| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and stationary points |
| Difficulty | Standard +0.3 Part (i) requires standard differentiation of a sum involving negative power and linear function, solving dy/dx=0, and using second derivative test—routine calculus. Part (ii) is a straightforward volume of revolution calculation using the standard formula with polynomial expansion. Both parts follow textbook procedures with no novel insight required, making this slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative4.08d Volumes of revolution: about x and y axes |
| Answer | Marks |
|---|---|
| (ii) | dy |
| Answer | Marks |
|---|---|
| =40π/3 oe or 41.9 | *M1 |
| Answer | Marks |
|---|---|
| A1 | [7] |
| [5] | (kx−3)−2 |
Question 11:
--- 11 (i)
(ii) ---
11 (i)
(ii) | dy
=−k(kx−3)−2
Attempt diffn. and equate to 0 +k =0
dx
(kx−3)2 =1 or k3x2 −6k2x+8k (=0)
2 4
x= or
k k
d2y
=2k2(kx−3)−3
dx2
2 d2y ( −2k2)
When x= , = <0 MAX All previous
k dx2
4 d2y ( 2k2)
When x= , = >0 MIN working correct
k dx2
2
V =(π)∫(x−3)−1 +(x−3) dx
=(π)∫[(x−3)−2 +(x−3)2
+2]dx
(x−3)3
=(π)−(x−3)−1 + (+2x) Condone missing 2x
3
1 1
=(π) 1− +4− −9+0
3 3
=40π/3 oe or 41.9 | *M1
DM1
*A1*A1
B1
DB1
DB1
*M1
A1
A1
DM1
A1 | [7]
[5] | (kx−3)−2
Must contain + other
term(s)
Simplify to a quadratic
Legitimately obtained
Ak2(kx−3)−3
Ft must contain
where A>0
Convincing alt. methods (values
either side) must show which
values used & cannot use
x=3/k
Attempt to expand y² and then
integrate
Or
x3
−(x−3)−1 + −3x2 +9x+2x
3
Apply limits 0→2
2 missing → 28π/3 scores
M1A0A1M1A0A curve has equation $y = (kx - 3)^{-1} + (kx - 3)$, where $k$ is a non-zero constant.
\begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinates of the stationary points in terms of $k$, and determine the nature of each stationary point. Justify your answers. [7]
\end{enumerate}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item \includegraphics{figure_3}
The diagram shows part of the curve for the case when $k = 1$. Showing all necessary working, find the volume obtained when the region between the curve, the $x$-axis, the $y$-axis and the line $x = 2$, shown shaded in the diagram, is rotated through $360°$ about the $x$-axis. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2016 Q11 [12]}}