Mathematics. Guide

Transcription

1 Mathematics Guide - Contents Question Item Objective Type Skill 05 ALG.0 Multiple-choice answer Concepts 009 ALG.0.05 Multiple-choice answer Concepts 07 ALG.0.05 Multiple-choice answer Concepts 068 ALG.0.0 Multiple-choice answer Concepts 5 06 ALG.0.0 Multiple-choice answer Concepts ALG.0.05 Multiple-choice answer Applications ALG.0.0 Short-constructed answer Concepts 8 08 ALG.0.05 Short-constructed answer Concepts ALG.0.05 Short-constructed answer Applications ALG.0.0 Extended answer Applications 0566 ALG.0.0 Extended answer Applications 0696 ALG.0 Extended answer Problem solving 0 ALG.0 Extended answer Problem solving - Correction key B

2 B D D 5 D 6 D 7 a) 50 b) c) d) The rule is f(t) 5 sin 9 000t or f(t) 5 sin t. Note : Since f then P f and B or P A 5 Therefore, the rule is f(t) 5 sin 9 000t or f(t) 5 sin t. 9 t The rule of the sinusoidal function is f(t) sin +. 0

3 0 Example of an appropriate method By letting f(x) 0, you obtain -sin x 0 sin x x - sin x or 7 x over [0, [ Over [0, [, there are also 5 + and Answer The zeros of this function over [0, [ are 5 7 5,, and. Example of an appropriate method t m(t) 5 sin t 00 5 sin t 0.8 sin sin (0.8) θ θ t 0.97 t.5 Answer: θ. t. t 8.6 During the period covered by the study, the mass of the humpback whales was exactly 00 tonnes at.5 months and 8.6 months.

4 Example of an appropriate solution Let t: time, in seconds, that has past since :00 f(t): height of the jet, in metres The rule of correspondence f(t) a sin b(t h) + k a p 60 b b 5 b b 60 0 f(t) (m) t(s) f 0 ( x) sin ( t h + k) Translation (h, k) (5, ) f(t) sin (t 5) + 0 Height at h min 0 s Since the function has a period of one minute, the jet will be at the same height in 0 seconds. f ( 0) sin ( 0 5) Answer: At hours minutes and 0 seconds, the water jet will be at a height of m.

5 Example of an appropriate solution Rule of distance of head to the ground Frequency Period 6 seconds/cycle Amplitude 0 P b y a cos b 0 cycles 60 seconds ( x h) max + min k b 6 + k y 0 cos t + 80 Distance to ground at t 5 y 0 cos y 60 cm ( 5) + 80 Answer: Five seconds after it is released, the head is at the height of 60 cm. Note: To account for rounding at different places, accept answers in the range of 59 to 6.

6 Name : Group : Date : Mathematics Question Booklet If x,,, which one of the following statements is TRUE? A) sin x > 0 and cos x > 0 C) sin x < 0 and cos x > 0 B) sin x > 0 and cos x < 0 D) sin x < 0 and cos x < 0 A sailboat on the sea follows the regular movement of the waves. Its vertical height h(t) can be expressed as a function of time t. This is represented by the graph below. h(t) - t Which rule defines this function? A) h(t) sin 8(t + ) C) h(t) sin (t ) B) h(t) sin (t + ) D) h(t) sin 6(t + )

7 Given the graph. h(t) t - 0 Which of the following rules can be applied to this graph? A) h ( t) cos t + C) h ( t) - cos t + B) h ( t) - cos t D) h ( t) cos t What is the period of the function defined by f(x) sin x? A) C) B) D)

8 5 Find the range and the period of the function defined by f(x) sin x. A) [-, ] and C) [-, ] and B) [-, ] and D) [-, ] and 6 Jamie is practicing for a skateboard competition at the neighbourhood park. The ramp is in the shape of a sinusoidal function. The following graph represents the height, f(x), of the ramp, in metres, as a function of the horizontal distance, x, in metres. The maximum and minimum points of the ramp are separated by 6 metres horizontally and.5 metres vertically. The minimum is 0.5 metres above ground level. f (x) x Which of the following rules represents the above situation? A) f(x) cos x C) f(x).75 cos x B) f(x) cos x +.5 D) f(x).75 cos x 6 +.5

9 7 An oscillatory movement is expressed by the equation f(t) 60 sin(500t 00). Find a) its frequency b) its period c) its phase shift d) its amplitude 8 Radio station CKOI-FM broadcasts through a frequency of 97 KHz, or cycles/s. The radio's volume is set at 5, thus determining the sound amplitude. Write a rule in the form f(t) A sin Bt of the sinusoidal curve representing the sound waves transmitted. 9 The screen of the oscilloscope below illustrates a sinusoidal function f, representing the amplitude of the vibration of a guitar string as a function of time t, in seconds. f ( t ) (5, 7) (0, ) (5, ) t What is the rule of this sinusoidal function?

10 0 Function f is defined by the rule f(x) -sin x. Determine the zeros of this function over [0, [. For two years, oceanographers have compiled data on the mass of humpback whales. They noted that the whale's mass varies according to the following sinusoidal function: t m(t) 5 sin + 80 where t [0, [ where t is the number of months gone by since the beginning of the study, and m(t) represents the mass of the whales in tonnes. When did the mass of the humpback whales reach exactly 00 tonnes during the period covered by the study? A fountain in a shopping centre has a single jet of water. The height of the jet of water varies according to a sinusoidal function. Joel notes that, in exactly one minute, the jet goes from a minimum height of m to a maximum height of 5 m and back to m. At :00, the jet of water is at a height of m. What will be the height of the jet of water, to the nearest tenth of a metre, when the clock reads ::0? ( hours, minutes, 0 seconds) The diagram below depicts the head of a Jack-in-a-box used in the display window of a department store. The head is connected to a motor, and its up-and-down movement follows a sinusoidal curve. The head is compressed to 0 cm at t 0 and it reaches a maximum height of 0 cm. It bounces with a frequency of 0 cycles per minute. y 0 cm 0 cm At what height is the head, 5 seconds after it is released? x