| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a straightforward integration question requiring finding a normal line equation (using chain rule differentiation) and calculating two areas by integration. While it involves multiple steps (differentiation, normal equation, two integrations), each technique is standard P1 material with no novel insights required. The 'show areas are equal' adds mild problem-solving but the path is clear. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals1.08h Integration by substitution |
| Answer | Marks |
|---|---|
| (ii) | 8 |
| Answer | Marks |
|---|---|
| 3 | B1 |
| Answer | Marks |
|---|---|
| [6] | Without the “×3” |
Question 10:
--- 10
(i)
(ii) ---
10
(i)
(ii) | 8
y =
3x+4
dy −4
= × 3 aef
dx (3x+4)2 3
3 2
→ m (x=0) = − Perpendicular m (x=0) =
2 3
2
Eqn of normal y−4= (x−0)
3
20
Meets x = 4 at B 4,
3
8 8 (3x+4))
∫ dx= ÷3
(3x+4) 1
2
32
Limits from 0 to 4 → Area P =
3
Area Q = Trapezium – P
Area of Trapezium =
1 20 64
4+ ×4=
2 3 3
32
→ Areas of P and Q are both
3 | B1
B1
M1
M1
A1
[5]
B1 B1
M1 A1
M1
A1
[6] | Without the “×3”
For “×3” even if 1st B mark lost.
Use of m 1 m 2 = −1 after attempting
dy
to find (x=0)
dx
Unsimplified line equation
cao
Without “÷3”. For “÷3”
Correct use of correct limits. cao
Correct method for area of
trapezium
All correct.\includegraphics{figure_10}
The diagram shows part of the curve $y = \frac{8}{\sqrt{(3x + 4)}}$. The curve intersects the $y$-axis at $A(0, 4)$. The normal to the curve at $A$ intersects the line $x = 4$ at the point $B$.
\begin{enumerate}[label=(\roman*)]
\item Find the coordinates of $B$. [5]
\item Show, with all necessary working, that the areas of the regions marked $P$ and $Q$ are equal. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2015 Q10 [11]}}