C H P T R 8 of Chapters 7 8. Multiple-choice questions The domain of the function whose graph is shown is: 3, ] B, 3] C [, 3] D [, 3), 3) 3 3 Which of the following sets of ordered pairs does not represent a function where is the value of the function?, ): =, } B, ): =, R\}} C, ): = 3 + 3, R} D, ): = 3 + 7, R}, ): = e, R} 3 The implied largest possible) domain for the function with the rule = is: R\} B, ) C, ) D, ] R + If f ) = then f ),insimplified form, is equal to: a a B C D a a a + 5 The graph shown has the equation:, >, = B =,, < Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard 8
Chapter 8 of Chapters 7 83, > C =, +, > D =,, > =, If f : [, ] R where f ) = sin and g:[, ] R where g) = sin, then ) 3 the value of f + g) is: B C D 7 If f ) = 3 + and g) =, then f g3)) equals: 3 B C 5 D 9 8 If f ) = 3, and g) =,, the domain of f + g is: [, ] B [, ] C, ] D R + } [, ] 9 If g) = + and f ) = 3 +, then the rule of the product function fg) equals: + 3 + 3 B 3 + + 3 + C 3 + 3 D 3 + + 3 3 + The domain of the function whose graph is shown is: [, 5] B, 5] C, 5] D, 5), 5) The implied domain for the function with equation = is: 5 3 3 5 [, ) B : < < } C [, ] D, ] R + The graph of the function with rule = f ) isshown. cont d.) Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
8 ssential Mathematical Methods 3&CS Which one of the following graphs is the graph of the inverse of f? D 3 The graph shown has the rule: ), = 3, <, B = 3, < ), C = 3, < ), < D = 3, ), = 3, < B C 3 The inverse, f,ofthe function f : [, 3] R, f ) = is: f : [, ] R, f ) = + B f : [3, ] R, f ) = + C f : [, 3] R, f ) = D f : [, ] R, f ) = f : [, ] R, f ) = + 5 f is the function defined b f ) = +, R. suitable restriction for f, f such that f eists, would be: f : [, ] R, f ) = + B f : R R, f ) = + Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 85 C f : [, ] R, f ) = + f : [, ) R, f ) = + D f :[, ) R, f ) = + Let h:[a, ] R where h) =.Ifais the smallest real value such that h has an inverse function, h, then a equals: B C D 7 If f ) = 3, R, then f ) equals: B 3 + C 3 3 ) D 3 + + ) 3 8 The solution of the equation = 3 is: B C 8 D 9 The graph shows: + = B = C + + = D = + = ) If + = 5, then equals: 3 5 B 7 C D 3 5 5 The equation of the line that passes through the points, 3) and, ) is: = + B = C + = D = = If the angle between the lines = 8 + and 3 = is, then tan is best approimated b:.7 B. C D.8 3 The line with equation = meets the -ais at and the -ais at B. IfO is the 5 origin, the area of the triangle OB is: 3 5 square units B 9 square units 5 C square units D 5 square units square units Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
8 ssential Mathematical Methods 3&CS If the equations 3 = and 3 = 3 are simultaneousl true, then + equals: 5 B C D 5 5 If the graphs of the relations 7 = and 3 + = are drawn on the same pair of aes, the -coordinate of the point of intersection is: B C D 3 possible equation for the graph shown is: 3 = B + 3 = + 3 C 3 = + D = 3 + = 3 7 The function given b f ) = has the range given b: + 3 R\ } B R C R\3} D R\} R\ 3} 8 parabola has its verte at, 3). possible equation for this parabola is: = + ) + 3 B = ) 3 C = + ) 3 D = ) + 3 = 3 + ) 9 Which one of the following is an even function of? f ) = 3 + B f ) = 3 C f ) = ) D f ) = f ) = 3 + 3 The graph of = 3 + can be obtained from the graph of = b: B C D a translation, ), ) followed b a dilation of factor 3 from the -ais a translation, ) +, ) followed b a dilation of factor from the -ais 3 a translation, ) + 3, ) followed b a dilation of factor 3 from the -ais a translation, ), ) followed b a dilation of factor 3 from the -ais a translation, ) +, ) followed b a dilation of factor 3 from the -ais 3 function with rule f ) = 3 + has maimal domain:, ) B [, ) C, ) D [, ) [, ) Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 87 3 possible equation of the graph shown is: = 3 + B = 3 + C = 3 + D = 3 + = 3 + 3, ), ) 3 33 The range of the function f : R\} R, f ) = ) + is: 3, ] B, ) C [3, ) D [, ), ) 3 If 3 + k + = when =, then k equals: B C D 35 The quadratic equation whose roots are 5 and 7 is: + 35 = B 35 = C + 35 = D 35 = + + 35 = 3 If 3 5 + + k is divisible b +, then k equals: 7 B 5 C D 5 7 37 Which one of the following could be the equation of the graph shown? = ) + ) B = + ) ) C = + ) ) D = ) + ) = ) 38 The graph shown is: + = + ) 3 B = ) 3 C = 3 + D = + )3 + = ) 3 39 P) = 3 + 5 has the factors: ) ) + 3) B + ) + ) + 3) C + ) ) + 3) D + ) ) 3) ) ) 3) If P) = 3 + 3 +, when P) isdivided b ) the remainder is: 3 B 5 C D 9 Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
88 ssential Mathematical Methods 3&CS The graph shown is that of the function f ) = m + 3, where m is a constant. The inverse f is defined as f) = m + 3 f : R R, f ) = a + b,where 3 a and b are constants. Which one of the following statements is true? a = 3 m, b = m B a < and b < C a = m, b = 3 D a > and b > a = m, b = 3 m If 3 + + a has a remainder when divided b +, then a equals: 8 B C D 3 Which of these equations is represented b the graph shown? = + ) ) B = C = ) D = + ) ) = The function f : R R where f ) = e + has an inverse function f. The domain of f is:, ) B R C [, ) D, ) [, ) 5 The function f : R + R where f ) = log e + has an inverse function f. The rule for f is given b: f ) = e B f ) = e ) C f ) = e D f ) = e + f ) = e Let f : R R where f ) = e and g:, ) R where g) = log e + ). The function with the rule = f g)) has the range:, ) B, ) C, ] D [, ) [, ] 7 The function g: R R where g) = e has an inverse whose rule is given b: f ) = B f ) = log e e + ) C f ) = log e ) D f ) = log e ) f ) = log e + ) 8 The function f :[, ) R where f ) = log e 3) has an inverse. The domain of this inverse is: [, ) B, ) C [, ) D 3, ) R Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 89 9 The function f : R R where f ) = e has an inverse whose rule is given b f ) = e ) B log e C + log e D log e + ) log e ) 5 The function f : R + R where f ) = log e has an inverse function f. The rule for f is given b f ) = e ) B log e C e D e log e 5 Forwhat values of is the function f with the rule f ) = + log e 3 ) defined? ) [ ), ) B 3, C [, ) D 3,, ) 5 The graphs of the function f :, ) R where f ) = + log e + ) and its inverse f are best shown b which one of the following? D = 53 For log 8 + log =, = B = C =.5 B ±.5 C D ±. 5 The equation log = log 3) + isequivalent to the equation: = 3 ) B = 3 C = 3 + D = 3 + = 3 55 The graph indicates that the relationship between N and t is: N = e t B N = e t C N = e t + D N = e t N = e t log e N t Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
9 ssential Mathematical Methods 3&CS 5 possible equation for the graph is: = e B = e C = + e D = + e = e 57 possible equation for the graph is: = log e ) B = log e + ) C = log e + ) D = log e + ) = log e + ) 58 possible equation for the graph shown is: = cos 3 + ) B = cos + ) C = sin 3 + ) π π 5π D = cos 3 + ) = cos 3 ) 59 The function f : R R where f ) = 3 cos + ) has range: [ 3, 5] B [, 5] C R D [, 5] [ 3, ] Twovalues between and for which sin + 3 = are: 3, B, C 3 3, 5 D 3 3, 5 3 possible equation for the graph shown is: = sin ) B = sin + ) C = sin ) D = cos ) = cos + ) π 7, 7π = θ Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 9 The function f : R R, f ) = 3 sin has: amplitude 3 and period B amplitude and period C amplitude and period D amplitude 3 and period amplitude and period 3 The function f : R R, where f ) = 3 sin has range: [, 3] B [, ] C [, 3] D [ 3, 3] [, 5] Consider the polnomial p) = a) + a) + a) where a >. The equation p) = has eactl: distinct real solution B distinct real solutions C 3 distinct real solutions D distinct real solutions 5 distinct real solutions 5 The gradient of a straight line perpendicular to the line shown is: B C D 3 The graph of a function f whose