Study Pure Mathematics worksheet for A-Level. Useful for revision, homework, and exam preparation in Zimbabwe.
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This is the A-Level Pure Mathematics worksheet from August 2020. It is a free worksheet for revision.
Upper $6{ }^{\text {th }}$
LOCKDOWN PURE MATHS TEST 16
Saturday 08 August 2020
40 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. If $a=x^{\frac{1}{2}}+x^{-\frac{1}{2}}$ and $b=x^{\frac{1}{2}}-x^{-\frac{1}{2}}$, expand $(a b)^{2}+2$.
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2. Let $0<\theta<180^{\circ}$ and define $A=\left(\begin{array}{cc}\cos \theta | -\sin \theta \sin \theta | \cos \theta\end{array}\right)$.
Prove by Mathematical induction that $A^{n}=\left(\begin{array}{cc}\cos n \theta | -\sin \theta \theta \sin \theta \theta | \cos \theta \theta\end{array}\right)$ for all $n \in \mathbb{Z}^{+}$.
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3. Express $\frac{18}{x^{3}-9 x}$ in partial fractions.
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4. Let $S=\{p, q, r\}$. The following table defines a binary operation $*$ on $S$.
$*$ | $p$ | $q$ | $r$
$p$ | $p$ | $q$ | $r$
$q$ | $q$ | $p$ | $r$
$r$ | $r$ | $q$ | $p$
Verify if $S$ forms an Abelian group under $*$.
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5. The coordinates of point $A$ are $(3,-4)$. The line $2 x-3 y=12$ cuts the $x$-axis and the $y$-axis at $B$ and $C$ respectively. Find the
(i) equation of the line through $A$ which is perpendicular to line $B C$,
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(ii) perpendicular distance from $A$ to the line $B C$, giving your answer in the form $k \sqrt{13}$, where $k$ is a rational number,
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(iii) area of triangle $A B C$.
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6. Find the Taylors expansion of $\sqrt[3]{x}$ in powers of ( $x-1$ ) up to and including the term in $(x-1)^{4}$. Hence find an approximation for $\sqrt[3]{1.5}$ up to 5 decimal places.
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Isaiah 35:1-2 ~ Tarakino N.P. (Trockers)
Talent is nothing without hard work. Never underestimate the power of practice.
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