| 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 a transcendental equation numerically. The combination of calculus techniques, algebraic manipulation, and numerical methods elevates this above standard textbook exercises. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| (i) 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] | |
| (ii) State or imply indefinite integral 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] |
**(i)** 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] |
**(ii)** State or imply indefinite integral 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]{5d76d7cb-c2cb-4fa4-8133-d5d43702b293-3_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]}}