maximum value of $h$.

minimum value of $h$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Write down the amplitude of the function.

Find the period of the function.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

Find the height of $\text{C}$ above the ground when $t=2$.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point $\text{C}$ is now higher than $110 \text{m}$ for $1$ second during each complete rotation, find the time for one complete rotation.

maximum $h=130$ metres             A1

[1 mark]

minimum $h=50$ metres             A1

[1 mark]

$\left(60÷12=\right) 5 \text{seconds}$             A1

[1 mark]

$360÷5$            (M1)

Note: Award (M1) for $360$ divided by their time for one revolution.

$=72°$             A1

[2 marks]

(amplitude =)  $40$         A1

[1 mark]

(period $=\frac{360}{72}=$)  $5$         A1

[1 mark]

Maximum point labelled with correct coordinates.         A1

At least one minimum point labelled. Coordinates seen for any minimum points must be correct.         A1

Correct shape with an attempt at symmetry and “concave up" evident as it approaches the minimum points. Graph must be drawn in the given domain.         A1

[3 marks]

$h=90-40 \cos \left(144°\right)$           (M1)

$\left(h=\right) 122 \left(\text{m}\right) \left(122.3606\dots \right)$           A1

[2 marks]

evidence of $h=100$ on graph  OR  $100=90-40 \cos \left(72t\right)$           (M1)

$t$ coordinates $3.55 (3.54892...)$  OR  $1.45 (1.45107...)$ or equivalent           (A1)

Note: Award A1 for either $t$-coordinate seen.

$=2.10$ seconds  $\left(2.09784\dots \right)$           A1

[3 marks]

METHOD 1

$90-40 \cos \left(at°\right)=110$           (M1)

$\cos \left(at°\right)=-0.5$

$at°=120, 240$           (A1)

$1=\frac{240}{a}-\frac{120}{a}$           (M1)

$a=120$           (A1)

period $=\frac{360}{120}=3$ seconds           A1

METHOD 2

attempt at diagram           (M1)

$\cos \alpha =\frac{20}{40}$ (or recognizing special triangle)           (M1)

angle made by $\text{C}$, $2\alpha =120°$           (A1)

one third of a revolution in $1$ second           (M1)

hence one revolution $=3$ seconds           A1

METHOD 3

considering $h\left(t\right)=110$ on original function           (M1)

$t=\frac{5}{3}$  or  $\frac{10}{3}$           (A1)

$\frac{10}{3}-\frac{5}{3}=\frac{5}{3}$           (A1)

Note: Accept $t=1.67$ or equivalent.

so period is $\frac{3}{5}$ of original period           (R1)

so new period is $3$ seconds           A1

[5 marks]

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.

This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.