# 2003 A Level H2 Math

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Paper 1
Paper 2
##### 2003 A Level H2 Math Paper 1 Question 5

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]

##### 2003 A Level H2 Math Paper 2 Question 4

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]

##### 2003 A Level H2 Math Paper 2 Question 5

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]

Last Part

Last Part