| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Standard +0.3 This is a structured multi-part C2 question involving standard differentiation of power functions, finding tangent equations using point-gradient form, solving simultaneous equations, basic integration, and calculating an area. All techniques are routine for C2 level with clear scaffolding through parts (a)-(d), making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08b Integrate x^n: where n != -1 and sums1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{3}{2}x^{\frac{1}{2}} - 3\) | M1, A1 | One term correct; Both correct |
| Total (a) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| When \(x=0\), \(\frac{dy}{dx} = -3\) | B1F\(\surd\) | Ft provided answer \(< 0\) |
| Equation of tangent at \(O\) is \(y = -3x\) | B1F\(\surd\) | OE Ft on \(y'(0)\) |
| Total (b)(i) | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| At \((9,0)\): \(\frac{dy}{dx} = \frac{3}{2}(9)^{\frac{1}{2}} - 3\) | M1 | Attempt to find \(y'(9)\) |
| Equation tangent at \(A\): \(y - 0 = y'(9)[x-9]\) | m1 | OE |
| \(\Rightarrow y = \frac{3}{2}(x-9) \Rightarrow 2y = 3x - 27\) | A1 | CSO. AG |
| Total (b)(ii) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Eliminating \(y \Rightarrow -6x = 3x - 27\) | M1 | OE method to one variable (eg \(2y = -y - 27\)) |
| \(9x = 27 \Rightarrow x = 3\) | A1F | A1F for each coordinate; only ft on \(y=kx\) tangent in (b)(i) for \(k<0\) |
| When \(x=3\), \(y=-9\). \(\{P(3,-9)\}\) | A1F | |
| Total (b)(iii) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int\left(x^{\frac{3}{2}} - 3x\right)dx = \frac{2}{5}x^{\frac{5}{2}} - \frac{3x^2}{2}\ (+c)\) | M1, A2,1,0 | One power correct. Condone absence of "\(+c\)" and unsimplified forms |
| Total (c) | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_0^9\left(x^{\frac{3}{2}} - 3x\right)dx\) | B1 | PI |
| \(= \frac{2}{5}\times9^{\frac{5}{2}} - \frac{3}{2}\times9^2 - 0 = -24.3\) | M1 | Correct use of limits following integration |
| Area of triangle \(OPA = \frac{1}{2}\times9\times | y_P | \) |
| Sh.Area \(= \frac{1}{2}\times9\times | y_P | - \left |
| \(= 40.5 - 24.3 = 16.2\) | A1 | |
| Total (d) | 5 | |
| TOTAL | 75 |
## Question 8(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = \frac{3}{2}x^{\frac{1}{2}} - 3$ | M1, A1 | One term correct; Both correct |
| **Total (a)** | **2** | |
## Question 8(b)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| When $x=0$, $\frac{dy}{dx} = -3$ | B1F$\surd$ | Ft provided answer $< 0$ |
| Equation of tangent at $O$ is $y = -3x$ | B1F$\surd$ | OE Ft on $y'(0)$ |
| **Total (b)(i)** | **2** | |
## Question 8(b)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| At $(9,0)$: $\frac{dy}{dx} = \frac{3}{2}(9)^{\frac{1}{2}} - 3$ | M1 | Attempt to find $y'(9)$ |
| Equation tangent at $A$: $y - 0 = y'(9)[x-9]$ | m1 | OE |
| $\Rightarrow y = \frac{3}{2}(x-9) \Rightarrow 2y = 3x - 27$ | A1 | **CSO**. AG |
| **Total (b)(ii)** | **3** | |
## Question 8(b)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Eliminating $y \Rightarrow -6x = 3x - 27$ | M1 | OE method to one variable (eg $2y = -y - 27$) |
| $9x = 27 \Rightarrow x = 3$ | A1F | A1F for each coordinate; only ft on $y=kx$ tangent in (b)(i) for $k<0$ |
| When $x=3$, $y=-9$. $\{P(3,-9)\}$ | A1F | |
| **Total (b)(iii)** | **3** | |
## Question 8(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int\left(x^{\frac{3}{2}} - 3x\right)dx = \frac{2}{5}x^{\frac{5}{2}} - \frac{3x^2}{2}\ (+c)$ | M1, A2,1,0 | One power correct. Condone absence of "$+c$" and unsimplified forms |
| **Total (c)** | **3** | |
## Question 8(d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_0^9\left(x^{\frac{3}{2}} - 3x\right)dx$ | B1 | PI |
| $= \frac{2}{5}\times9^{\frac{5}{2}} - \frac{3}{2}\times9^2 - 0 = -24.3$ | M1 | Correct use of limits following integration |
| Area of triangle $OPA = \frac{1}{2}\times9\times|y_P|$ | M1 | |
| Sh.Area $= \frac{1}{2}\times9\times|y_P| - \left|\int_0^9\left(x^{\frac{3}{2}}-3x\right)dx\right|$ | M1 | OE |
| $= 40.5 - 24.3 = 16.2$ | A1 | |
| **Total (d)** | **5** | |
| **TOTAL** | **75** | |8 A curve, drawn from the origin $O$, crosses the $x$-axis at the point $A ( 9,0 )$. Tangents to the curve at $O$ and $A$ meet at the point $P$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-5_778_901_461_571}
The curve, defined for $x \geqslant 0$, has equation
$$y = x ^ { \frac { 3 } { 2 } } - 3 x$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $O$ and hence write down an equation of the tangent at $O$.
\item Show that the equation of the tangent at $A ( 9,0 )$ is $2 y = 3 x - 27$.
\item Hence find the coordinates of the point $P$ where the two tangents meet.
\end{enumerate}\item Find $\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x$.
\item Calculate the area of the shaded region bounded by the curve and the tangents $O P$ and $A P$.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q8 [18]}}