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Trigonometric Functions · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SSine, cosine and tangent functions6 marksPaper 1~9 min
The depth, dd metres, of water in a harbour is modelled by d(t)=3cos⁡(πt6)+5d(t) = 3\cos\left(\frac{\pi t}{6}\right) + 5 where tt is the time in hours after midnight, 0≤t≤240 \leq t \leq 24.
(a)
Write down the minimum depth of water in the harbour. [1 mark]
(b)
Find the depth of water at 2:00 AM. Give your answer correct to 3 significant figures. [2 marks]
(c)
Find the first time after midnight at which the depth of water is 3.5 m3.5\,\text{m}. Give your answer correct to the nearest minute. (Total: 6 marks) [3 marks]
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Solutions

2FoundationSAQ-SGraphing trigonometric functions10 marksPaper 1~15 min
The depth of water, dd metres, in a harbour is modelled by the function d(t)=3.2sin⁡ ⁣(π6t)+4.8d(t) = 3.2\sin\!\left(\frac{\pi}{6}t\right) + 4.8 where tt is the time in hours after midnight.
(a)
State the amplitude and the period of d(t)d(t). [2 marks]
(b)
Calculate the minimum depth of water in the harbour and state the first time after midnight at which this minimum occurs. A vessel requires a minimum depth of 3.0 m3.0\,\text{m} to enter the harbour safely. [3 marks]
(c)
Determine the values of tt, in the interval 0≤t≤120 \leq t \leq 12, at which the depth equals 3.0 m3.0\,\text{m}, and hence state the length of time during this interval for which the harbour is unsafe for the vessel. [5 marks]
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Solutions

3MasterySAQ-SGraphing trigonometric functions5 marksPaper 1~8 min
The temperature TT in degrees Celsius inside a greenhouse over a 24-hour period is modelled by T(t)=18−5sin⁡ ⁣(π12(t−6))T(t) = 18 - 5\sin\!\left(\frac{\pi}{12}(t - 6)\right) where tt is the time in hours after midnight, 0≤t<240 \leq t < 24.
(a)
Show that the temperature at 14:00 is 13.7∘C13.7^\circ\text{C}, correct to one decimal place. [2 marks]
(b)
Find the two times during the day when the temperature is exactly 15.5∘C15.5^\circ\text{C}. Give your answers in hours and minutes, correct to the nearest minute. [3 marks]
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Solutions

4ChallengeSAQ-LTrigonometric identities and equations8 marksPaper 1~12 min
The depth of water, dd metres, at a harbour entrance is modelled by d(t)=4.5+2.5cos⁡ ⁣(π6t)d(t) = 4.5 + 2.5\cos\!\left(\frac{\pi}{6}t\right) where tt is the time in hours after midnight.
(a)
A boat requires a minimum depth of 5 m5\,\text{m} to enter the harbour safely. Determine the two smallest positive values of tt for which d(t)=5d(t) = 5. Give your answers correct to three significant figures. [4 marks]
(b)
A second boat requires a minimum depth of 6 m6\,\text{m}. Using your answers from part (a), determine the total length of time during 0≤t≤120 \leq t \leq 12 when both boats can safely enter the harbour simultaneously. Give your answer in hours correct to three significant figures. [4 marks]
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Solutions

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