| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve with Horizontal Line |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring standard techniques: (a) differentiate using power rule and find tangent equation at a point, (b) integrate to find area between curve and horizontal line with given limits. Both parts are routine A-level calculus applications with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left[\frac{dy}{dx}=\right]\frac{1}{2}x^{-1/2}-2x^{-3/2}\) | B1 B1 | Allow unsimplified versions. |
| At \(x=1\), \(\frac{dy}{dx}=\frac{1}{2}-2=-\frac{3}{2}\) | M1 | Substitute \(x=1\) into a differentiated \(y\). |
| Equation of tangent is \(y-5=-\frac{3}{2}(x-1)\) | A1 | WWW Or \(y=-\frac{3}{2}x+\frac{13}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{x^{3/2}}{3/2}+8x^{1/2}\) | B1 | OE Integrate to find area under curve, allow unsimplified versions. |
| \(\left[\left(\frac{128}{3}+32\right)-\left(\frac{2}{3}+8\right)\right]\) | M1 | Apply limits \(1\to 16\) to an integrated expression. |
| Area under line \(=15\times 5=75\) | B1 | Or by \(\int_1^{16}5\,dx\) |
| Required area \(=75-66=9\) | A1 |
## Question 8(a):
$\left[\frac{dy}{dx}=\right]\frac{1}{2}x^{-1/2}-2x^{-3/2}$ | **B1 B1** | Allow unsimplified versions.
At $x=1$, $\frac{dy}{dx}=\frac{1}{2}-2=-\frac{3}{2}$ | **M1** | Substitute $x=1$ into a differentiated $y$.
Equation of tangent is $y-5=-\frac{3}{2}(x-1)$ | **A1** | WWW Or $y=-\frac{3}{2}x+\frac{13}{2}$
**4 total**
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## Question 8(b):
$\frac{x^{3/2}}{3/2}+8x^{1/2}$ | **B1** | OE Integrate to find area under curve, allow unsimplified versions.
$\left[\left(\frac{128}{3}+32\right)-\left(\frac{2}{3}+8\right)\right]$ | **M1** | Apply limits $1\to 16$ to an integrated expression.
Area under line $=15\times 5=75$ | **B1** | Or by $\int_1^{16}5\,dx$
Required area $=75-66=9$ | **A1** |
**4 total**
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\includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-12_577_1088_260_523}
The diagram shows the curve with equation $y = x ^ { \frac { 1 } { 2 } } + 4 x ^ { - \frac { 1 } { 2 } }$. The line $y = 5$ intersects the curve at the points $A ( 1,5 )$ and $B ( 16,5 )$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the tangent to the curve at the point $A$.
\item Calculate the area of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q8 [8]}}