Download and practise with this A-Level Pure Mathematics September 2020 question paper. Useful for revision, practice, and exam 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 51
Thursday 24 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. The complex number $z$ is represented by the point $P(x, y)$ in the Argand diagram and
\[
|z+3|=k|z-i|
\]
where $k$ is a real positive constant.
(a) When $k \neq 1$, the loci of $P$ is a circle. Find in terms of $k$,
(i) the equation of the circle,
(ii) the coordinates of the centre of the circle.
[7]
(b) (i) Write down the equation of the locus of $P$ when $k=1$.
(ii) Give a geometrical interpretation of this locus.
[2]
2. The transformation $T$ in the plane consists of a reflection in the line $y=x$ followed by a reflection in the line $y=-x$.
(a) Determine the $2 \times 2$ matrix which represents $T$.
[4]
(b) Identify the single transformation that is equivalent to $T$.
[1]
3. The curve $C$ has equation $x^{4}+3 x^{2} y-2 y^{2}=34$.
(i) Show that $\frac{d y}{d x}=\frac{4 x^{3}+6 x y}{4 y-3 x^{2}}$.
[3]
(ii) Find the coordinates of each of the points on $C$ where the tangent is parallel to the $y$-axis.
[4]
4. Given that $x=\tan ^{-1} t, y=$ Int , where $t>0$,
(a) Find an expression for $\frac{d y}{d x}$ in terms of $t$,
[4]
(b) find the value of $x$ for which $\frac{d^{2} y}{d x^{2}}=0$.
[5]
Romans 12:12 ~ Tarakino N.P. (Trockers)
When mediocrity becomes the accepted norm, excellence dies a painful death. Endurance is
one of the most difficult disciplines, but victory comes to those who endure.
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