| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Parametric or Inverse Function Area |
| Difficulty | Standard +0.3 This question requires finding intersection points and calculating area between curves. Part (i) is straightforward substitution to find point A. Part (ii) involves setting up and evaluating a definite integral, requiring students to express both curves as functions of y (since one is already in that form), find limits, and integrate. While it tests multiple skills (solving equations, curve sketching understanding, integration), these are all standard A-level techniques with no novel insight required. The working is methodical rather than conceptually challenging. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = (\frac{1}{2}, 0)\) | B1 [1] | Accept \(x = 0\) at \(y = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int(1-2x)^{\frac{1}{2}}\,dx = \left[\dfrac{(1-2x)^{3/2}}{3/2}\right] \div (-2)\) | B1B1 | May be seen in a single expression |
| \(\int(2x-1)^2\,dx = \left[\dfrac{(2x-1)^3}{3}\right] \div 2\) | B1B1 | May use \(\int_a^{\cdot} x\,dy\), may expand \((2x-1)^2\) |
| \([0-(-1/3)] - [0-(-1/6)]\) | M1 | Correct use of *their* limits |
| \(1/6\) | A1 [6] |
## Question 7:
### Part (i):
$A = (\frac{1}{2}, 0)$ | **B1** [1] | Accept $x = 0$ at $y = 0$
### Part (ii):
$\int(1-2x)^{\frac{1}{2}}\,dx = \left[\dfrac{(1-2x)^{3/2}}{3/2}\right] \div (-2)$ | **B1B1** | May be seen in a single expression
$\int(2x-1)^2\,dx = \left[\dfrac{(2x-1)^3}{3}\right] \div 2$ | **B1B1** | May use $\int_a^{\cdot} x\,dy$, may expand $(2x-1)^2$
$[0-(-1/3)] - [0-(-1/6)]$ | **M1** | Correct use of *their* limits
$1/6$ | **A1** [6] |
---7\\
\includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-3_704_558_258_790}
The diagram shows parts of the curves $y = ( 2 x - 1 ) ^ { 2 }$ and $y ^ { 2 } = 1 - 2 x$, intersecting at points $A$ and $B$.\\
(i) State the coordinates of $A$.\\
(ii) Find, showing all necessary working, the area of the shaded region.
\hfill \mbox{\textit{CAIE P1 2016 Q7 [7]}}