A2 June 2019 Q2
2. Given that \(k\) is a real non-zero constant and that
\[y = x^3\sin kx\]use Leibnitz’s theorem to show that
\[\frac{\mathrm{d}^5y}{\mathrm{d}x^5} = (k^2x^2 + A)k^3x\cos kx + B(k^2x^2 + C)k^2\sin kx\]where \(A\), \(B\) and \(C\) are integers to be determined.
(4)
| Scheme | Marks | AO |
|---|---|---|
| \(u = x^3 \Rightarrow \dfrac{\mathrm{d}u}{\mathrm{d}x} = 3x^2,\ \dfrac{\mathrm{d}^2u}{\mathrm{d}x^2} = 6x,\ \dfrac{\mathrm{d}^3u}{\mathrm{d}x^3} = 6\) | M1 | 1.1b |
| \(\begin{aligned}&v = \sin kx \Rightarrow \dfrac{\mathrm{d}v}{\mathrm{d}x} = k\cos kx,\ \dfrac{\mathrm{d}^2v}{\mathrm{d}x^2} = -k^2\sin kx,\ \dfrac{\mathrm{d}^3v}{\mathrm{d}x^3} = -k^3\cos kx,\\[6pt] &\dfrac{\mathrm{d}^4v}{\mathrm{d}x^4} = k^4\sin kx,\ \dfrac{\mathrm{d}^5v}{\mathrm{d}x^5} = k^5\cos kx\end{aligned}\) | M1 | 2.1 |
| \(\begin{aligned}&\dfrac{\mathrm{d}^5y}{\mathrm{d}x^5} = x^3k^5\cos kx + 5 \times 3x^2 \times k^4\sin kx + \dfrac{5 \times 4}{2} \times 6x \times \left(-k^3\cos kx\right) +\\[4pt] &\dfrac{5 \times 4 \times 3}{3!} \times 6 \times \left(-k^2\sin kx\right)\end{aligned}\) | M1 | 2.1 |
| \(= \left(k^2x^2 - 60\right)k^3x\cos kx + 15\left(k^2x^2 - 4\right)k^2\sin kx\) | A1 | 1.1b |
| (4) | ||
| (4 marks) |
Notes
M1: Differentiates \(u = x^3\) three times. Need to see \(x^3 \to \ldots x^2 \to \ldots x \to k\)
M1: Uses \(v = \sin kx\) to establish the form of the derivatives. Need to see at least alternating \(k^{\cdots}\sin kx\) and \(k^{\cdots}\cos kx\) with increasing powers of \(k\) for at least 3 derivatives.
M1: Uses a correct formula with 2 and 3! (or 6) with terms shown to disappear after the fourth term. This needs to be a correct application of the theorem so that the correct binomial coefficients need to go with the correct pairings of their derivatives. If there is any doubt, at least 3 terms should have the correct structure. Allow equivalent notation for the binomial coefficients e.g. \(\dbinom{5}{0}, \dbinom{5}{1}\) etc. or \({}^5\mathrm{C}_0, {}^5\mathrm{C}_1\) etc.
A1: Correct expression in the required form with correct values of \(A\), \(B\) and \(C\). Apply isw if necessary e.g. if a correct expression is followed by \(A = 60\), \(B = 15\), \(C = -4\)
(NB \(A = -60\), \(B = 15\), \(C = -4\))
If there is no use Leibnitz’s theorem e.g. repeated differentiation of products, this scores no marks.