Lockdown Pure Maths Test 40. Free PDF download (706 KB) from the ExamineAI past papers library.
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Questions: Lockdown Pure Maths Test 51 Question Paper
This is the A-Level Pure Mathematics marking scheme from September 2020. It shows how marks are awarded, so you can mark your own answers to the matching question paper. See the question paper.
Upper $6{ }^{\text {th }}$
LOCKDOWN PURE MATHS TEST 40
Wednesday 09 September 2020
ANSWERS
25 MARKS
NB: If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to $\mathbf{2}$ significant figures.
1. Express $\cos ^{5} \theta$ in terms of multiples of $\cos \theta$.
[6]
\[
\begin{array}{l}
z+\frac{1}{z}=2 \cos \theta
\Rightarrow\left(z+\frac{1}{z}\right)^{5}=(2 \cos \theta)^{5}
\Rightarrow\left(z+\frac{1}{z}\right)^{5}=32 \cos ^{5} \theta
\Rightarrow \cos ^{5} \theta=\frac{1}{32}\left(z+\frac{1}{z}\right)^{5}
\Rightarrow \cos ^{5} \theta=\frac{1}{32}\left[z^{5}+5 z^{4}\left(\frac{1}{z}\right)+\frac{5(4) z^{3}}{2 \times 1}\left(\frac{1}{z}\right)^{2}+\frac{5(4)(3) z^{2}}{3 \times 2 \times 1}\left(\frac{1}{z}\right)^{3}+\cdots\right]
\quad \quad\left[\ldots+\frac{5(4)(3)(2) z}{4 \times 3 \times 2 \times 1}\left(\frac{1}{z}\right)^{4}+\frac{5(4)(3)(2)(1) z^{0}}{5 \times 4 \times 3 \times 2 \times 1}\left(\frac{1}{z}\right)^{5}\right]
\Rightarrow \cos ^{5} \theta=\frac{1}{32}\left(z^{5}+5 z^{3}+10 z+\frac{10}{z}+\frac{5}{z^{3}}+\frac{1}{z^{5}}\right)
\Rightarrow \cos ^{5} \theta=\frac{1}{32}\left(z^{5}+\frac{1}{z^{5}}+5 z^{3}+\frac{5}{z^{3}}+10 z+\frac{10}{z}\right)
\Rightarrow \cos ^{5} \theta=\frac{1}{32}\left[\left(z^{5}+\frac{1}{z^{5}}\right)+5\left(z^{3}+\frac{1}{z^{3}}\right)+10\left(z+\frac{1}{z}\right)\right]
\text { Remember that } z+\frac{1}{z}=2 \cos \theta \text { and } z^{n}+\frac{1}{z^{n}}=2 \operatorname{cosn} \theta
\end{array}
\]
DON'T stop when you are tired. STOP when you are done.
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