IB DP Math AA : Topic : SL 3.7 :Composite functions of the form: IB Style Questions HL Paper 2

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]

b.Solve \(f(x) = 3\) for \(0 \le x \le \pi \).[2]
 
▶️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]

a.

\(x = 1.74,{\text{ }}2.97\;\;\;\left( {x = \frac{1}{2}\left( {\pi  + \arcsin \frac{1}{3}} \right),{\text{ }}\frac{1}{2}\left( {2\pi  – \arcsin \frac{1}{3}} \right)} \right)\)     (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]

 

 

 

 

 
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