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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