rule is = f ) has eactl one asmptote for which the equation is =. The inverse function f eists. The inverse function will have: a horizontal asmptote with equation = B avertical asmptote with equation = C avertical asmptote with equation = D a horizontal asmptote with equation = no asmptote 7 The function f : R R, f ) = a sinb) + c where a, b and c are positive constants has period: a B b C c D a b 8 The functions f : [8, 3] R, f ) = and g: R + R, g) = log are used to define the composite [ function ) g f. The range of g f is: 3 [, ) B, C [5, ] D R + R 9 The rule for the inverse relation of the function with rule = + 5 and domain R is: = ± + B = + 5 D = 5 = 5 C = ± 7 The range of the function with rule = 3 sin +3 is: [, 3] B [, ] C [ 3, 3] D [, ] [ 3, ] Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
9 ssential Mathematical Methods 3&CS 8. tended-response questions n arch is constructed as shown. D C F 7 m Z H O G B O = OB The height of the arch is 9 metres OZ = 9 m). The width of the arch is metres B = m). The equation of the curve is of the form = a + b, taking aes as shown. a Find values of a and b. b man of height.8 m stands at C OC = 7 m). How far above his head is the point on the arch? That is, find the distance D.) c horizontal bar FG is placed across the arch as shown. The height, OH, ofthe bar above the ground is.3 m. Find the length of the bar. a The epression 3 + a 7 8 leaves a remainder of 7 when divided b + 5. Determine the value of a. b Solve the equation: 3 = + 5 + c i Given that the epression 5 + 7 leaves the same remainder whether divided b b or c,where b c, show that b + c = 5 ii Given further that bc = and b > c, find the values of b and c. 3 s a pendulum swings, its horizontal position, measured from the central position, varies from cmat) tocmatb). is given b the rule: = sin t a Sketch the graph of against t for t [, ]. b Find the horizontal position of the pendulum for: i t = ii t = iii t = c Find the first time that the pendulum has horizontal position =. d Find the period of the pendulum, i.e. the time it takes to go from to B and back to. B Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 93 Two people are rotating a skipping rope. The rope is held.5 m above the ground.it reaches a height of.5 m above the ground, and just touches the ground..5 m.5 m The vertical position, m, of the point P on the rope at time t seconds is given b the rule: =.5 cos t) +.5 a Find when: i t = ii t = iii t = b How long does it take for one revolution of the rope? c Sketch the graph of against t. d Find the first time that the point P on the rope is. metres above the ground. 5 The population of a countr is found to be growing continuousl at an annual rate of.9% after Januar 95. The population t ears after Januar 95 is given b the formula: pt) = 5 e kt a Find the value of k. b Find the population on Januar 95. c Find the epected population on Januar. d fter how man ears would the population be 3? football is kicked so that it leaves the plaer s foot with a velocit of V m/s. The horizontal distance travelled b the football after being kicked is given b the formula: = V sin where is the angle of projection. a Find the distance the ball is kicked if V = 5 m/s and = 5. b For V =, sketch the graph of against for 9. c If the ball goes 3 m and the initial velocit is m/s, find the angle of projection. 7 large urn was filled with water. It was turned on, and the water was heated until its temperature reached 95 C. This occurred at eactl : p.m., at which time the urn was turned off and the water began to cool. The temperature of the room where the urn was located remained constant at 5 C. Commencing at : p.m. and finishing at midnight, Jenn measured the temperature of the water ever hour on the hour for the net hours and recorded the results. t : p.m. Jenn recorded the temperature of the water to be 55 C. She found that the temperature T degrees Celsius) of the water could be described b the equation: P T = e kt + 5, where t where t is the number of hours after : p.m. Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
