These Ten-Year-Series (TYS) worked solutions with video explanations for 2006 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-7d94f9@troppus.
The function $\text{g}$ is defined by $\text{g}:x\mapsto \frac{3}{x},x>0$.
Find, in a similar form, ${{\text{g}}^{2}}$ and ${{\text{g}}^{35}}$.
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Verify that if
${{v}_{n}}=n\left( n+1 \right)\left( n+2 \right)…\left( n+m \right)$.
then
${{v}_{n+1}}-{{v}_{n}}=\left( m+1 \right)\left( n+1 \right)\left( n+2 \right)…\left( n+m \right)$.
[2]
Given now that
${{u}_{n}}=\left( n+1 \right)\left( n+2 \right)…\left( n+m \right)$,
find $\sum\limits_{n=1}^{N}{{{u}_{n}}}$ in terms of $m$ and $N$.
[3]
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Show that the equation
${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-8z+8=0$
has a root of the form $k\text{i}$, where $k$ is real.
[3]
Hence solve the equation
${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-8z+8=0$.
[3]
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Given that $z=\frac{x}{{{\left( {{x}^{2}}+32 \right)}^{\frac{1}{2}}}}$, show that $\frac{\text{d}z}{\text{d}x}=\frac{32}{{{\left( {{x}^{2}}+32 \right)}^{\frac{3}{2}}}}$.
[3]
Find the exact value of the area of the region bounded by the curve $y=\frac{1}{{{\left( {{x}^{2}}+32 \right)}^{\frac{3}{2}}}}$, the $x$-axis and the lines $x=2$ and $x=7$.
[3]
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