IB DP Maths Topic 3.4 The circular functions sinx , cosx and tanx SL Paper 1

 

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Question

Consider \(g(x) = 3\sin 2x\) .

Write down the period of g.

[1]
a.

On the diagram below, sketch the curve of g, for \(0 \le x \le 2\pi \) .


[3]
b.

Write down the number of solutions to the equation \(g(x) = 2\) , for \(0 \le x \le 2\pi \) .

[2]
c.
Answer/Explanation

Markscheme

\({\text{period}} = \pi \)     A1     N1

[1 mark]

a.

      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]

b.

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]

c.

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.

[5]
a.

Find \(f'(x)\) , giving your answer in the form \(a{\sin ^p}x{\cos ^q}x\) where \(a{\text{, }}p{\text{, }}q \in \mathbb{Z}\) .

[2]
b.

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.

[7]
c.
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]

a.

\(f'(x) = 3{\sin ^2}x\cos x\)     A2     N2

[2 marks]

b.

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]

c.

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.

[2]
a(i) and (ii).

(i)     Show that \(h(8) = 90.5\).

(ii)    Find \(h(21)\) .

[4]
b(i) and (ii).

Sketch the graph of h , for \(0 \le t \le 40\) .

[3]
c.

Given that h can be expressed in the form \(h(t) = a\cos bt + c\) , find a , b and c .

[5]
d.
Answer/Explanation

Markscheme

(i) 100 (metres)     A1     N1

(ii) 50 (metres)     A1     N1

[2 marks]

a(i) and (ii).

(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]

b(i) and (ii).


     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]

c.

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]

d.

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 .

[2]
a.

(i)     Show that the period of f is \(\pi \) .

(ii)    Hence, find the value of b .

[4]
b(i) and (ii).

Given that \(0 < c < \pi \)  , write down the value of c .

[1]
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]

a.

(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]

b(i) and (ii).

\(c = \frac{\pi }{4}\)     A1     N1

[1 mark]

c.

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 .

[3]
a(i) and (ii).

Write down the gradient of the curve at P.

[1]
b.

Write down the equation of the normal to the curve at P.

[2]
c.
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]

a(i) and (ii).

0     A1     N1

[1 mark]

b.

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]

c.

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)     \(r\) .

[6]
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]

 

a.

valid approach to find \(q\)     (M1)

eg   period = 4 , \(q = \frac{{2\pi }}{{{\rm{period}}}}\) 

\(q = \frac{\pi }{2}\)     A1     N2

[2 marks]

b.

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]

c.

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 \(k\).

[3]
a.

Find the minimum value of \(f(x)\).

[2]
b.

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 \(q\).

[2]
c.
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] 

a.

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]

b.

\(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]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Question

Let \(f(x) = 3\sin (\pi x)\).

Write down the amplitude of \(f\).

[1]
a.

Find the period of \(f\).

[2]
b.

On the following grid, sketch the graph of \(y = f(x)\), for \(0 \le x \le 3\).

[4]
c.
Answer/Explanation

Markscheme

amplitude is 3     A1     N1

a.

valid approach     (M1)

eg\(\;\;\;{\text{period}} = \frac{{2\pi }}{\pi },{\text{ }}\frac{{360}}{\pi }\)

period is 2     A1     N2

b.

      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.

c.

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 \(f\).

[3]
a.

On the following grid sketch the graph of \(f\).

M16/5/MATME/SP1/ENG/TZ1/03.b

[4]
b.
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]

a.

M16/5/MATME/SP1/ENG/TZ1/03.b/M     A1A1A1A1     N4

[4 marks]

b.

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 \(a,{\text{ }}b \in \mathbb{Z}\).

[5]
a.

Hence find the range of \(h\).

[2]
b.
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]

a.

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]

b.
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