June 2022 Paper 1 Q6
6
(a) Find the first two terms, in ascending powers of \(x\), of the binomial expansion of\[\left(1 - \frac{x}{2}\right)^{\frac{1}{2}}\] [2 marks]
(b) Hence, for small values of \(x\), show that\[\sin 4x + \sqrt{\cos x} \approx A + Bx + Cx^2\]where \(A\), \(B\) and \(C\) are constants to be found. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Expands to obtain the first two terms Can be unsimplified Condone sign error | M1 | 1.1a |
| Obtains \(1 - \dfrac{1}{4}x\) OE Accept if listed as two separate terms. Ignore any extra terms | A1 | 1.1b |
| (2) |
Typical solution
\[\left(1 - \frac{x}{2}\right)^{\frac{1}{2}} \approx 1 + \left(\frac{1}{2}\right)\left(-\frac{x}{2}\right)\]\[\approx 1 - \frac{1}{4}x\]| Scheme | Marks | AO |
|---|---|---|
| States or uses at least one small angle approximation correctly either \(\sin kx \approx kx\) or \(\sqrt{\cos x} \approx \sqrt{1 - \dfrac{x^2}{2}}\) | M1 | 3.1a |
| Uses both small angle approximations correctly for sine and cosine \(\sin kx \approx kx\) and \(\sqrt{\cos x} \approx \sqrt{1 - \dfrac{x^2}{2}}\) Must have eliminated all trig expressions Inconsistent variables for angles must eventually be consistent to be awarded A1 | A1 | 1.1b |
| Uses their expansion from (a) Must have replaced \(x\) with \(x^2\) or Applies binomial theorem correctly to \(\left(1 - \dfrac{x^2}{2}\right)^{\frac{1}{2}}\) ignore any extra terms | M1 | 3.1a |
| Completes argument to obtain \(4x + \left(1 - \dfrac{x^2}{4}\right)\) or \(1 + 4x - \dfrac{1}{4}x^2\) Accept any order of terms Ignore higher powers of \(x\) Must be in terms of \(x\) Do not ISW | R1 | 2.1 |
| (4) | ||
| (6 marks) |