These Ten-Year-Series (TYS) worked solutions with video explanations for 2003 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at gs.ude.htamnagmitobfsctd-8b9d0e@troppus.
Referred to the origin $O$ , the position vectors of points $A$ and $B$ are $4\mathbf{i}-11\mathbf{j}+4\mathbf{k}$ and $7\mathbf{i}+\mathbf{j}+7\mathbf{k}$ respectively.
(i)
Find a vector equation for the line $l$ passing through $A$ and $B$ .
[2]
(i) Find a vector equation for the line $l$ passing through $A$ and $B$ .
[2]
(ii)
Find the position vector of the point $P$ on $l$ such that $OP$ is perpendicular to $l$.
[4]
(ii) Find the position vector of the point $P$ on $l$ such that $OP$ is perpendicular to $l$.
[4]
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The first, second and forth terms of a convergent geometric progression are consecutive terms of an arithmetic progression. Prove that the common ratio of the geometric progression is $\frac{-1+\sqrt{5}}{2}$ .
[5]
The first term of the geometric progression is positive. Show that the sum of the first 5 terms of this progression is greater than nine tenths of the sum of infinity.
[3]
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It is given that $y=\sin \left[ \ln \left( 1+x \right) \right]$. Show that
(i)
$\left( 1+x \right)\frac{\text{d}y}{\text{d}x}=\cos \left[ \ln \left( 1+x \right) \right]$,
[1]
(i) $\left( 1+x \right)\frac{\text{d}y}{\text{d}x}=\cos \left[ \ln \left( 1+x \right) \right]$,
[1]
(ii)
${{\left( 1+x \right)}^{2}}\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+\left( 1+x \right)\frac{\text{d}y}{\text{d}x}+y=0$.
[3]
(ii) ${{\left( 1+x \right)}^{2}}\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}+\left( 1+x \right)\frac{\text{d}y}{\text{d}x}+y=0$.
[3]
Find the Maclaurin series for $y$, up to and including the term in ${{x}^{3}}$.
[5]
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