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Questions: Trockers A-Level Pure Mathematicsugust 2020 Question Paper
This is the A-Level Pure Mathematics marking scheme from August 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 17
Sunday 09 August 2020
ANSWERS
30 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. By means of substitution $2 u=\cos x$, show that
\[
\int_{0}^{\frac{1}{2}} 4 u^{2} \sqrt{\left(1-4 u^{2}\right)} d u=\frac{1}{8} \int_{0}^{\frac{\pi}{2}} \sin ^{2} 2 x d x
\]
and hence find the exact value of $\int_{0}^{\frac{1}{2}} 4 u^{2} \sqrt{\left(1-4 u^{2}\right)} d u$.
[4]
\[
\int_{0}^{\frac{1}{2}} 4 u^{2} \sqrt{\left(1-4 u^{2}\right)} d u
\]
Let $2 u=\cos x$
\[
\begin{array}{l}
\Rightarrow 2 \frac{d u}{d x}=-\sin x
\Rightarrow d u=-\frac{\sin x}{2} d x
\end{array}
\]
Changing of limits:
\multicolumn{3}{c}{$x=\cos ^{-1}(2 u)$}
$\boldsymbol{u}$ | $\mathbf{0}$ | $\frac{\mathbf{1}}{\mathbf{2}}$
$\boldsymbol{x}$ | $\frac{\pi}{2}$ | 0
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