These Ten-Year-Series (TYS) worked solutions with video explanations for 2012 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
In the triangle $ABC$, $AB=1$, angle $BAC=\theta $ radians and angle $ABC=\frac{3}{4}\pi $ radians (see diagram).
(i)
Show that $AC=\frac{1}{\cos \theta -\sin \theta }$.
[4]
(i) Show that $AC=\frac{1}{\cos \theta -\sin \theta }$.
[4]
(ii)
Given that $\theta $ is a sufficiently small angle, show that
$AC\approx 1+a\,\theta +b\,{{\theta }^{2}}$,
for constants $a$ and $b$ to be determined.
[4]
(ii) Given that $\theta $ is a sufficiently small angle, show that
$AC\approx 1+a\,\theta +b\,{{\theta }^{2}}$,
for constants $a$ and $b$ to be determined.
[4]
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The curve $C$ has equation $x-y={{\left( x+y \right)}^{2}}$. It is given that $C$ has only one turning point.
(i)
Show that $1+\frac{\text{d}y}{\text{d}x}=\frac{2}{2x+2y+1}$.
[4]
(i) Show that $1+\frac{\text{d}y}{\text{d}x}=\frac{2}{2x+2y+1}$.
[4]
(ii)
Hence, or otherwise, show that $\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=-{{\left( 1+\frac{\text{d}y}{\text{d}x} \right)}^{3}}$.
[3]
(ii) Hence, or otherwise, show that $\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=-{{\left( 1+\frac{\text{d}y}{\text{d}x} \right)}^{3}}$.
[3]
(iii)
Hence state, with one reason, whether the turning point is a maximum or a minimum.
[2]
(iii) Hence state, with one reason, whether the turning point is a maximum or a minimum.
[2]
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(i)
Find a vector equation of the line through the points $A$ and $B$ with position vectors $\text{7}\mathbf{i}\,+\,8\mathbf{j}\,+\,9\mathbf{k}$ and $-\mathbf{i}-8\mathbf{j}+\mathbf{k}$ respectively.
[3]
(i) Find a vector equation of the line through the points $A$ and $B$ with position vectors $\text{7}\mathbf{i}\,+\,8\mathbf{j}\,+\,9\mathbf{k}$ and $-\mathbf{i}-8\mathbf{j}+\mathbf{k}$ respectively.
[3]
(ii)
The perpendicular to this line from the point $C$ with position vector $\mathbf{i}+8\mathbf{j}+3\mathbf{k}$ meets the line at point $N$. Find the position vector of $N$ and the ratio $AN:NB$.
[5]
(ii) The perpendicular to this line from the point $C$ with position vector $\mathbf{i}+8\mathbf{j}+3\mathbf{k}$ meets the line at point $N$. Find the position vector of $N$ and the ratio $AN:NB$.
[5]
(iii)
Find the Cartesian equation of the line which is a reflection of the line $AC$ in the line $AB$.
[4]
(iii) Find the Cartesian equation of the line which is a reflection of the line $AC$ in the line $AB$.
[4]
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[It is given that a sphere of radius $r$ has surface area $4\pi {{r}^{2}}$ and volume $\frac{4}{3}\pi {{r}^{3}}$.]
A model of a concert hall is made up of three parts.
• The roof is modelled by the curved surface of a hemisphere of radius $r$ cm.
• The walls are modelled by the curved surface of a cylinder of radius $r$ cm and height $h$ cm.
• The floor is modelled by a circular disc of radius $r$ cm.
The three parts are joined together as shown in the diagram. The model is made of material of negligible thickness.
(i)
It is given that the volume of the model is a fixed value $k$ cm$^{3}$, and the external surface area is a minimum. Use differentiation to find the values of $r$ and $h$ in terms of $k$. Simplify your answers.
[7]
(i) It is given that the volume of the model is a fixed value $k$ cm$^{3}$, and the external surface area is a minimum. Use differentiation to find the values of $r$ and $h$ in terms of $k$. Simplify your answers.
[7]
(ii)
It is given instead that the volume of the model is $200$ cm$^{3}$ and its external surface area is $180$ cm$^{2}$. Show that there are two possible values of $r$. Given also that $r<h$, find the value of $r$ and the value of $h$.
[5]
(ii) It is given instead that the volume of the model is $200$ cm$^{3}$ and its external surface area is $180$ cm$^{2}$. Show that there are two possible values of $r$. Given also that $r<h$, find the value of $r$ and the value of $h$.
[5]
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On a remote island a zoologist measures the tail lengths of a random sample of $20$ squirrels. In a species of squirrel known to her, the tail lengths have mean $14.0$ cm. She carries out a test, at the $5\%$ significance level, of whether squirrels on the island have the same mean tail length as the species known to her. She assumes that the tail lengths of squirrels on the island are normally distributed with standard deviation $3.8$ cm.
(i)
State appropriate hypotheses for the test.
[1]
(i) State appropriate hypotheses for the test.
[1]
The sample mean tail length is denoted by $\overline{x}$ cm.
(ii)
Use an algebraic method to calculate the set of values of $\overline{x}$ for which the null hypothesis would not be rejected. (Answers obtained by trial and improvement from a calculator will obtain no marks.)
[3]
(ii) Use an algebraic method to calculate the set of values of $\overline{x}$ for which the null hypothesis would not be rejected. (Answers obtained by trial and improvement from a calculator will obtain no marks.)
[3]
Five years later, the zoologist took a sample of $20$ squirrels and found that $\overline{x}=15.8$. He wants to make a claim that the mean is at least ${{\mu }_{0}}$ and he does not want his claim to be rejected at $5\%$ significance level.
(iii)
Find the range of values of ${{\mu }_{0}}$ that he can claim.
[4]
(iii) Find the range of values of ${{\mu }_{0}}$ that he can claim.
[4]
(iv)
In order to carry out this test, the zoologist made two assumptions. State these two assumptions.
[2]
(iv) In order to carry out this test, the zoologist made two assumptions. State these two assumptions.
[2]
The assumptions are:
The assumptions are:
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In an opinion poll before an election, a sample of $30$ voters is obtained.
(i)
The number of voters in the sample who support the Alliance Party is denoted by $A$.
State, in context, what must be assumed for $A$ to be well modelled by a binomial distribution.
[2]
(i)
The number of voters in the sample who support the Alliance Party is denoted by $A$.
State, in context, what must be assumed for $A$ to be well modelled by a binomial distribution.
[2]
Assume now that $A$ has the distribution $\text{B}\left( 30,p \right)$.
(ii)
Given that $p=0.15$, find $\text{P}\left( A=3\,\,\text{or}\,\,4 \right)$.
[2]
(ii) Given that $p=0.15$, find $\text{P}\left( A=3\,\,\text{or}\,\,4 \right)$.
[2]
(iii)
For an unknown value of $p$, it is given that $\text{P}\left( A=15 \right)=0.06864$ correct to $5$ decimal places. Show that $p$ satisfies an equation of the form $p\left( 1-p \right)=k$, where $k$ is a constant to be determined. Hence find the value of $p$ to a suitable degree of accuracy. Given that $p<0.5$.
[3]
(iii) For an unknown value of $p$, it is given that $\text{P}\left( A=15 \right)=0.06864$ correct to $5$ decimal places. Show that $p$ satisfies an equation of the form $p\left( 1-p \right)=k$, where $k$ is a constant to be determined. Hence find the value of $p$ to a suitable degree of accuracy. Given that $p<0.5$.
[3]
Assumptions:
Assumptions:
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