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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 41
Thursday 10 September 2020
ANSWERS
35 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. Let $y=\sqrt[3]{x}$. Write down $\frac{d y}{d x}$, and hence find an expression for the approximate small change in $y$ for which $x$ changes by a small amount $\delta x$.
Use your result to estimate the cube root of 1001 .
[5]
\[
\begin{array}{l}
y=\sqrt[3]{x}
\Rightarrow y=x^{\frac{1}{3}}
\Rightarrow \frac{d y}{d x}=\frac{1}{3} x^{\frac{1}{3}-1}
\Rightarrow \frac{d y}{d x}=\frac{1}{3} x^{-\frac{2}{3}}
\Rightarrow \frac{d y}{d x}=\frac{1}{3 x^{\frac{2}{3}}}
\Rightarrow \frac{d y}{d x}=\frac{1}{3(\sqrt[3]{x})^{2}}
\end{array}
\]
Now:
\[
\begin{array}{l}
\frac{\delta y}{\delta x} \approx \frac{d y}{d x}
\Rightarrow \delta y \approx\left(\frac{d y}{d x}\right) \delta x
\Rightarrow \delta y \approx\left(\frac{1}{3(\sqrt[3]{x})^{2}}\right) \delta x
\end{array}
\]
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