8Revision of Chapters 1 7

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8 of hapters 7 8 Technolog-free questions State the maimal domain and range of

the function with the rule f = + and sketch both functions on the one set of aes.

c Sketch the graph of = f f for its maimal domain.

5 Simplif log 0 5 + log 0 log 0 0. Find in terms of a if log a = + log a.

9 a The graph of the function f with rule f = log e + intersects the aes at the

1 8 of hapters 7 8 Technolog-free questions State the maimal domain and range of each of the following: a f = + f = b c f = d h = e f = 5 + Find the inverse of the function with the rule f = + and sketch both functions on the one set of aes. Find the inverse of the function with the rule f = +. Let f : R R, f = e. a Find the rule and domain of f. b Sketch the graphs of f and f on the one set of aes. c Sketch the graph of = f f for its maimal domain. d Sketch the graph of = f f for its maimal domain. e Find = f f. 5 Simplif log log 0 log 0 0. Find in terms of a if log a = + log a. 7 Solve = 0. 8 Solve the equation e = 9 for. 9 a The graph of the function f with rule f = log e + intersects the aes at the points a, 0 and 0, b. Find the eact values of a and b. b Hence sketch the graph of = f. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

2 hapter 8: of hapters 7 0 Solve the equation 5 + = 0 for. Solve the equation sin = for [ π, π]. π a State the range and period of the function h: R R, h = 5 cos. b Solve the equation cos + π = for [0, π]. onsider the simultaneous equations m + = + m = Find the values of m such that the sstem of equations has: a a unique solution b no solution c infinitel man solutions. If a graph has rule = a + b and passes through the points, and,, find the values of a and b. 5 Find the values of m for which the equation + m + = 0 has: a one solution b two solutions Two points and have coordinates a, and,. a Find the values of a if: i the midpoint of is 0, ii the length of is iii the gradient of is. c no solution. b Find the equation of the line passing through and if a =, and find the angle the line makes with the positive direction of the -ais. 7 Let f : R R, f =. a State whether the function f is even, odd or neither. b Find the inverse function f. c Find: i f ii f iii { : f = f } 8 Let f = and g =. Find: a f b g c f a d ga e { : f = 0 } f { : g = 0 } g { : f > 0 } 9 Let f = and g = +. a Find: i f g ii g f iii g f b Find a transformation that takes the graph of = g to the graph of = g f. c Find a transformation that takes the graph of = to the graph of = g. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

3 8 Multiple-choice questions 5 0 Solve cos =, giving the general solution. function has rule = e kt. Given that = when t = and that = 0 when t =, find the values of and k. Solve each of the following inequalities for : a + 0 b + > 0 8 Multiple-choice questions The domain of the function whose graph is shown is, ], ] [, ] [,, Which of the following sets of ordered pairs does not represent a function where is the value of the function? {, : =, 0 } {, : =, R \{0} } {, : = +, R } {, : = + 7, R } {, : = e, R } The implied largest possible domain for the function with the rule = is R \{},,, ] R + If f = can be simplified as a, then f a 0 a a a + 5 If f : [0, π] R where f = sin and g: [0, π] R where g = sin, then π the value of f + g is 0 If f = + and g =, then f g equals If f =,0 and g =,, then the domain of f + g is [0, ] [0, ], ] R + {0} [, ] ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

4 hapter 8: of hapters 7 8 The graph shown has the equation, > 0 =, 0, 0 =, < 0, > 0 =, 0 +, > 0 =, 0, > 0 =, 0 9 If g = + and f = +, then the rule of the product function fg equals The implied domain for the function with rule = is [, { : < < } [, ], R + The graph of the function with rule = f is shown. Which one of the following graphs is the graph of the inverse of f? ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

5 8 Multiple-choice questions 7 The domain of the function whose graph is shown is [, 5], 5], 5], 5, The graph shown has the rule, =, <, =, <, =, <, < =,, =, < The inverse, f, of the function f : [, ] R, f = is f : [0, ] R, f = + f : [, ] R, f = + f : [, ] R, f = f : [0, ] R, f = f : [0, ] R, f = + 5 Let f be the function defined b f = +, R. suitable restriction f of f such that f eists would be f :[, ] R, f = f : R R, f = + + f :[, ] R, f = f : [0, R, f = + f :[, R, f = + + ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

6 8 hapter 8: of hapters 7 Let h: [a, ] R where h =. If a is the smallest real value such that h has an inverse function, h, then a equals 0 7 If f =, R, then f equals The solution of the equation = is 8 9 The graph shows + = = + + = 0 = + = If 5 + = 5, then equals The equation of the line that passes through the points, and, 0 is = + = + = = = The straight line with equation = meets the -ais at and the -ais at. 5 If is the origin, the area of the triangle is 5 square units 5 square units 9 5 square units 0 square units 0 square units If the equations = and = are simultaneousl true, then + equals The graphs of the relations 7 = 0 and + = are drawn on the same pair of aes. The -coordinate of the point of intersection is ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

