Ten-Year-Series (TYS) Solutions | Past Year Exam Questions

2012 A Level H2 Math

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.

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2012 A Level H2 Math Paper 1 Question 4
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2012 A Level H2 Math Paper 1 Question 8

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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2012 A Level H2 Math Paper 1 Question 9
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2012 A Level H2 Math Paper 1 Question 10

[It is given that a sphere of radius $r$ has surface area $4\pi {{r}^{2}}$ and volume $\frac{4}{3}\pi {{r}^{3}}$.]

2012 TYS 2012

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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