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Question
Consider \(g(x) = 3\sin 2x\) .
Write down the period of g.
On the diagram below, sketch the curve of g, for \(0 \le x \le 2\pi \) .
Write down the number of solutions to the equation \(g(x) = 2\) , for \(0 \le x \le 2\pi \) .
Answer/Explanation
Markscheme
\({\text{period}} = \pi \) A1 N1
[1 mark]
A1A1A1 N3
Note: Award A1 for amplitude of 3, A1 for their period, A1 for a sine curve passing through \((0{\text{, }}0)\) and \((0{\text{, }}2\pi )\) .
[3 marks]
evidence of appropriate approach (M1)
e.g. line \(y = 2\) on graph, discussion of number of solutions in the domain
4 (solutions) A1 N2
[2 marks]
Question
Let \(f:x \mapsto {\sin ^3}x\) .
(i) Write down the range of the function f .
(ii) Consider \(f(x) = 1\) , \(0 \le x \le 2\pi \) . Write down the number of solutions to this equation. Justify your answer.
Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where \(a{\text{, }}p{\text{, }}q \in \mathbb{Z}\) .
Let \(g(x) = \sqrt 3 \sin x{(\cos x)^{\frac{1}{2}}}\) for \(0 \le x \le \frac{\pi }{2}\) . Find the volume generated when the curve of g is revolved through \(2\pi \) about the x-axis.
Answer/Explanation
Markscheme
(i) range of f is \([ – 1{\text{, }}1]\) , \(( – 1 \le f(x) \le 1)\) A2 N2
(ii) \({\sin ^3}x \Rightarrow 1 \Rightarrow \sin x = 1\) A1
justification for one solution on \([0{\text{, }}2\pi ]\) R1
e.g. \(x = \frac{\pi }{2}\) , unit circle, sketch of \(\sin x\)
1 solution (seen anywhere) A1 N1
[5 marks]
\(f'(x) = 3{\sin ^2}x\cos x\) A2 N2
[2 marks]
using \(V = \int_a^b {\pi {y^2}{\rm{d}}x} \) (M1)
\(V = \int_0^{\frac{\pi }{2}} {\pi (\sqrt 3 } \sin x{\cos ^{\frac{1}{2}}}x{)^2}{\rm{d}}x\) (A1)
\( = \pi \int_0^{\frac{\pi }{2}} {3{{\sin }^2}x\cos x{\rm{d}}x} \) A1
\(V = \pi \left[ {{{\sin }^3}x} \right]_0^{\frac{\pi }{2}}\) \(\left( { = \pi \left( {{{\sin }^3}\left( {\frac{\pi }{2}} \right) – {{\sin }^3}0} \right)} \right)\) A2
evidence of using \(\sin \frac{\pi }{2} = 1\) and \(\sin 0 = 0\) (A1)
e.g. \(\pi \left( {1 – 0} \right)\)
\(V = \pi \) A1 N1
[7 marks]
Question
The following diagram represents a large Ferris wheel, with a diameter of 100 metres.
Let P be a point on the wheel. The wheel starts with P at the lowest point, at ground level. The wheel rotates at a constant rate, in an anticlockwise (counter-clockwise) direction. One revolution takes 20 minutes.
Let \(h(t)\) metres be the height of P above ground level after t minutes. Some values of \(h(t)\) are given in the table below.
Write down the height of P above ground level after
(i) 10 minutes;
(ii) 15 minutes.
(i) Show that \(h(8) = 90.5\).
(ii) Find \(h(21)\) .
Sketch the graph of h , for \(0 \le t \le 40\) .
Given that h can be expressed in the form \(h(t) = a\cos bt + c\) , find a , b and c .
Answer/Explanation
Markscheme
(i) 100 (metres) A1 N1
(ii) 50 (metres) A1 N1
[2 marks]
(i) identifying symmetry with \(h(2) = 9.5\) (M1)
subtraction A1
e.g. \(100 – h(2)\) , \(100 – 9.5\)
\(h(8) = 90.5\) AG N0
(ii) recognizing period (M1)
e.g. \(h(21) = h(1)\)
\(h(21) = 2.4\) A1 N2
[4 marks]
A1A1A1 N3
Note: Award A1 for end points (0, 0) and (40, 0) , A1 for range \(0 \le h \le 100\) , A1 for approximately correct sinusoidal shape, with two cycles.
[3 marks]
evidence of a quotient involving 20, \(2\pi \) or \({360^ \circ }\) to find b (M1)
e.g. \(\frac{{2\pi }}{b} = 20\) , \(b = \frac{{360}}{{20}}\)
\(b = \frac{{2\pi }}{{20}}\) \(\left( { = \frac{\pi }{{10}}} \right)\) (accept \(b = 18\) if working in degrees) A1 N2
\(a = – 50\) , \(c = 50\) A2A1 N3
[5 marks]
Question
The diagram below shows part of the graph of \(f(x) = a\cos (b(x – c)) – 1\) , where \(a > 0\) .
