October 2021 Paper 3 Q9
9 The diagram shows the curve \(y = 3 - \sqrt{x}\).
Determine the exact area of the region bounded by the curve \(y = 3 - \sqrt{x}\), the lines \(y = 5x - 1\) and \(x = 4\) and the \(x\)-axis. [10]
| Scheme | Marks | AO |
|---|---|---|
 | B1 | 1.1 |
| [1] |
Notes
B1: Correct line through \(((0, -1)\) and \((1, 4)\)
If \(x\)-intercept marked and \((1, 4)\) slightly out may award B1 BOD
| Scheme | Marks | AO |
|---|---|---|
| DR \(3 - \sqrt{x} = 5x - 1\) \(4 - 5x = \sqrt{x} \Rightarrow (4 - 5x)^2 = x\) or \(y = 5(3 - y)^2 - 1\) | M1 | 3.1a |
| \(25x^2 - 41x + 16 = 0\) or \(5y^2 - 31y + 44 = 0\) | M1 | 1.1 |
| \((x - 1)(25x - 16) = 0\) or \((5y - 11)(y - 4) = 0\) | M1 | 1.1 |
| \(x = \dfrac{16}{25} = 0.64\) or \(y = 2.2\) | A1 | 2.2a |
| \(y = 5 \times 0.64 - 1 = 2.2\) or \(x = (3 - 2.2)^2 = 0.64\) | M1 | 1.1 |
| \(\frac{1}{2}(0.64 - 0.2)(2.2)\ \ [= 0.484 = \frac{121}{250}]\) | M1 | 3.1a |
| \(\displaystyle\int_{0.64}^{4} (3 - \sqrt{x})\,\mathrm{d}x\) | M1* | 2.1 |
| \(\left[3x - \frac{2}{3}x^{\frac{3}{2}}\right]_{0.64}^{4}\) | M1** | 1.1 |
| \(\left(12 - \frac{16}{3}\right) - \left(1.92 - \frac{128}{375}\right) = \frac{636}{125} = 5.088\) | M1 | 1.1 |
| Total area \(= 0.484 + 5.088 = 5.572 = \frac{1393}{250} = 5\frac{143}{250}\) | A1 | 1.1 |
| [10] |
Notes
DR: This question included the instruction: In this question you must show detailed reasoning.
M1: Getting into suitable form for solution ie = 0
Or as a quadratic in \(\sqrt{x}\): \(5x + \sqrt{x} - 4 = 0\)
M1: Attempt to solve quadratic by formula or factorising (oe, via \(\sqrt{x}\) quadratic)
\((5\sqrt{x} - 4)(\sqrt{x} + 1) = 0\)
A1: Correct root chosen
Or replace M1A1 with SC1 if \(x = 0.64\) or \(y = 2.2\) seen with no method for solving
\(\sqrt{x} = \frac{4}{5}\) so \(x = 0.64\)
M1: FT their positive root
M1: Attempt to find area of triangle
\(\frac{1}{2} \times 0.64 \times 2.2\) is M0
Alternative method for area of triangle
| Scheme | Marks |
|---|---|
| \(\displaystyle\int_{0.2}^{0.64} (5x - 1)\,\mathrm{d}x\) | M1 M1 |
M1: Correct integral
M1: Correct limits
FT their positive root
M1*: Allow any limits \(0 \leqslant x \leqslant 4\)
Allow if clearly embedded eg \(\displaystyle\int_{0.64}^{4} (5x - 4 + \sqrt{x})\,\mathrm{d}x\)
M1**: Integration of M1* integral \(\sqrt{x}\) term correct (ignore limits)
Dep on M1* only
M1: Evaluation of M1* integral substitution seen
Dep on M1**
A1: All correct, other partitions possible
Dep on all 9 previous marks