DP IB Maths: AA SL

Topic Questions

3.5 Trigonometric Functions & Graphs

1a
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3 marks

A graph has the equation y equals cos space 3 x  for the interval  negative straight pi over 3 less or equal than x less or equal than 2 over 3 space pi.

Sketch the graph on the axes below.

q1a-3-5-hard-ib-aa-sl-maths

1b
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5 marks

A straight line with equation y space equals space 1 half  intersects the graph of y equals cos space 3 x

(i)
Sketch the line y space equals space 1 half on to the same set of axes.
(ii)
Find the coordinates of the points of intersection between y equals cos space 3 x and  y equals begin inline style begin display style 1 half end style end style.

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2
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5 marks
(i)
Sketch the graph of space y equals sin left parenthesis 2 theta minus 60 right parenthesis in the interval negative 180 degree less or equal than theta less or equal than 180 degree .
(ii)
Write down all the values for theta, where y equals sin left parenthesis 2 theta minus 60 right parenthesis degree equals 0 in the given interval.

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3a
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4 marks

An average heart contains a volume of approximately 140 millilitres and pumps out one half of its volume with each beat. A healthy adult has a heart rate of about 70 beats per minute.

Assuming that the heart starts at full capacity, the volume of blood, V left parenthesis t right parenthesis , in the heart can be modelled as a function of the time, t, in seconds.

Write down a model for the volume of blood, V left parenthesis t right parenthesis , giving your answer in the form V left parenthesis t right parenthesis equals A space cos left parenthesis B t right parenthesis plus D, where A, B and C are constants to be found.

3b
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3 marks

Show that the heart is at its minimum capacity at t space equals space 3 over 7 seconds.

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4a
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3 marks

A snake moves along a horizontal surface following a line that is 3 m from, and parallel to, the edge of a straight section of river, as shown in the diagram below.

q4-3-5-hard-ib-aa-sl-maths

The perpendicular distance of the tip of the snake’s tail from the edge of the river, y cm, can be modelled by the function

y equals 8 space sin space left parenthesis 15 x right parenthesis degree plus 300 for the interval  0 less or equal than x less or equal than 120

where x is the horizontal distance, in cm, moved by the tip of the tail from the start of the movement.

(i)
State the maximum perpendicular distance that the tip of the tail will be from  the edge of the river.
(ii)
State the number of times that the tip of the tail will be at the maximum distance in the given domain.
4b
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3 marks

A stone is located at a perpendicular distance of 294 cm from the river when the snake has travelled 1 over 6 of the total horizontal distance.

Show that the tip of the tail will collide with the stone.

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5a
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2 marks

A hamster runs in its exercise wheel, rotating the wheel at a constant speed. The wheel has a diameter of 14 centimetres and the top of the wheel is positioned at a height of k centimetres above the floor of the cage.

A point at the top of the wheel is marked before the hamster starts to run, turning the wheel clockwise. The hamster takes 4 seconds to turn the wheel one complete revolution.

After t seconds, the height of the mark on the wheel above the floor of the cage is given by

h left parenthesis t right parenthesis equals 10 plus a space cos open parentheses straight pi over 2 space t close parentheses  for  0 less or equal than t less or equal than 150

After 26 seconds, the mark is 3 cm above the cage floor. Find k .

5b
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3 marks

Find the value of a.

5c
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4 marks

Find the value of t spacewhen the mark is 8 cm above the floor of the cage for the 5th time in the given time period and state whether the mark is ascending or descending.

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6a
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3 marks

A student sets up an experiment with a model car moving along a horizontal surface in a straight line. The car has a trailer attached by a pin joint. A momentary force is applied to the end of the trailer, perpendicular to the direction of travel, which causes it to move sideways back and forth as the car continues to move forwards. A diagram can be seen below.

q6-3-5-hard-ib-aa-sl-maths

Point P is situated at the midpoint of the end of the trailer. The displacement, d in cm, of point P relative to the centre line of the car in the direction of motion can be modelled by the function

d equals e to the power of negative 2 over 5 t space plus 8 over 15 pi end exponent space sin space left parenthesis 3 t minus 4 pi right parenthesis

where t is the time in seconds since the application of the force.