The point \({\rm{P}}\left( {\frac{\pi }{4},2} \right)\) is a maximum point and the point \({\rm{Q}}\left( {\frac{{3\pi }}{4}, – 4} \right)\) is a minimum point.
Find the value of a .
(i) Show that the period of f is \(\pi \) .
(ii) Hence, find the value of b .
Given that \(0 < c < \pi \) , write down the value of c .
Answer/Explanation
Markscheme
evidence of valid approach (M1)
e.g. \(\frac{{{\text{max }}y{\text{ value}} – {\text{min }}y{\text{ value}}}}{2}\) , distance from \(y = – 1\)
\(a = 3\) A1 N2
[2 marks]
(i) evidence of valid approach (M1)
e.g. finding difference in x-coordinates, \(\frac{\pi }{2}\)
evidence of doubling A1
e.g. \(2 \times \left( {\frac{\pi }{2}} \right)\)
\({\text{period}} = \pi \) AG N0
(ii) evidence of valid approach (M1)
e.g. \(b = \frac{{2\pi }}{\pi }\)
\(b = 2\) A1 N2
[4 marks]
\(c = \frac{\pi }{4}\) A1 N1
[1 mark]
Question
The following diagram shows the graph of \(f(x) = a\cos (bx)\) , for \(0 \le x \le 4\) .
There is a minimum point at P(2, − 3) and a maximum point at Q(4, 3) .
(i) Write down the value of a .
(ii) Find the value of b .
Write down the gradient of the curve at P.
Write down the equation of the normal to the curve at P.
Answer/Explanation
Markscheme
(i) \(a = 3\) A1 N1
(ii) METHOD 1
attempt to find period (M1)
e.g. 4 , \(b = 4\) , \(\frac{{2\pi }}{b}\)
\(b = \frac{{2\pi }}{4}\left( { = \frac{\pi }{2}} \right)\) A1 N2
[3 marks]
METHOD 2
attempt to substitute coordinates (M1)
e.g. \(3\cos (2b) = – 3\) , \(3\cos (4b) = 3\)
\(b = \frac{{2\pi }}{4}\left( { = \frac{\pi }{2}} \right)\) A1 N2
[3 marks]
0 A1 N1
[1 mark]
recognizing that normal is perpendicular to tangent (M1)
e.g. \({m_1} \times {m_2} = – 1\) , \(m = – \frac{1}{0}\) , sketch of vertical line on diagram
\(x = 2\) (do not accept 2 or \(y = 2\) ) A1 N2
[2 marks]
Question
The diagram below shows part of the graph of a function \(f\) .
The graph has a maximum at A(\(1\), \(5\)) and a minimum at B(\(3\), \( -1\)) .
The function \(f\) can be written in the form \(f(x) = p\sin (qx) + r\) . Find the value of
(a) \(p\)
(b) \(q\)
(c) \(r\) .
Answer/Explanation
Markscheme
(a) valid approach to find \(p\) (M1)
eg amplitude \( = \frac{{{\rm{max}} – {\rm{min}}}}{2}\) , \(p = 6\)
\(p = 3\) A1 N2
[2 marks]
(b) valid approach to find \(q\) (M1)
eg period = 4 , \(q = \frac{{2\pi }}{{{\rm{period}}}}\)
\(q = \frac{\pi }{2}\) A1 N2
[2 marks]
(c) valid approach to find \(r\) (M1)
eg axis = \(\frac{{{\rm{max}} + {\rm{min}}}}{2}\) , sketch of horizontal axis, \(f(0)\)
\(r = 2\) A1 N2
[2 marks]
Total [6 marks]
valid approach to find \(p\) (M1)
eg amplitude \( = \frac{{{\rm{max}} – {\rm{min}}}}{2}\) , \(p = 6\)
\(p = 3\) A1 N2
[2 marks]
valid approach to find \(q\) (M1)
eg period = 4 , \(q = \frac{{2\pi }}{{{\rm{period}}}}\)
\(q = \frac{\pi }{2}\) A1 N2
[2 marks]
valid approach to find \(r\) (M1)
eg axis = \(\frac{{{\rm{max}} + {\rm{min}}}}{2}\) , sketch of horizontal axis, \(f(0)\)
\(r = 2\) A1 N2
[2 marks]
Total [6 marks]
Question
Let \(f(x) = \sin \left( {x + \frac{\pi }{4}} \right) + k\). The graph of f passes through the point \(\left( {\frac{\pi }{4},{\text{ }}6} \right)\).
Find the value of \(k\).
Find the minimum value of \(f(x)\).
