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.
A free account is needed to download. Sign in or create one
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 50
Wednesday 23 September 2020
30 MARKS ANSWER ALL QUESTIONS
***PRESENT YOUR WORK NEATI.Y***
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. Solve the equation
\[
\left(\frac{3-i}{2+i}\right) z=\frac{5+5 i}{-1+2 i}
\]
giving your answer in the form $a+i b$.
[5]
2. Given that $\sec \theta \tan \theta=2$, show that $\sin \theta=\frac{\sqrt{17}-1}{4}$.
Hence write down the solution of $\sec \theta \tan \theta=2$, where $0 \leq \theta \leq 2 \pi$.
[7]
3. The parametric equations of a curve are $x=a \sin \theta$ and $y=a \theta \cos \theta$, where $a$ is a positive constant and $0<\theta<\frac{1}{2} \pi$. Find $\frac{d y}{d x}$ in terms of $\theta$, and hence show that the gradient of the curve is zero where $\theta=\cot \theta$.
[6]
By sketching a suitable pair of graphs, show that the equation $\theta=\cot \theta$ is satisfied by just one value of $\theta$ in the relevant range.
[2]
Determine, with reasons, whether this value of $\theta$ is greater than or less than $\frac{1}{4} \pi$.
[2]
4.
\[
f(x) \equiv 5 \cos x-12 \sin x, \quad x \in \Re .
\]
Given that $f(x)=R \cos (x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$,
(a) find the values of $R$ and $\alpha$,
[2]
(b) find the range of value of $f(x)$,
[2]
(c) show that
\[
\int_{0}^{\frac{\pi}{2}-\alpha} f(x) d x=1
\]
2 Thessalonians 3:10 ~ Tarakino N.P. (Trockers)
You don't need to be a great student to start working hard. Once you start, you become great.
Lost time is never found again and there is no compensation for that...
Page 1
(C) Tarakino N.P. 0772978155/0717267175
ntarakino@gmail.com