| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.3 Part (i) is a straightforward volume of revolution requiring the standard formula V = π∫y² dx with a simple integrand (x³ + x²) that integrates routinely. Part (ii) involves finding dy/dx using the chain rule, evaluating at x=3, finding the normal gradient, and determining the y-intercept—all standard A-level techniques with no novel insight required. This is slightly easier than average due to the algebraic simplicity. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| 9(i) | ( ) | |
| V =( π ) ∫ x3 +x2 ( dx ) | M1 | Attempt ∫y2dx |
| Answer | Marks |
|---|---|
| 0 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | DM1 | May be implied by a correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | A1 | Accept 91.9 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 9(ii) | dy 1( )−1/2× ( ) |
| Answer | Marks | Guidance |
|---|---|---|
| dx 2 | B2,1,0 | Omission of 3x2 +2x is one error |
| (At x = 3,) y = 6 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 6 4 | DB1ft | Ft on their dy/dx providing differentiation attempted |
| Answer | Marks | Guidance |
|---|---|---|
| 11 | DM1 | Equation through (3, their 6) and with gradient ‒1/their |
| Answer | Marks |
|---|---|
| 11 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | ( )
V =( π ) ∫ x3 +x2 ( dx ) | M1 | Attempt ∫y2dx
3
( π ) x4 + x3
4 3
0 | A1
( π )81 +9 ( −0 )
4 | DM1 | May be implied by a correct answer
117π
oe
4 | A1 | Accept 91.9
If additional areas rotated about x-axis, maximum of
M1A0DM1A0
4
Question | Answer | Marks | Guidance
--- 9(ii) ---
9(ii) | dy 1( )−1/2× ( )
= x3 +x2 3x2 +2x
dx 2 | B2,1,0 | Omission of 3x2 +2x is one error
(At x = 3,) y = 6 | B1
1 1 11
At x = 3, m= × ×33 = soi
2 6 4 | DB1ft | Ft on their dy/dx providing differentiation attempted
Equation of normal is y−6=− 4 ( x−3 )
11 | DM1 | Equation through (3, their 6) and with gradient ‒1/their
m
1
When x = 0, y = 7 oe
11 | A1
6
Question | Answer | Marks | Guidance\includegraphics{figure_9}
The diagram shows part of the curve with equation $y = \sqrt{x^3 + x^2}$. The shaded region is bounded by the curve, the $x$-axis and the line $x = 3$.
\begin{enumerate}[label=(\roman*)]
\item Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360°$ about the $x$-axis. [4]
\item $P$ is the point on the curve with $x$-coordinate $3$. Find the $y$-coordinate of the point where the normal to the curve at $P$ crosses the $y$-axis. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2019 Q9 [10]}}