Study Pure Mathematics revision pack for A-Level. Useful for revision, homework, and exam preparation in Zimbabwe.
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This is the A-Level Pure Mathematics revision pack from 2022. It is a free revision pack for revision. Time allowed: 1 hour 30 minutes.
A' LEVEL
PURE MATHEMATICS
2022 FINAL REVISION TEST 3
TIME: 1 hour 30 minutes
(...) Marks
Answer all questions.
[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 correct to the nearest degree, and in other cases it should be given to 2 significant figures.]
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Sharewell B. C
1. Given that, $y=\tan x$
Show that,
a) $y^{\prime \prime \prime}=2\left(y^{2}+1\right)\left(3 y^{2}+1\right)$
b) Given that $y=\tan \left(3 x+\frac{\pi}{4}\right)$, show that,
i. $\quad \frac{d y}{d x}=3\left(1+y^{2}\right)$
ii. $\quad \frac{d^{3} y}{d x^{3}}=6 y\left(\frac{d^{2} y}{d x^{2}}\right)+6\left(\frac{d y}{d x}\right)^{2}$
Find Maclaurin's series for $\tan \left(3 x+\frac{\pi}{4}\right)$, up to and including the term in $x^{3}$.
[7]
2. Given that $\frac{d^{2} y}{d x^{2}}+y \frac{d y}{d x}=x$, at $x=1, y=0$, and $\frac{d y}{d x}=2$. Use Taylor's series to find a solution of the differential equation in ascending powers of ( $x-1$ ) up to and including the term in $(x-1)^{3}$.
[6]
3. The set of values given by the iterative formula $x_{n+1}=\frac{2 x_{n}}{3}+\frac{7}{\left(x_{n}\right)^{3}}$
Taking $x_{1}=2$ as the initial value, converges to $\alpha$, find $\alpha$ correct to 3 decimal places, give the result of each iteration to 4 decimal places.
[4]
4.