Let \(g(x) = \sin x\). The graph of g is translated to the graph of \(f\) by the vector \(\left( {\begin{array}{*{20}{c}} p \\ q \end{array}} \right)\).
Write down the value of \(p\) and of \(q\).
Answer/Explanation
Markscheme
METHOD 1
attempt to substitute both coordinates (in any order) into \(f\) (M1)
eg \(f\left( {\frac{\pi }{4}} \right) = 6,{\text{ }}\frac{\pi }{4} = \sin \left( {6 + \frac{\pi }{4}} \right) + k\)
correct working (A1)
eg \(\sin \frac{\pi }{2} = 1,{\text{ }}1 + k = 6\)
\(k = 5\) A1 N2
[3 marks]
METHOD 2
recognizing shift of \(\frac{\pi }{4}\) left means maximum at \(6\) R1)
recognizing \(k\) is difference of maximum and amplitude (A1)
eg \(6 – 1\)
\(k = 5\) A1 N2
[3 marks]
evidence of appropriate approach (M1)
eg minimum value of \(\sin x\) is \( – 1,{\text{ }} – 1 + k,{\text{ }}f'(x) = 0,{\text{ }}\left( {\frac{{5\pi }}{4},{\text{ }}4} \right)\)
minimum value is \(4\) A1 N2
[2 marks]
\(p = – \frac{\pi }{4},{\text{ }}q = 5{\text{ }}\left( {{\text{accept \(\left( \begin{array}{c} – {\textstyle{\pi \over 4}}\\5\end{array} \right)\)}}} \right)\) A1A1 N2
[2 marks]
Examiners report
[N/A]
[N/A]
[N/A]
Question
Let \(f(x) = 3\sin (\pi x)\).
Write down the amplitude of \(f\).
Find the period of \(f\).
On the following grid, sketch the graph of \(y = f(x)\), for \(0 \le x \le 3\).
Answer/Explanation
Markscheme
amplitude is 3 A1 N1
valid approach (M1)
eg\(\;\;\;{\text{period}} = \frac{{2\pi }}{\pi },{\text{ }}\frac{{360}}{\pi }\)
period is 2 A1 N2
A1
A1A1A1 N4
Note: Award A1 for sine curve starting at (0, 0) and correct period.
Only if this A1 is awarded, award the following for points in circles:
A1 for correct x-intercepts;
A1 for correct max and min points;
A1 for correct domain.
Question
Let \(f(x) = 3\sin \left( {\frac{\pi }{2}x} \right)\), for \(0 \leqslant x \leqslant 4\).
(i) Write down the amplitude of \(f\).
(ii) Find the period of \(f\).
On the following grid sketch the graph of \(f\).
Answer/Explanation
Markscheme
(i) 3 A1 N1
(ii) valid attempt to find the period (M1)
eg\(\,\,\,\,\,\)\(\frac{{2\pi }}{b},{\text{ }}\frac{{2\pi }}{{\frac{\pi }{2}}}\)
period \( = 4\) A1 N2
[3 marks]
A1A1A1A1 N4
[4 marks]
Question
Let \(f(x) = 6x\sqrt {1 – {x^2}} \), for \( – 1 \leqslant x \leqslant 1\), and \(g(x) = \cos (x)\), for \(0 \leqslant x \leqslant \pi \).
Let \(h(x) = (f \circ g)(x)\).
Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).
Hence find the range of \(h\).
Answer/Explanation
Markscheme
attempt to form composite in any order (M1)
eg\(\,\,\,\,\,\)\(f\left( {g(x)} \right),{\text{ }}\cos \left( {6x\sqrt {1 – {x^2}} } \right)\)
correct working (A1)
eg\(\,\,\,\,\,\)\(6\cos x\sqrt {1 – {{\cos }^2}x} \)
correct application of Pythagorean identity (do not accept \({\sin ^2}x + {\cos ^2}x = 1\)) (A1)
eg\(\,\,\,\,\,\)\({\sin ^2}x = 1 – {\cos ^2}x,{\text{ }}6\cos x\sin x,{\text{ }}6\cos x \left| \sin x\right|\)
valid approach (do not accept \(2\sin x\cos x = \sin 2x\)) (M1)
eg\(\,\,\,\,\,\)\(3(2\cos x\sin x)\)
\(h(x) = 3\sin 2x\) A1 N3
[5 marks]
valid approach (M1)
eg\(\,\,\,\,\,\)amplitude \( = 3\), sketch with max and min \(y\)-values labelled, \( – 3 < y < 3\)
correct range A1 N2
eg\(\,\,\,\,\,\)\( – 3 \leqslant y \leqslant 3\), \([ – 3,{\text{ }}3]\) from \( – 3\) to 3
Note: Do not award A1 for \( – 3 < y < 3\) or for “between \( – 3\) and 3”.
[2 marks]