply indefinite integral is of the form \(k\ln(1+x^3)\) | M1 | |
| State indefinite integral \(\frac{1}{3}\ln(1+x^3)\) | A1 | |
| Substitute limits correctly in an integral of the form \(k\ln(1+x^3)\) | M1 | |
| State or imply that the area of \(R\) is equal to \(\frac{1}{3}\ln(1+p^3) - \frac{1}{3}\ln 2\), or equivalent | A1 | |
| Use a correct method for finding \(p\) from an equation of the form \(\ln(1+p^3) = a\) or \(\ln((1+p^3)/2) = b\) | M1 | |
| Obtain answer \(p = 3.40\) | A1 | [2] |
## Question 10:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use the quotient rule | **M1** | |
| Obtain correct derivative in any form | **A1** | |
| Equate derivative to zero and solve for $x$ | **M1** | |
| Obtain answer $x = \sqrt[3]{2}$, or exact equivalent | **A1** | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply indefinite integral is of the form $k\ln(1+x^3)$ | **M1** | |
| State indefinite integral $\frac{1}{3}\ln(1+x^3)$ | **A1** | |
| Substitute limits correctly in an integral of the form $k\ln(1+x^3)$ | **M1** | |
| State or imply that the area of $R$ is equal to $\frac{1}{3}\ln(1+p^3) - \frac{1}{3}\ln 2$, or equivalent | **A1** | |
| Use a correct method for finding $p$ from an equation of the form $\ln(1+p^3) = a$ or $\ln((1+p^3)/2) = b$ | **M1** | |
| Obtain answer $p = 3.40$ | **A1** | [2] |10\\
\includegraphics[max width=\textwidth, alt={}, center]{efa44efb-9350-4d54-b5e1-4f722781a5f3-3_366_764_1914_687}
The diagram shows the curve $y = \frac { x ^ { 2 } } { 1 + x ^ { 3 } }$ for $x \geqslant 0$, and its maximum point $M$. The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = p$.\\
(i) Find the exact value of the $x$-coordinate of $M$.\\
(ii) Calculate the value of $p$ for which the area of $R$ is equal to 1 . Give your answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2015 Q10 [10]}}