9 ssential Mathematical Methods 3&CS a Find the values of and k. b Find the temperature at midnight. c t what time did Jenn first record a temperature less than C? d Sketch the graph of T against t. 8 On an overnight interstate train an electrical fault meant that the illumination in two carriages, and B,was affected. Before the fault occurred the illumination in carriage was I units and.i units in carriage B. ver time the train stopped the illumination in carriage reduced b 7% and b % in carriage B. a Write down eponential epressions for the epected illumination in each carriage after the train had stopped for the nth time. b t some time after the fault occurred the illumination in both carriages was approimatel the same. t how man stations did the train stop before this occurred? 9 The diagram shows a conical glass fibre. The circular cross-sectional area at end B is. mm. The cross-sectional area diminishes b afactor of.9) per metre length of the fibre. The total length is 5 m. B a Write down a rule for the cross-sectional area of the fibre at a distance m from B. b What is the cross-sectional area of the fibre at a point one-third of its length from B? c The fibre is constructed in such a wa that the strength increases in the direction B to.tadistance m from B the strength is given b the rule S =.9) 3. If the load the fibre will take at each point before breaking is given b load = strength cross-sectional area, write down an epression, in terms of, for the load the fibre will stand at a distance m from B. d piece of glass fibre that will have to carr loads of up to..9).5 units is needed. How much of the5mfibre could be used with confidence for this purpose? pizza is divided b a number of straight cuts as shown. The table shows the largest number of pieces f n) into which it is divided b n cuts. n 3 5 f n) 7 a b c Find a quadratic model for this data. Use our model to find the greatest number of pizza pieces produced b: i straight cuts ii 5 straight cuts Check our answers to b b drawing diagrams. Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard
Chapter 8 of Chapters 7 95 a The graph is of one complete ccle of: ) t = h k cos i How man units long is OP? ii press OQ, OR in terms of h and k. b c The number of hours of dalight on the st of each month in a cit in the northern hemisphere is given b the table: Dec Jan Feb Mar pr Ma Jun Jul ug Sep Oct Nov Dec 7.5 8. 9.9.. 5.8.5 5.9.3. 9.8 8. 7.5 Using suitable scales, plot these points and draw a curve through them. Call December month, Januar month, etc., and treat all months as of equal length. Find the values of h and k so that ) our graph is approimatel that of: t = h k cos a In the figure = a 3) intersects the -ais at and B.Point C is the verte of the curve and a is a positive constant. i Find the coordinates of and B in terms of a. ii Find the area of triangle BC in terms of a. b The graph shown has rule: = a) a) + a where a > R Q O Pa, a) i Use a calculator to sketch the graph for a =,, 3 S 5a Q, 3 7 a3 + a ii Find the values of a for which 7 a3 + a = iii Find the values of a for which 7 a3 + a < iv Find the value of a for which 7 a3 + a = v Find the value of a for which 7 a3 + a = vi Plot the graphs = a) a) + a for the values of a obtained in iv and v. Triangle PSQ is a right-angled triangle. i Give the coordinates of S. ii Find the length of PS and SQ in terms of a. iii Give the area of triangle PSQ in terms of a. iv Find the value of a for which the area of the triangle is. v Find the value of a for which the area of the triangle is 5. c C P B t Cambridge Universit Press Uncorrected Sample Pages 8 vans, Lipson, Jones, ver, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard