| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Volume of revolution with substitution |
| Difficulty | Standard +0.3 This question involves standard integration techniques (substitution for an improper integral, volume of revolution formula) and finding a normal line equation. Part (a) is a routine improper integral with a straightforward substitution u=3x-2. Part (b) applies the standard volume formula π∫y²dx over finite limits. Part (c) requires finding the derivative, calculating the normal gradient, and finding a y-intercept. All techniques are standard P1/C3 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08c Improper integrals: infinite limits or discontinuous integrands4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left\{\dfrac{(3x-2)^{-\frac{1}{2}}}{-1/2}\right\} \div \{3\}\) | B2, 1, 0 | Attempt to integrate |
| \(-\dfrac{2}{3}[0 - 1]\) | M1 | M1 for applying limits \(1 \to \infty\) to an integrated expression (either correct power or dividing by their power) |
| \(\dfrac{2}{3}\) | A1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([\pi]\int y^2\,dx = [\pi]\int(3x-2)^{-3}\,dx = [\pi]\dfrac{(3x-2)^{-2}}{-2\times 3}\) | \*M1 A1 | M1 for attempt to integrate \(y^2\) (power increases); allow 1 error. A1 for correct result in any form |
| \([\pi]\left[-\dfrac{1}{6}\right]\left[\dfrac{1}{16} - 1\right]\) | DM1 | Apply limits 1 and 2 to an integrated expression and subtract correctly; allow 1 error |
| \(\dfrac{5\pi}{32}\) | A1 | OE |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{\text{d}y}{\text{d}x} = -\dfrac{3}{2} \times 3(3x-2)^{-\frac{5}{2}}\) | M1 | M1 for attempt to differentiate (power decreases); allow 1 error |
| At \(x = 1\), \(\dfrac{\text{d}y}{\text{d}x} = -\dfrac{9}{2}\) | \*M1 | Substitute \(x = 1\) into *their* differentiated expression; allow 1 error |
| Equation of normal: \(y - 1 = \dfrac{2}{9}(x - 1)\) OR evaluates \(c\) | DM1 | Forms equation of line or evaluates \(c\) using \((1, 1)\) and gradient \(\dfrac{-1}{\text{their}\,\frac{\text{d}y}{\text{d}x}}\) |
| At \(A\), \(y = \dfrac{7}{9}\) | A1 | OE e.g. AWRT 0.778; must clearly identify \(y\)-intercept |
| Total | 4 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left\{\dfrac{(3x-2)^{-\frac{1}{2}}}{-1/2}\right\} \div \{3\}$ | B2, 1, 0 | Attempt to integrate |
| $-\dfrac{2}{3}[0 - 1]$ | M1 | M1 for applying limits $1 \to \infty$ to an integrated expression (either correct power or dividing by their power) |
| $\dfrac{2}{3}$ | A1 | |
| **Total** | **4** | |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[\pi]\int y^2\,dx = [\pi]\int(3x-2)^{-3}\,dx = [\pi]\dfrac{(3x-2)^{-2}}{-2\times 3}$ | \*M1 A1 | M1 for attempt to integrate $y^2$ (power increases); allow 1 error. A1 for correct result in any form |
| $[\pi]\left[-\dfrac{1}{6}\right]\left[\dfrac{1}{16} - 1\right]$ | DM1 | Apply limits 1 and 2 to an integrated expression and subtract correctly; allow 1 error |
| $\dfrac{5\pi}{32}$ | A1 | OE |
| **Total** | **4** | |
---
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{\text{d}y}{\text{d}x} = -\dfrac{3}{2} \times 3(3x-2)^{-\frac{5}{2}}$ | M1 | M1 for attempt to differentiate (power decreases); allow 1 error |
| At $x = 1$, $\dfrac{\text{d}y}{\text{d}x} = -\dfrac{9}{2}$ | \*M1 | Substitute $x = 1$ into *their* differentiated expression; allow 1 error |
| Equation of normal: $y - 1 = \dfrac{2}{9}(x - 1)$ OR evaluates $c$ | DM1 | Forms equation of line or evaluates $c$ using $(1, 1)$ and gradient $\dfrac{-1}{\text{their}\,\frac{\text{d}y}{\text{d}x}}$ |
| At $A$, $y = \dfrac{7}{9}$ | A1 | OE e.g. AWRT 0.778; must clearly identify $y$-intercept |
| **Total** | **4** | |10
\begin{enumerate}[label=(\alph*)]
\item Find $\int _ { 1 } ^ { \infty } \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$.\\
\includegraphics[max width=\textwidth, alt={}, center]{af7aeda9-2ded-4db4-9ff3-ed6adc67859f-16_499_689_1322_726}
The diagram shows the curve with equation $y = \frac { 1 } { ( 3 x - 2 ) ^ { \frac { 3 } { 2 } } }$. The shaded region is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 2$. The shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis.
\item Find the volume of revolution.\\
The normal to the curve at the point $( 1,1 )$ crosses the $y$-axis at the point $A$.
\item Find the $y$-coordinate of $A$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q10 [12]}}