| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question with standard components: (a) routine quotient rule differentiation to verify a known result, (b) standard sketch of sec x, and (c) direct application of the volume formula requiring integration of sec²x, which students know from standard results. The question is slightly above average only because it combines multiple techniques, but each step is textbook-standard with no novel insight required. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.07q Product and quotient rules: differentiation4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| \(z = \frac{\sin x}{\cos x}\); \(\frac{dz}{dx} = \frac{\cos x \cos x - \sin x(-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x\) | M1 A1 A1 | use of quotient rule \(\left(\frac{\pm\cos^2 x \pm \sin^2 x}{\cos^2 x}\right)\); AG (be convinced); 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Graph with correct shape including asymptotic behaviour and symmetrical about \(x = 0\) and \(y > 0\) | M1 A1 | use of 1; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = (k)\int \sec^2 x\, dx = (k)[\tan x]_0^{\pi/3} = 4.89\) | M1 A1 A1 | CAO; 3 marks total |
### 7(a)
$z = \frac{\sin x}{\cos x}$; $\frac{dz}{dx} = \frac{\cos x \cos x - \sin x(-\sin x)}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$ | M1 A1 A1 | use of quotient rule $\left(\frac{\pm\cos^2 x \pm \sin^2 x}{\cos^2 x}\right)$; AG (be convinced); 3 marks total
### 7(b)
Graph with correct shape including asymptotic behaviour and symmetrical about $x = 0$ and $y > 0$ | M1 A1 | use of 1; 2 marks total
### 7(c)
$V = (k)\int \sec^2 x\, dx = (k)[\tan x]_0^{\pi/3} = 4.89$ | M1 A1 A1 | CAO; 3 marks total
**Total for Question 7:** 8 marks7
\begin{enumerate}[label=(\alph*)]
\item Given that $z = \frac { \sin x } { \cos x }$, use the quotient rule to show that $\frac { \mathrm { d } z } { \mathrm {~d} x } = \sec ^ { 2 } x$.
\item Sketch the curve with equation $y = \sec x$ for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.
\item The region $R$ is bounded by the curve $y = \sec x$, the $x$-axis and the lines $x = 0$ and $x = 1$.
Find the volume of the solid formed when $R$ is rotated through $2 \pi$ radians about the $x$-axis, giving your answer to three significant figures.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q7 [8]}}