[Maximum number: 16]

Thai cushions are designed with a triangular cross-section and are made from layers of smaller cushions. These cushions can be modelled as triangular prisms.
This is shown in the diagram.

Thai cushion with 4 layers

Cross-section of Thai cushion with 5 layers

(a)

Write down the number of triangular prisms in the bottom layer of the cushion with

[ 2 ]

(i)

4 layers.

(ii)

5 layers.

Mayumi notices that the number of triangular prisms in the bottom layer of the cushions forms an arithmetic sequence.

[ 2 ]

(b)

(i)

Write down the common difference of this sequence.

(ii)

Find an expression for the number of triangular prisms in the bottom layer of a cushion with n layers.

Mayumi wants to extend this design to create a cushion with 9 layers.

[ 3 ]

(c)

(i)

Find the number of triangular prisms in the bottom layer of Mayumi's cushion.

(ii)

Calculate the total number of triangular prisms in Mayumi's cushion.

[ 3 ]

(d)

Find an expression for the total number of triangular prisms in a cushion with n layers, giving your answer in its simplest form.

[ 2 ]

(e)

The cross-section of the cushion consists of black triangles and white triangles.

This cushion with 4 layers has a total of 6 white triangles.

This cushion with 5 layers has 4 white triangles in its bottom layer.

Write down the total number of black triangles in a cushion with 4 layers.

The number of black triangles in each layer forms an arithmetic sequence.

[ 1 ]

(f)

Find and simplify an expression for the total number of black triangles in a cushion with n layers.

The total number of white triangles in a cushion with n layers is $\frac{n(n-1)}{2}$ .

[ 2 ]

(g)

Using both the given expression and your answer to part (f), find and simplify an expression for the total number of black and white triangles in a cushion with n layers.

[ 3 ]

[Maximum number: 5]

The first three terms of a geometric sequence are 2,6 and 18.

(a)

Write down the common ratio, r.

[ 1 ]

(b)

Find the 8th term of the sequence.

[ 2 ]

(c)

Find the sum of the first 10 terms of the sequence.

[ 2 ]

[Maximum number: 5]

The amount, in milligrams, of a medicinal drug in the body t hours after it was injected is given by $D(t)=23(0.85)^{t}, t \geq 0$ . Before this injection, the amount of the drug in the body was zero.

(a)

Write down

[ 3 ]

(i)

the initial dose of the drug.

(ii)

the percentage of the drug that leaves the body each hour.

[ 3 ]

(b)

Calculate the amount of the drug remaining in the body 10 hours after the injection.

[ 2 ]

[Maximum number: 2]

Gabriel is investigating the shape of model airplane wings. A cross-section of one of the wings is shown, graphed on the coordinate axes.

The shaded part of the cross-section is the area between the x-axis and the curve with equation

y=2 \sqrt{x}-\frac{x}{5}+1, \text { for } 0 \leq x \leq 100

where x is the distance, in cm , from the front of the wing and y is the height, in cm , above the horizontal axis through the wing, as shown in the diagram.

(a)

Calculate the percentage error of Gabriel's estimate in part (b).

[ 2 ]

[Maximum number: 1]

The following data show the heights, in metres, of six players in a basketball team.

(a)

Write down the shortest possible height of Gheorghe.

[ 1 ]

[Maximum number: 5]

Jan is investigating the shape of model helicopter propeller blades. A cross-section of one of the blades is shown, graphed on the coordinate axes.

The shaded part of the cross-section is the area between the x-axis and the curve with equation

y=4 \sqrt{x}-\frac{x}{2}+1, \text { for } 0 \leq x \leq 64

where x is the distance, in mm , from the edge of the blade and y is the height, in mm , above the horizontal axis through the blade, as shown in the diagram.

(a)

Find the values of a, b and c, shown in the table.

Jan uses the trapezoidal rule with four intervals to estimate the shaded area of the cross-section of the blade.

[ 3 ]

(b)

Calculate the percentage error of Jan's estimate in part (b).