Find the total distance that point P has moved perpendicular to the line of motion, in the first 3 seconds.

6b
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3 marks

When the maximum displacement from the centre line does not exceed 2.5 mm, the trailer is considered to be stable.

State the time after which the trailer can be considered stable.

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7a
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3 marks

Let  f left parenthesis x right parenthesis equals 5 space sin space open parentheses pi over 4 x close parentheses minus 4 for x element of straight real numbers.

Let g left parenthesis x right parenthesis equals 3 f left parenthesis 2 x right parenthesis.

The function g can be written in the form g left parenthesis x right parenthesis equals 15 space sin left parenthesis b x right parenthesis plus c.

The range of f is k less or equal than f left parenthesis x right parenthesis less or equal than m. Find k and m.

7b
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2 marks

Find the range of g.

7c
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5 marks
(i)
Find the value of b and c.
(ii)
Find the period of g .
7d
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3 marks

The equation g left parenthesis x right parenthesis equals f left parenthesis x right parenthesis has two solutions where pi less or equal than x less or equal than 2 pi. Find both solutions.

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8a
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7 marks

Let f left parenthesis x right parenthesis equals p space cos space left parenthesis q cross times x right parenthesis plus r

The diagram below shows the graph of f, for 0 less or equal than x less or equal than 10

q12-3-5-hard-ib-aa-sl-maths

The first local minimum is at point Aleft parenthesis 1.5 comma 3.2 right parenthesis  and the next local maximum is at point Bleft parenthesis 3 comma 7.4 right parenthesis .

Find the value of

(i)
p
(ii)
q
(iii)
r.
8b
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4 marks

Let g left parenthesis x right parenthesis equals 4 space tan space left parenthesis 2 x right parenthesis

Find the two solutions for f left parenthesis x right parenthesis equals g open parentheses 1 fourth x minus 3 close parentheses plus 4, for 0 less or equal than x less or equal than 10.

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1a
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1 mark

The graph below shows the curve with equation y space equals space sin space 2 x in the interval negative 60 degree less or equal than x less or equal than 270 degree.

q1a-3-5-medium-ib-aa-sl-maths

Point A has coordinates left parenthesis negative 45 degree comma negative 1 right parenthesis and is the minimum point closest to the origin. Point B is the maximum point closest to the origin. State the coordinates of B.

1b
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2 marks

A straight line with equation y space equals space minus 1 half meets the graph of y equals sin space 2 x at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates open parentheses negative 15 degree comma negative 1 half close parentheses, use graph symmetries to determine the coordinates of Q and R.

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2
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4 marks
(i)
Sketch the graph of space y equals cos space left parenthesis theta plus 30 degree right parenthesis in the interval negative 180 degree less or equal than theta less or equal than 360 degree.

(ii)
Write down all the values where cos space left parenthesis theta plus 30 degree right parenthesis equals 0 in the given interval.

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3
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3 marks

A dolphin is swimming such that it is diving in and out of the water at a constant speed.

On each jump and dive the dolphin reaches a height of 2 m above sea level and a depth of 2 m below sea level.

Starting at sea level, the dolphin takes fraction numerator 2 straight pi over denominator 3 end fraction seconds to jump out of the water, dive back in and return to sea level.

Write down a model for the height, h straight m, of the dolphin, relative to sea level, at time t, in the form h equals A space sin space left parenthesis B t right parenthesis where A space and space B  are constants to be found.

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4
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6 marks

A section of a new rollercoaster has a series of rises and falls. The vertical displacement of the rollercoaster carriage, y, measured in metres relative to a fixed reference height, can be modelled using the function y equals 30 space cos space left parenthesis 24 t right parenthesis degree,  where t is the time in seconds.

Sketch the function for the interval space 0 less or equal than t less or equal than 30.

How many times will the rollercoaster carriage fall during the 30 seconds?

How long does the model suggest it will take for the rollercoaster carriage to reach the bottom of the first fall?

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5a
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3 marks

The height, h space straight m, of water in a reservoir is modelled by the function

h left parenthesis t right parenthesis equals A plus B space sin space open parentheses straight pi over 6 t close parentheses comma space t greater or equal than 0

where t is the time in hours after midnight. A and B are positive constants.

