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 gs.ude.htamnagmitobfsctd-647612@troppus.
(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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(i)
Show that $\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}=\frac{A}{{{n}^{3}}-n}$, where $A$ is a constant to be found.
[2]
(i) Show that $\frac{1}{n-1}-\frac{2}{n}+\frac{1}{n+1}=\frac{A}{{{n}^{3}}-n}$, where $A$ is a constant to be found.
[2]
(ii)
Hence find $\sum\limits_{r=2}^{n}{\frac{1}{{{r}^{3}}-r}}$, (There is no need to express your answer as a single algebraic fraction.)
[3]
(ii) Hence find $\sum\limits_{r=2}^{n}{\frac{1}{{{r}^{3}}-r}}$, (There is no need to express your answer as a single algebraic fraction.)
[3]
(iii)
Give a reason why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{3}}-r}}$ converges, and write down its value.
[2]
(iii) Give a reason why the series $\sum\limits_{r=2}^{\infty }{\frac{1}{{{r}^{3}}-r}}$ converges, and write down its value.
[2]
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It is given that
$\text{f}(x)=\left\{\begin{matrix} 7-{{x}^{2}}\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,0<x\le 2,\\\ 2x – 1\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,2<x\le 4, \\\end{matrix}\right.$
and that $\text{f}(x)=\text{f}(x+4)$ for all real values of $x$.
(i)
Evaluate $\text{f}(27)+\text{f}(45)$.
[2]
(i) Evaluate $\text{f}(27)+\text{f}(45)$.
[2]
(ii)
Sketch the graph of $y=\text{f}(x)$ for $-7\le x\le 10$.
[3]
(ii) Sketch the graph of $y=\text{f}(x)$ for $-7\le x\le 10$.
[3]
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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 planes ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}2 \\1 \\3 \\\end{matrix} \right)=1$ and $\mathbf{r}\cdot \left( \begin{matrix}-1 \\2 \\1 \\\end{matrix} \right)=2$ respectively, and meet in a line $l$.
(i)
Find the acute angle between ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$.
[3]
(i) Find the acute angle between ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$.
[3]
(ii)
Find a vector equation of $l$.
[4]
(ii) Find a vector equation of $l$.
[4]
(iii)
The plane ${{\text{p}}_{3}}$ has equation $2x+y+3z-1+k\left( -x+2y+z-2 \right)=0$. Explain why $l$ lies in ${{\text{p}}_{3}}$ for any constant $k$. Hence, or otherwise, find a cartesian equation of the plane in which both $l$ and the point $\left( 2,3,4 \right)$ lie.
[5]
(iii) The plane ${{\text{p}}_{3}}$ has equation $2x+y+3z-1+k\left( -x+2y+z-2 \right)=0$. Explain why $l$ lies in ${{\text{p}}_{3}}$ for any constant $k$. Hence, or otherwise, find a cartesian equation of the plane in which both $l$ and the point $\left( 2,3,4 \right)$ lie.
[5]
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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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Two scientists are investigating the change of a certain population of size $n$ thousand at time $t$ years.
(i)
One scientist suggests that $n$ and $t$ are related by the differential equation $\frac{{{\text{d}}^{2}}n}{\text{d}{{t}^{2}}}=10-6t$. Find the general solution of this differential equation. Sketch three members of the family of solution curves, given that $n=100$ when $t=0$.
[5]
(i) One scientist suggests that $n$ and $t$ are related by the differential equation $\frac{{{\text{d}}^{2}}n}{\text{d}{{t}^{2}}}=10-6t$. Find the general solution of this differential equation. Sketch three members of the family of solution curves, given that $n=100$ when $t=0$.
[5]
(ii)
The other scientist suggests that $n$ and $t$ are related by the differential equation $\frac{\text{d}n}{\text{d}t}=3-0.02n$. Find $n$ in terms of $t$, given again that $n=100$ when $t=0$. Explain in simple terms what will eventually happen to the population using this model.
[7]
(ii) The other scientist suggests that $n$ and $t$ are related by the differential equation $\frac{\text{d}n}{\text{d}t}=3-0.02n$. Find $n$ in terms of $t$, given again that $n=100$ when $t=0$. Explain in simple terms what will eventually happen to the population using this model.
[7]
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Find the number of ways in which the letters of the word ELEVATED can be arranged if
(i)
there are no restrictions,
(i) there are no restrictions,
(ii)
T and D must not be next to one another,
(ii) T and D must not be next to one another,
(iii)
consonants (L, V, T, D) and vowels (E, A) must alternate,
(iii) consonants (L, V, T, D) and vowels (E, A) must alternate,
(iv)
between any two Es there must be at least $2$ other letters.
(iv) between any two Es there must be at least $2$ other letters.
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