These Ten-Year-Series (TYS) worked solutions with video explanations for 2010 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
A bottle containing liquid is taken from a refrigerator and placed in a room where the temperature is a constant $20{}^\circ \text{C}$. As the liquid warms up, the rate of increase of it temperature $\theta {}^\circ \text{C}$ after time $t$ minutes is proportional to the temperature different $\left( 20-\theta \right){}^\circ \text{C}$. Initially the temperature of the liquid is $10{}^\circ \text{C}$ and the rate of increase of the temperature is $1{}^\circ \text{C}$ per minute. By setting up and solving a differential equation, show that $\theta =20-10{{e}^{-\frac{1}{10}t}}$Â
Find the time it takes the liquid to reach a temperature of $15{}^\circ \text{C}$, and state what happens to $\theta $ for large values of $t$. Sketch a graph of $\theta $ against $t$.
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A company requires a box made of cardboard of negligible thickness to hold $300$ cm$^{3}$ of powder when full. The length of the box is $3x$ cm, the width is $x$ cm and the height is $y$ cm. The lid has depth $ky$ cm, where $0<k\le 1$ (see diagram).
(i)
Use differentiation to find, in terms of $k$, the value of $x$ which gives a minimum total external surface area of the box and the lid.
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(i) Use differentiation to find, in terms of $k$, the value of $x$ which gives a minimum total external surface area of the box and the lid.
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(ii)
Find also the ratio of the height to the width, $\frac{y}{x}$, in this case, simplifying your answer.
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(ii) Find also the ratio of the height to the width, $\frac{y}{x}$, in this case, simplifying your answer.
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(iii)
Find the values between which $\frac{y}{x}$ must lie.
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(iii) Find the values between which $\frac{y}{x}$ must lie.
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(iv)
Find the value of $k$ for which the box has square ends.
[2]
(iv) Find the value of $k$ for which the box has square ends.
[2]
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For events $\text{A}$ and $\text{B}$, it is given that $\text{P}(A)=0.7$, $\text{P}(B)=0.6$ and $\text{P}(A\left| B’ \right.)=0.8$. Find
(i)
$\text{P}(A\cap B’)$,
(i) $\text{P}(A\cap B’)$,
(ii)
$\text{P}(A\cup B)$,
(ii) $\text{P}(A\cup B)$,
(iii)
$\text{P}(B’\left| A \right.)$,
(iii) $\text{P}(B’\left| A \right.)$,
Find a third event $C$, it is given that $\text{P}(C)=0.5$ and that $A$ and $C$ are independent.
(iv)
Find $\text{P}\left( A’\cap C \right)$
(iv) Find $\text{P}\left( A’\cap C \right)$
(v)
Hence state an inequality satisfied by $\text{P}(A’\cap B\cap C)$.
(v) Hence state an inequality satisfied by $\text{P}(A’\cap B\cap C)$.
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In this question you should state clearly the values of the parameters of any normal distribution you use.
Over a three-month period Ken makes $X$ minutes of peak-rate telephone calls and $Y$ minutes of cheap-rate calls. $X$ and $Y$ are independent random variables with the distributions $\text{N}\left( 180,{{30}^{2}} \right)$ and $\text{N}\left( 400,{{60}^{2}} \right)$ respectively.
(i)
Find the probability that, over a three-month period, the number of minutes of cheap-rate calls made by Ken is more than twice the number of minutes of peak-rate calls.
(i) Find the probability that, over a three-month period, the number of minutes of cheap-rate calls made by Ken is more than twice the number of minutes of peak-rate calls.
Peak-rate calls cost $\$0.12$ per minute and cheap-rate calls cost $\$0.05$ per minute.
(ii)
Find the probability that, over a three-month period, the total cost of Ken’s calls is greater than $\$45$.
(ii) Find the probability that, over a three-month period, the total cost of Ken’s calls is greater than $\$45$.
(iii)
Find the probability that the total cost of Ken’s peak-rate calls over two independent three-month periods is greater than $\$45$.
(iii) Find the probability that the total cost of Ken’s peak-rate calls over two independent three-month periods is greater than $\$45$.
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