These Ten-Year-Series (TYS) worked solutions with video explanations for 2009 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
(i)
The first three terms of a sequence are given by ${{u}_{1}}=10$, ${{u}_{2}}=6$, ${{u}_{3}}=5$. Given that ${{u}_{n}}$ is a quadratic polynomial in $n$, find ${{u}_{n}}$ in terms of $n$.
[4]
(i) The first three terms of a sequence are given by ${{u}_{1}}=10$, ${{u}_{2}}=6$, ${{u}_{3}}=5$. Given that ${{u}_{n}}$ is a quadratic polynomial in $n$, find ${{u}_{n}}$ in terms of $n$.
[4]
(ii)
Find the set of values of $n$ for which ${{u}_{n}}$ is greater than $100$.
[2]
(ii) Find the set of values of $n$ for which ${{u}_{n}}$ is greater than $100$.
[2]
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Two musical instruments, $A$ and $B$, consist of metal bars of decreasing lengths.
(i)
The first bar of instrument $A$ has length $20$ cm and the lenghts of the bars form a geometric progression. The $25$th bar has length $5$ cm. Show that the length of all the bars must be less than $357$ cm, no matter how many bars there are.
(i) The first bar of instrument $A$ has length $20$ cm and the lenghts of the bars form a geometric progression. The $25$th bar has length $5$ cm. Show that the length of all the bars must be less than $357$ cm, no matter how many bars there are.
Instrument $B$ consists of only $25$ bars which are identical to the first $25$ bars of instrument $A$.Â
(ii)
Find the total length, $L$ cm, of all the bars of instrument $B$ and the length of the $13$ th bar.
(ii) Find the total length, $L$ cm, of all the bars of instrument $B$ and the length of the $13$ th bar.
(iii)
Unfortunately the manufacturer misunderstands the instructions and constructs instrument $B$ wrongly, so that the lengths of the bars are in arithmetic progression with common difference $d$ cm. If the total length of the $25$ bars is still $L$ cm and the length of the $25$th bar is still $5$ cm, find the value of $d$ and the length of the longest bar.
(iii) Unfortunately the manufacturer misunderstands the instructions and constructs instrument $B$ wrongly, so that the lengths of the bars are in arithmetic progression with common difference $d$ cm. If the total length of the $25$ bars is still $L$ cm and the length of the $25$th bar is still $5$ cm, find the value of $d$ and the length of the longest bar.
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The curve $C$ has parametric equations
$x={{t}^{2}}+4t$, $y={{t}^{3}}+{{t}^{2}}$.
(i)
Sketch the curve for $-2\le t\le 1$.
[1]
(i) Sketch the curve for $-2\le t\le 1$.
[1]
The tangent to the curve at point $P$ where $t=2$ is denoted by $l$.
(ii)
Find the cartesian equation of $l$.
[3]
(ii) Find the cartesian equation of $l$.
[3]
(iii)
The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.
[4]
(iii) The tangent $l$ meets $C$ again at the point $Q$. Use a non-calculator method to find the coordinates of $Q$.
[4]
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