| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | March |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a standard integration by substitution question with a straightforward trigonometric substitution (u = cos x) that leads to a polynomial integral, followed by a routine optimization problem using differentiation. Both parts require only direct application of A-level techniques with no novel insight, making it slightly easier than average. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(du = -\sin x\, dx\) | B1 | |
| Using Pythagoras express the integral in terms of \(u\) | M1 | |
| Obtain integrand \(\pm\sqrt{u}(1 - u^2)\) | A1 | |
| Integrate and obtain \(-\frac{2}{3}u^{\frac{3}{2}} + \frac{2}{7}u^{\frac{7}{2}}\), or equivalent | A1 | |
| Change limits correctly and substitute correctly in an integral of the form \(au^{\frac{3}{2}} + bu^{\frac{7}{2}}\) | M1 | Or substitute original limits correctly in an integral of the form \(a(\cos x)^{\frac{3}{2}} + b(\cos x)^{\frac{7}{2}}\) |
| Obtain answer \(\frac{8}{21}\) | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use product rule and chain rule at least once | M1 | |
| Obtain correct derivative in any form | A1 + A1 | |
| Equate derivative to zero and obtain a horizontal equation in integral powers of \(\sin x\) and \(\cos x\) | M1 | |
| Use correct methods to obtain an equation in one trig function | M1 | |
| Obtain \(\tan^2 x = 6\), \(7\cos^2 x = 1\) or \(7\sin^2 x = 6\), or equivalent, and obtain answer \(1.183\) | A1 | |
| Total | 6 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $du = -\sin x\, dx$ | B1 | |
| Using Pythagoras express the integral in terms of $u$ | M1 | |
| Obtain integrand $\pm\sqrt{u}(1 - u^2)$ | A1 | |
| Integrate and obtain $-\frac{2}{3}u^{\frac{3}{2}} + \frac{2}{7}u^{\frac{7}{2}}$, or equivalent | A1 | |
| Change limits correctly and substitute correctly in an integral of the form $au^{\frac{3}{2}} + bu^{\frac{7}{2}}$ | M1 | Or substitute original limits correctly in an integral of the form $a(\cos x)^{\frac{3}{2}} + b(\cos x)^{\frac{7}{2}}$ |
| Obtain answer $\frac{8}{21}$ | A1 | |
| **Total** | **6** | |
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use product rule and chain rule at least once | M1 | |
| Obtain correct derivative in any form | A1 + A1 | |
| Equate derivative to zero and obtain a horizontal equation in integral powers of $\sin x$ and $\cos x$ | M1 | |
| Use correct methods to obtain an equation in one trig function | M1 | |
| Obtain $\tan^2 x = 6$, $7\cos^2 x = 1$ or $7\sin^2 x = 6$, or equivalent, and obtain answer $1.183$ | A1 | |
| **Total** | **6** | |10\\
\includegraphics[max width=\textwidth, alt={}, center]{dcfbe7af-c212-42b1-8a90-8e0418cf0ffd-16_330_689_264_726}
The diagram shows the curve $y = \sin ^ { 3 } x \sqrt { } ( \cos x )$ for $0 \leqslant x \leqslant \frac { 1 } { 2 } \pi$, and its maximum point $M$.\\
(i) Using the substitution $u = \cos x$, find by integration the exact area of the shaded region bounded by the curve and the $x$-axis.\\
(ii) Showing all your working, find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2019 Q10 [12]}}