Download and practise with this Trockers A-Level Pure Mathematics September 2020 question paper. Useful for school exam revision, classroom practice, homework support, and preparation in Zimbabwe.
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Answers: Lockdown Pure Maths Test 51 Answers Marking Scheme
This is the A-Level Pure Mathematics question paper from September 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 36
Wednesday 02 September 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. The complex number $w$ is defined by $w=2+i$.
(i) Showing your working, express $w^{2}$ in the form $x+i y$, where $x$ and $y$ are real.
Find the modulus of $w^{2}$.
[3]
(ii) Shade on an Argand diagram the region whose points represent the complex numbers $z$ which satisfy $\left|z-w^{2}\right| \leq\left|w^{2}\right|$.
[3]
2. Let
\[
I=\int_{0}^{1} \frac{x^{2}}{\sqrt{\left(4-x^{2}\right)}} d x
\]
(a) Using the substitution $x=2 \sin \theta$, show that
\[
I=\int_{0}^{\frac{\pi}{6}} 4 \sin ^{2} \theta d \theta
\]
(b) Hence find the exact value of $I$.
[4]
3. A curve has equation $y=\frac{1}{x-3}+x$.
(i) Find $\frac{d y}{d x}$ and $\frac{d^{2} y}{d x^{2}}$.
[2]
(ii) Find the coordinates of the maximum point $A$ and the minimum point $B$ on the curve.
[5]
4. Solve the equation $8+\cot \theta=2 \operatorname{cosec}^{2} \theta$ for $0^{\circ} \leq \theta \leq 360^{\circ}$.
[6]
5. (i) Sketch the graph of $y=2 \sin x$ for $0 \leq x \leq 2 \pi$.
[1]
(ii) By adding a suitable straight line to your sketch, determine the number of real roots of the equation $2 \pi \sin x=\pi-x$.
State the equation of the straight line.
[3]
Proverbs 21:31 ~ Tarakino N.P. (Trockers)
Proper preparation prevents poor performance.
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