Question
The depth, h(t) metres, of water at the entrance to a harbour at t hours after midnight on a particular day is given by
\[h(t) = 8 + 4\sin \left( {\frac{{\pi t}}{6}} \right),{\text{ }}0 \leqslant t \leqslant 24.\]
(a) Find the maximum depth and the minimum depth of the water.
(b) Find the values of t for which \(h(t) \geqslant 8\).
▶️Answer/Explanation
Markscheme
(a) Either finding depths graphically, using \(\sin \frac{{\pi t}}{6} = \pm 1\) or solving \(h'(t) = 0\) for t (M1)
\(h{(t)_{\max }} = 12{\text{ (m), }}h{(t)_{\min }} = 4{\text{ (m)}}\) A1A1 N3
(b) Attempting to solve \(8 + 4\sin \frac{{\pi t}}{6} = 8\) algebraically or graphically (M1)
\(t \in [{\text{0}},{\text{6}}] \cup [{\text{12}},{\text{18}}] \cup \{ {\text{24}}\} \) A1A1 N3
[6 marks]
Examiners report
Not as well done as expected with most successful candidates using a graphical approach. Some candidates confused t and h and subsequently stated the values of t for which the water depth was either at a maximum and a minimum. Some candidates simply gave the maximum and minimum coordinates without stating the maximum and minimum depths.
In part (b), a large number of candidates left out t = 24 from their final answer. A number of candidates experienced difficulties solving the inequality via algebraic means. A number of candidates specified incorrect intervals or only one correct interval.
Question
The graph below shows \(y = a\cos (bx) + c\).
Find the value of a, the value of b and the value of c.
▶️Answer/Explanation
Markscheme
\(a = 3\) A1
\(c = 2\) A1
period \( = \frac{{2\pi }}{b} = 3\) (M1)
\(b = \frac{{2\pi }}{3}{\text{ }}( = 2.09)\) A1
[4 marks]
Examiners report
Most candidates were able to find a and c, but many had difficulties with finding b.
Question
A function is defined by \(f(x) = A\sin (Bx) + C,{\text{ }} – \pi \le x \le \pi \), where \(A,{\text{ }}B,{\text{ }}C \in \mathbb{Z}\). The following diagram represents the graph of \(y = f(x)\).
a.Find the value of
(i) \(A\);
(ii) \(B\);
(iii) \(C\).[4]
▶️Answer/Explanation
Markscheme
(i) \(A = – 3\) A1
(ii) period \( = \frac{\pi }{B}\) (M1)
\(B = 2\) A1
Note: Award as above for \(A = 3\) and \(B = – 2\).
(iii) \(C = 2\) A1
[4 marks]
\(x = 1.74,{\text{ }}2.97\;\;\;\left( {x = \frac{1}{2}\left( {\pi + \arcsin \frac{1}{3}} \right),{\text{ }}\frac{1}{2}\left( {2\pi – \arcsin \frac{1}{3}} \right)} \right)\) (M1)A1
Note: Award (M1)A0 if extra correct solutions eg \(( – 1.40,{\text{ }} – 0.170)\) are given outside the domain \(0 \le x \le \pi \). Do not award FT in (b).
[2 marks]
Total [6 marks]