[ 2 ]

[Maximum number: 13]

Cathie is a financial analyst studying the growth of two investment accounts, Account 1 and Account 2, for a new client.
Account 1 has an initial amount of 5000 US Dollars (USD). Interest is added to the amount in Account 1 at the end of each year in the following manner: 200 USD at the end of the first year, 260 USD at the end of the second year, 320 USD at the end of the third year, 380 USD at the end of the fourth year and 440 USD at the end of the fifth year.
Assume the amount of interest continues to increase each year so that it follows an arithmetic sequence.

(a)

Find

[ 3 ]

(i)

the common difference.

(ii)

the amount of interest, in USD, added at the end of the 10th year.

[ 3 ]

(b)

Show that the amount of money in Account 1 after n years may be expressed as

5000+\frac{n}{2}(340+60 n)

(c)

Hence or otherwise, find the amount of money in Account 1 at the end of 10 years.

Account 2 has the same initial amount of 5000 USD . Account 2 pays 6.5 % interest compounded annually. The interest is added to the amount in the account at the end of each year.

The amount in Account 2 after n years can be expressed as $5000 \times B^{n}$ where $B \in \mathbb{R}$ .

(d)

(i)

Write down the value of B.

(ii)

Hence or otherwise, show that Account 1 will have more money than Account 2 at the end of 10 years.

The client is interested in a longer-term investment. Cathie finds that it will take at least m complete years for the amount in Account 2 to exceed the amount in Account 1.

[ 4 ]

(e)

Find the value of m.

[ 3 ]

(f)

Determine the total interest added to Account 2 at the end of m years.

Give your answer correct to the nearest dollar.

[ 3 ]

[Maximum number: 18]

A marathon is a race over a distance of 42.19 km .

(a)

(i)

Write down 42.19 km correct to the nearest kilometre.

[ 3 ]

(ii)

Find the percentage error if 42.19 km is rounded to the nearest kilometre.

Rolando and Davi are training to run a marathon. They both start their training in the same week.
Rolando runs 6 km in the first week, 8 km in the second week and 10 km in the third week. In each subsequent week he runs 2 km further than the previous week.

[ 3 ]

(b)

Find the week during which Rolando runs 42 km .

[ 3 ]

(c)

Find the total distance that Rolando runs in the first 10 weeks.

Davi runs 3 km in the first week. In each subsequent week he runs 20 % further than the previous week. These distances form the terms of a geometric sequence.

[ 2 ]

(d)

Find the first week during which Davi runs more than 42 km .

[ 4 ]

(e)

Find the first week that the total distance that Davi has run, since the start of training, exceeds 150 km .

[ 3 ]

[Maximum number: 4]

Katya approximates $\pi$ , correct to four decimal places, by using the following expression.

3+\frac{1}{6+\frac{13}{16}}

(a)

Calculate Katya's approximation of $\pi$ , correct to four decimal places.

[ 2 ]

(b)

Calculate the percentage error in using Katya's four decimal place approximation of $\pi$ , compared to the exact value of $\pi$ in your calculator.

[ 2 ]

[Maximum number: 5]

Zaha is designing a bridge to cross a river. She believes that the weight of the steel needed for this bridge is approximately 53632000 kg .
The exact weight of the steel needed for the bridge is 55625000 kg .

(a)

Find the percentage error in Zaha's approximation.

Zaha's design is used to build five identical bridges.

[ 2 ]

(b)

(i)

Find the weight of the steel needed for these five bridges, to three significant figures.

(ii)

Write down your answer to part (b)(i) in the form $a \times 10^{k}$ , where $1 \leq a<10$ , $k \in \mathbb{Z}$ .

[ 3 ]

Question 1

Question 1(a)

Question 1(a)(i)

Question 1(a)(ii)

Question 1(b)

Question 1(b)(i)

Question 1(b)(ii)

Question 1(c)

Question 1(c)(i)

Question 1(c)(ii)

Question 1(d)

Question 1(e)

Question 1(f)

Question 1(g)

Question 1

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Question 1(c)

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Question 1(a)(i)

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Question 1(d)

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Question 1(c)

Question 1(d)

Question 1(d)(i)

Question 1(d)(ii)

Question 1(e)

Question 1(f)

Question 1

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