FP3 June 2016 Q4
4.
| Scheme | Marks |
|---|---|
| \(15 + 2x - x^2 = 16 - (x - 1)^2\) | B1 |
| \(\displaystyle\int\frac{1}{\sqrt{16 - (x - 1)^2}}\,\mathrm{d}x = \arcsin\left(\frac{x - 1}{4}\right)\) | M1A1 |
| \(\left[\arcsin\left(\dfrac{x - 1}{4}\right)\right]_3^5 = \arcsin 1 - \arcsin\dfrac{1}{2}\) | dM1 |
| \(= \dfrac{\pi}{3}\) | A1 |
| (5) |
Notes
B1: Correct completion of the square. Allow e.g. \(15 + 2x - x^2 = -\left[(x - 1)^2 - 16\right]\) Allow 42 for 16
M1: \(k\arcsin(\mathrm{f}(x))\)
A1: Correct integration
dM1: Correct use of correct limits
May see
| Scheme | Marks |
|---|---|
| \(15 + 2x - x^2 = 16 - (1 - x)^2\) | B1 |
| \(\displaystyle\int\frac{1}{\sqrt{16 - (1 - x)^2}}\,\mathrm{d}x = -\arcsin\left(\frac{1 - x}{4}\right)\) | M1A1 |
| \(\left[-\arcsin\left(\dfrac{1 - x}{4}\right)\right]_3^5 = -\arcsin(-1) + \arcsin\left(-\dfrac{1}{2}\right)\) | dM1 |
| \(= \dfrac{\pi}{3}\) | A1 |
B1: Correct completion of the square. Allow e.g. \(15 + 2x - x^2 = -\left[(1 - x)^2 - 16\right]\) Allow 42 for 16
M1: \(k\arcsin(\mathrm{f}(x))\)
A1: Correct integration
dM1: Correct use of correct limits
By substitution 1
| Scheme | Marks |
|---|---|
| \(15 + 2x - x^2 = 16 - (x - 1)^2\) | B1 |
| \(\displaystyle x - 1 = 4\sin\theta \Rightarrow \int\frac{1}{\sqrt{16 - (x - 1)^2}}\,\mathrm{d}x = \int\frac{1}{\sqrt{16 - (4\sin\theta)^2}}4\cos\theta\,\mathrm{d}\theta\) | |
| \(\displaystyle = \int\mathrm{d}\theta = \theta\) | M1A1 |
| \(\left[\theta\right]_{\frac{\pi}{6}}^{\frac{\pi}{2}} = \dfrac{\pi}{2} - \dfrac{\pi}{6}\) | dM1 |
| \(= \dfrac{\pi}{3}\) | A1 |
B1: Correct completion of the square. Allow e.g. \(15 + 2x - x^2 = -\left[(1 - x)^2 - 16\right]\) Allow 42 for 16
M1: A full substitution leading to \(k\theta\) or \(k \times\) their variable
A1: Correct integration
dM1: Correct use of correct limits
By substitution 2
| Scheme | Marks |
|---|---|
| \(15 + 2x - x^2 = 16 - (x - 1)^2\) | B1 |
| \(\displaystyle x - 1 = u \Rightarrow \int\frac{1}{\sqrt{16 - (x - 1)^2}}\,\mathrm{d}x = \int\frac{1}{\sqrt{16 - u^2}}\,\mathrm{d}u\) | |
| \(\displaystyle\int\frac{1}{\sqrt{16 - u^2}}\,\mathrm{d}x = \arcsin\left(\frac{u}{4}\right)\) | M1A1 |
| \(\left[\arcsin\left(\dfrac{u}{4}\right)\right]_2^4 = \arcsin 1 - \arcsin\dfrac{1}{2}\) | dM1 |
| \(= \dfrac{\pi}{3}\) | A1 |
B1: Correct completion of the square. Allow e.g. \(15 + 2x - x^2 = -\left[(1 - x)^2 - 16\right]\) Allow 42 for 16
M1: \(k\arcsin(\mathrm{f}(u))\)
A1: Correct integration
dM1: Correct use of correct limits
By substitution 3
| Scheme | Marks |
|---|---|
| \(15 + 2x - x^2 = 16 - (x - 1)^2\) | B1 |
| \(\displaystyle x - 1 = 4\cos\theta \Rightarrow \int\frac{1}{\sqrt{16 - (x - 1)^2}}\,\mathrm{d}x = \int\frac{1}{\sqrt{16 - (4\cos\theta)^2}}\,{-4\sin\theta}\,\mathrm{d}\theta\) | |
| \(\displaystyle = \int -\mathrm{d}\theta = -\theta\) | M1A1 |
| \(\left[-\theta\right]_{\frac{\pi}{3}}^{0} = 0 + \dfrac{\pi}{3}\) | dM1 |
| \(= \dfrac{\pi}{3}\) | A1 |
B1: Correct completion of the square. Allow e.g. \(15 + 2x - x^2 = -\left[(1 - x)^2 - 16\right]\) Allow 42 for 16
M1: A full substitution leading to \(k\theta\) or \(k \times\) their variable
A1: Correct integration
dM1: Correct use of correct limits
| Scheme | Marks |
|---|---|
| \(5\cosh x - 4\sinh x = 5\left(\dfrac{e^x + e^{-x}}{2}\right) - 4\left(\dfrac{e^x - e^{-x}}{2}\right)\) | B1 |
| \(= \dfrac{e^x + 9e^{-x}}{2}\) or \(\dfrac{e^x}{2} + \dfrac{9e^{-x}}{2}\) | M1 |
| \(= \dfrac{e^{2x} + 9}{2e^x}\) * | A1* |
| (3) |
Notes
B1: Substitutes correct exponential forms
M1: Expands and collects terms in \(e^x\) and \(e^{-x}\)
A1*: Correct completion with no errors
More working may be shown but allow e.g. \(\dfrac{e^x + 9e^{-x}}{2} = \dfrac{e^{2x} + 9}{2e^x}\) or \(\dfrac{e^x}{2} + \dfrac{9e^{-x}}{2} = \dfrac{e^{2x} + 9}{2e^x}\)
| Scheme | Marks |
|---|---|
| \(u = e^x \Rightarrow \dfrac{\mathrm{d}u}{\mathrm{d}x} = e^x\) | B1 |
| \(\displaystyle\int\frac{2e^x}{e^{2x} + 9}\,\mathrm{d}x = \int\frac{2u}{u^2 + 9}\cdot\frac{\mathrm{d}u}{u}\) | M1 |
| \(= \dfrac{2}{3}\arctan\left(\dfrac{u}{3}\right)(+c)\) | dM1 |
| \(= \dfrac{2}{3}\arctan\left(\dfrac{e^x}{3}\right)(+c)\) | A1 |
| (4) | |
| (12 marks) |
Notes
B1: Correct derivative. Allow equivalents e.g. \(\dfrac{\mathrm{d}x}{\mathrm{d}u} = \dfrac{1}{u},\ \mathrm{d}u = e^x\mathrm{d}x\)
M1: Complete substitution into \(\displaystyle\int\frac{2e^x}{e^{2x} + 9}\,\mathrm{d}x\). Condone omission of “\(\mathrm{d}u\)” provided the substitution is otherwise complete apart from this. May be implied by e.g. \(\displaystyle\int\frac{2}{u^2 + 9}\,\mathrm{d}u\)
dM1: \(k\arctan(\mathrm{f}(u))\) only. Dependent on the first method mark.
A1: Cao (+c not required)