In terms of A and B, write down the natural height of the water in the reservoir, as well as its maximum and minimum heights.

5b
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3 marks

The maximum level of water is 3 m higher than its natural level.

The level of water is three times higher at its maximum than at its minimum.

Find the maximum, minimum and natural water levels.

5c
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3 marks
(i)
How many times per day does the water reach its maximum level?
(ii)
Find the times of day when the water level is at its minimum?

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6a
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2 marks

A lifejacket falls over the side of a boat from a height of 3 space straight m.
The height, h space straight m, of the lifejacket above or below sea level open parentheses h equals 0 close parentheses, at time t seconds after falling, is modelled by the equation  h equals 3 e to the power of negative 0.7 t end exponent space cos space 4 t.

The lifejacket reaches its furthest point below sea level after 0.742 seconds.
Find the total distance it has fallen, giving your answer to three significant figures.

6b
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2 marks

Write down the value of t for the first three times the lifejacket is at sea level.

6c
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3 marks
(i)
Find the value of 3 e to the power of negative 0.7 t end exponent when t equals 6.2.

(ii)
Hence justify why, from 6.2 seconds on, the lifejacket will always be within 4 centimetres of sea level.

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7a
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5 marks

The number of daylight hours, h, in the UK, during a day d days after the spring equinox (the day in spring when the number of daylight hours is 12), is modelled using the function

h equals 12 plus 9 over 2 space sin space open parentheses fraction numerator 2 pi over denominator 365 end fraction space d close parentheses

(i)
Find the number of daylight hours during the day that is 100 days after the spring equinox.
(ii)
Find the number of days after the spring equinox that the two days occur during which the number of daylight hours is closest to 9.
7b
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3 marks

For how many days of the year does the model suggest that the number of daylight hours exceeds 15 hours?  Give your answer as a whole number of days.

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8a
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3 marks

Felicity is a keen ice skater and has entered a competition that requires her to skate in a circular pathway in front of three judges. Her distance, d meters, away from the judges table, t spaceseconds after commencing her routine can be modelled by the function

d equals 12 space cos space pi over 30 t space plus 15

(i)
State the distance Felicity is away from the judges table at the start of her                 
(ii)
State the distance Felicity is away from the judges table after 15 seconds.
8b
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2 marks

Find, in terms of pi, the circumference of Felicity’s circular pathway on the ice rink.

8c
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2 marks

Find, in terms of pi, Felicity’s average speed for each lap on the ice rink.

8d
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3 marks

Felicity’s routine took three laps in total around the ice rink.

Find the times during Felicity’s routine where she was at a distance of 21 spacemetres from the judges table.

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1
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6 marks

Let f left parenthesis x right parenthesis equals sin space left parenthesis 2 left parenthesis x plus 5 right parenthesis right parenthesis minus 4

Sketch the graph of 2 f left parenthesis x minus 3 right parenthesis for the interval  0 less or equal than x less or equal than 2 pi. Label the coordinates of the local maxima and minima. The coordinates should be given as exact values.

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2
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7 marks
(i)
Sketch the graph of space y equals 4 space sin left parenthesis theta plus 45 right parenthesis minus fraction numerator 4 over denominator square root of 2 end fraction in the interval negative 180 degree less or equal than theta less or equal than 360 degree.
(ii)
Write down all the values for theta, where 4 space sin space open parentheses theta plus 45 close parentheses minus fraction numerator 4 over denominator square root of 2 end fraction equals 0 in the given interval.

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3
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7 marks

A particle is travelling horizontally whilst moving in and out of a body of water at a constant speed. The particle reaches a maximum height of 1.3 m above the water level and and a depth of 2.6 m.

The particle starts at a depth of  13 over 8m and takes 4 over 3 pi seconds to move up through the water, reach the maximum height, dive to the minimum depth and return to its starting depth.

Write down a model for the height, h m, of the particle, relative to water level, at a time t seconds, in the form h equals A space sin left parenthesis B t plus C right parenthesis plus D, where A comma B comma C and D are constants to be found.

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4a
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3 marks

A section of a model railway track has a series of rises and falls. The vertical displacement of a train carriage moving along the track, h in m, relative to the horizontal floor, can be modelled by

h left parenthesis t right parenthesis equals sin left parenthesis 6 t cubed minus 2 t squared minus 10 t right parenthesis plus 1.5, for  0 less or equal than t less or equal than straight pi over 3

where t is the time in minutes.

Find the average vertical speed of the train carriage when it experiences the maximum change in height within the given section of track.

4b
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3 marks

Vertical metal supports are required to ensure that the track is stable. A support is required at either end of the track, as well as at each local maximum and minimum.

Given that there is 7.9 m of metal available to create the supports, show that this is not sufficient to place the supports in the locations required.

4c
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4 marks

A support attached to the side of the track is also required. The support must attach to the model at four separate points, including the end point of the section of the track and can be modelled by the function y equals x plus c, where c is a constant.

Find the length of the side support from the first point of connection with the track to the fourth.

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5
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7 marks

The diagram below shows a circle with centre O and radius 2 cm. Points A and B lie on the circumference of the circle and angle straight A straight O with hat on top straight B equals 3 theta, where 0 less than theta less than pi over 3.

The tangents to the circle at points A and B intersect at point C.

q5-3-5-very-hard-ib-aa-sl-maths

Find the value of theta when the shaded area is equal to the area of sector OAB.

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6a
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3 marks

A particle, A, starts at a fixed point, O, before being set into motion. The vertical displacement of the particle,  h subscript A cm, from point O can be modelled by the equation

h subscript A left parenthesis x right parenthesis equals x squared space cos space 5 x space sin space x,   for  0 less or equal than x less or equal than 5

where x is the horizontal displacement, in cm, of the particle from O.

Find the straight line distance of the particle from O at end of the motion.

6b
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4 marks

A second particle, B, starts moving at the same time as particle A. The motion of particle B can be described by the function

h subscript B left parenthesis x right parenthesis equals 2 x plus a,   for   0 less or equal than x less or equal than 5 comma space space space a element of straight real numbers,   

Given that the particles stop moving at the point of collision, find the value of a.

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7a
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2 marks

A Ferris wheel with centre O and diameter 100 metres, comprises 32 individual compartments and rotates clockwise at a constant speed of 0.96 kilometres per hour.

q7a-3-5-very-hard-ib-aa-sl-maths

Passengers board the Ferris wheel at point A. The height, h metres, of a compartment above the ground after it passes through point A is modelled by the function

h left parenthesis t right parenthesis equals 58 plus 40 space cos space open parentheses 25 over 72 left parenthesis t plus 8 right parenthesis close parentheses minus 30 space sin space open parentheses 25 over 72 left parenthesis t plus 8 right parenthesis close parentheses, for t greater or equal than 0

where t is the time elapsed in minutes.

Find the height of point A above the ground.

7b
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3 marks
(i)
Calculate the number of minutes it takes for the Ferris wheel to complete one revolution.
(ii)
Hence find the number of revolutions the Ferris wheel makes in one hour.
7c
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4 marks

Points B and C indicate the edges of the region in which a person gains the best view of the city during the rotation of the Ferris wheel. B is reached 10 mins after boarding the wheel. Point C is located at a vertical distance of 26.2 m below point B.

Find

(i)
the length of time for which the person experiences the best view of the city.
(ii)
Find the angle straight B straight O with hat on top straight C.

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8a
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4 marks

A large clock face is mounted on a tower, with the centre of the clock face at a height of 19 m above ground level and the tip of the minute hand reaching the circumference of the clock face.

The clock is started at 12 pm and the tip of the minute hand travels a total distance of 5.8 m from its initial position in 35 minutes.

Find a model for the height of the tip of the minute hand, h m, above ground level, in  the form h left parenthesis x right parenthesis equals A space sin space left parenthesis B x right parenthesis plus C, where x is the angle measured clockwise between the number 12 and the minute hand and A comma space B and C are constants to be found.

8b
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8 marks

Given that the minute hand has travelled a total distance of 31.8 m before it is stopped, find

(i)
the final angle between the hour hand and the minute hand,
(ii)
the final height of the tip of the minute hand above the ground.

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