| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Combined region areas |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: finding a normal line (differentiate, find perpendicular gradient) and calculating area between curve and line using integration. All steps are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(y = 2\sqrt{x}\). \(dy/dx = x^{\frac{1}{2}}\). If \(x = 4\), \(m = \frac{1}{2}\). Perpendicular \(= -2\). Eqn of \(y = -2x + 12\) or \(y = 4 - 2(x - 4)\) | M1, A1, DM1, A1 | Realising the need to differentiate \(\div\) use; Correct only; \(m, m_2 = -1\) numerical needed; correct only |
| (ii) Area P = \(\int_1^4 2\sqrt{x} dx = 2x^{3/2}/1.5\). Evaluated from 1 to 4. Answer = \(32/3 - 4/3 = 28/3\) | M1 A1, DM1, A1 | Knowing to integrate. Correct unsimplified; Correct use of 1 to 4 – not for 2 to 4; Correct only |
**(i)** $y = 2\sqrt{x}$. $dy/dx = x^{\frac{1}{2}}$. If $x = 4$, $m = \frac{1}{2}$. Perpendicular $= -2$. Eqn of $y = -2x + 12$ or $y = 4 - 2(x - 4)$ | M1, A1, DM1, A1 | Realising the need to differentiate $\div$ use; Correct only; $m, m_2 = -1$ numerical needed; correct only
**(ii)** Area P = $\int_1^4 2\sqrt{x} dx = 2x^{3/2}/1.5$. Evaluated from 1 to 4. Answer = $32/3 - 4/3 = 28/3$ | M1 A1, DM1, A1 | Knowing to integrate. Correct unsimplified; Correct use of 1 to 4 – not for 2 to 4; Correct only10\\
\includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_595_800_1548_669}
The diagram shows the points $A ( 1,2 )$ and $B ( 4,4 )$ on the curve $y = 2 \sqrt { } x$. The line $B C$ is the normal to the curve at $B$, and $C$ lies on the $x$-axis. Lines $A D$ and $B E$ are perpendicular to the $x$-axis.\\
(i) Find the equation of the normal $B C$.\\
(ii) Find the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2002 Q10 [8]}}