These Ten-Year-Series (TYS) worked solutions with video explanations for 2008 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
The diagram shows the curve with equation $y={{x}^{2}}$. The area of the region bounded by the curve, the lines $x=1$, $x=2$ and the $x$-axis is equal to the area of the region bounded by the curve, the lines $y=a$, $y=4$ and the $y$-axis, where $a<4$. Find the value of $a$.
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A student puts $\$10$ on 1 January 2009 into a bank account which pays compound interest at a rate of $2\%$ per month on the last day of each month. She puts a further $\$10$ into the account on the first day of each subsequent month.
(a)
How much in total is in the account at the end of $2$ years?
(a) How much in total is in the account at the end of $2$ years?
(b)
After how many complete months will the total in the account first exceed $\$2000$?
(b) After how many complete months will the total in the account first exceed $\$2000$?
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The equations of three planes ${{p}_{1}}$, ${{p}_{2}}$, ${{p}_{3}}$ are
$2x-5y+3z=3$
$3x+2y-5z=-5$
$5x+\lambda y+17z=\mu $
respectively, where $\lambda $ and $\mu $ are constants. The planes ${{p}_{1}}$ and ${{p}_{2}}$ intersect in a line $\ell $.
(i)
Find a vector equation of $\ell $.
(i) Find a vector equation of $\ell $.
(ii)
Given that the line $\ell $ is contained in the plane ${{p}_{3}}$, find $\lambda $ and $\mu $.
(ii) Given that the line $\ell $ is contained in the plane ${{p}_{3}}$, find $\lambda $ and $\mu $.
(iii)
Given instead that the line $\ell $ does not intersect ${{p}_{3}}$, what can be said about the values of $\lambda $ and $\mu $?
(iii) Given instead that the line $\ell $ does not intersect ${{p}_{3}}$, what can be said about the values of $\lambda $ and $\mu $?
(iv)
If the equation of ${{p}_{3}}$ is $5x-22y+17z=5$, find the shortest distance between $\ell $ and plane ${{p}_{3}}$.
(iv) If the equation of ${{p}_{3}}$ is $5x-22y+17z=5$, find the shortest distance between $\ell $ and plane ${{p}_{3}}$.
(v)
Find the Cartesian equation of the plane which contains $\ell $ and the point $(1,\,-1,\,3)$.
(v) Find the Cartesian equation of the plane which contains $\ell $ and the point $(1,\,-1,\,3)$.
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Let $\text{f}\left( x \right)={{\text{e}}^{x}}\sin x$.
(i)
Sketch the graph of $y=\text{f}\left( x \right)$ for $-3\le x\le 3$.
(i) Sketch the graph of $y=\text{f}\left( x \right)$ for $-3\le x\le 3$.
(ii)
Using MF26, find the series expansion of $\text{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.
(ii) Using MF26, find the series expansion of $\text{f}\left( x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{3}}$.
Denote the answer to part (ii) by $\text{g}\left( x \right)$.
(iii)
On the same diagram as in part (i), sketch the graph of $y=\text{g}\left( x \right)$. Label the two graphs clearly.
(iii) On the same diagram as in part (i), sketch the graph of $y=\text{g}\left( x \right)$. Label the two graphs clearly.
(iv)
Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\text{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\text{f}\left( x \right)$.
(iv) Find, for $-3\le x\le 3$, the set of values of $x$ for which the value of $\text{g}\left( x \right)$ is within $\pm 0.5$ of the value of $\text{f}\left( x \right)$.
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A group of diplomats is to be chosen to represent three islands, $K$, $L$ and $M$. The group is to consist of $8$ diplomats and is chosen from a set of $12$ diplomats consisting of $3$ from $K$, $4$ from $L$ and $5$ from $M$. Find the number of ways in which the group can be chosen if it includes
(i)
$2$ diplomats from $K$, $3$ from $L$ and $3$ from $M$,
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(i) $2$ diplomats from $K$, $3$ from $L$ and $3$ from $M$,
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(ii)
diplomats from $L$ and $M$ only,
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(ii) diplomats from $L$ and $M$ only,
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(iii)
at least $4$ diplomats from $M$,
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(iii) at least $4$ diplomats from $M$,
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(iv)
at least $1$ diplomat from each island.
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(iv) at least $1$ diplomat from each island.
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