Download and practise with this A-Level Pure Mathematicsugust 2020 question paper. Useful for revision, practice, and exam preparation in Zimbabwe.
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Answers: Lockdown Pure Maths Test 31 Answers Marking Scheme
This is the A-Level Pure Mathematics question paper from August 2020. Use it for exam practice under timed conditions, then check your work against the marking scheme. See the marking scheme.
Upper $6{ }^{\text {th }}$
LOCKDOWN PURE MATHS TEST 18
Monday 10 August 2020
30 MARKS ANSWER ALL QUESTIONS
***PRESENT YOUR WORK NEATLY***
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]
2. Given that $\frac{1}{y-x}, \frac{1}{2 y}$ and $\frac{1}{y-z}$ are consecutive terms of an arithmetic progression, prove that $x, y$ and $z$ are consecutive terms of a geometric progression.
[5]
3. The function $f$ and $g$ are defined by : $x \mapsto e^{2 x}, x \in \mathbb{R}, g: x \mapsto x^{2}, x \in \mathbb{R}$. Find $f^{-1}(x)$ and $g f(x)$, clearly state the domains. Sketch the graph of $y=f^{-1}(x-2)$.
[6]
4. Prove by induction that for all positive integers $n$
\[
\sum_{r=1}^{n} 3^{r-1}=\frac{1}{2}\left(3^{n}-1\right)
\]
5. A semi-circle with centre $(1 ; 0)$ and radius 2 , lies on the $x$-axis as shown. Find the volume of the solid formed when the shaded region is rotated completely about the $x$-axis.
[5]
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