7 8 Multiple-choice questions 9 5 possible equation for the graph shown is = + = + = + = + = The function given b f = has the range + R \ { } R R \{} R \{} R \ { } 7 parabola has its verte at,. possible equation for this parabola is = + + = = + = + = + 8 Which one of the following is an even function of? f = + f = f = f = f = + 9 The graph of = + can be obtained from the graph of = b a translation,, followed b a dilation of factor from the -ais a translation, +, followed b a dilation of factor from the -ais a translation, +, followed b a dilation of factor from the -ais a translation,, followed b a dilation of factor from the -ais a translation, +, followed b a dilation of factor from the -ais 0 function with rule f = + has maimal domain, [,, [, [, possible equation for the graph shown is = +, = + = + = + = +, 0 The range of the function f : R \{} R, f = + is, ], [, [,, ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

8 0 hapter 8: of hapters 7 If + k + = 0 when =, then k equals 0 The quadratic equation with solutions 5 and 7 is + 5 = 0 5 = = 0 5 = = 0 5 If k is divisible b +, then k equals Which one of the following could be the equation of the graph shown? = + = + = + = + = 7 The graph shown is + = + = = + = + + = 8 P = + 5 has the factorisation The graph shown is that of the function f = m +, where m is a constant. The inverse function is f : R R, f = a + b, where a and b are constants. Which one of the following statements is true? a = m, b = m a < 0 and b < 0 a = m, b = a > 0 and b > 0 a = m, b = m f = m + ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

9 8 Multiple-choice questions 0 Let P = + +. When P is divided b, the remainder is 5 If + + a has remainder when divided b +, then a equals 8 Which of these equations is represented b the graph shown? = + = = = + = 0 9 The function f : R R, f = e + has an inverse function f. The domain of f is 0, R [,, [0, The function f : R + R, f = log e + has an inverse function f. The rule for f is given b f = e f = e f = e f = e + f = e 5 Let f : R R where f = e and let g:, R where g = log e +. The function with the rule = f g has the range, 0, 0, ] [, [0, ] The function g: R R, g = e has an inverse whose rule is given b f = e f = log e + f = log e f = log e f = log e + 7 The function f : [, R, f = log e has an inverse. The domain of this inverse is [0, 0, [,, R 8 The function f : R R, f = e has an inverse whose rule is given b f = e log e + log e log e + log e 9 The function f : R + R, f = log e has an inverse function f. The rule for f is given b f = e log e e e log e ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

10 hapter 8: of hapters 7 50 For which values of is the function f with the rule f = + log e defined? [,, [,,, 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? = = = 5 If log 8 + log =, then =.5 ±.5 ±. 5 The equation log 0 = log 0 + is equivalent to the equation = 0 = + 0 = 0 = 0 = The graph indicates that the relationship between N and t is N = e t N = e t 00 N = e t N = e t + N = e t log e N t 55 possible equation for the graph is = e = e = + e = + e = e = ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

11 8 Multiple-choice questions 5 possible equation for the graph is = log e = log e + = log e + = log e + = log e + 57 possible equation for the graph shown is = cos θ + π = cos θ + π = sin θ + π = cos θ + π = cos θ π π π 5π θ 58 The function f : R R where f = cos θ + π has range [, 5] [, 5] R [, 5] [, ] 59 Two values between 0 and π for which sin θ + = 0 are π, π 0, 0 π, 5π π, 5π 7π, π 0 possible equation for the graph shown is = sin π = sin + π = sin π = cos π = cos + π π 7π The function f : R R, f = sin has amplitude and period π amplitude and period π amplitude and period π amplitude and period π amplitude and period π ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

12 hapter 8: of hapters 7 The function f : R R, f = sin has range [0, ] [, ] [, ] [, ] [, 5] onsider the polnomial p = a + a + a where a > 0. The equation p = 0 has eactl distinct real solution distinct real solutions distinct real solutions 5 distinct real solutions distinct real solutions The gradient of a straight line perpendicular to the line shown is 5 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 = a vertical asmptote with equation = a vertical asmptote with equation = a horizontal asmptote with equation = no asmptote The function f : R R, f = a sinb + c, where a, b and c are positive constants, has period π π a b c a b 7 The functions f : [8, ] R, f = and g: R + R, g = log are used to define the composite function g f. The range of g f is [ [,, [5, ] R + R 8 The rule for the inverse relation of the function with rule = + 5 and domain R is = ± + = + 5 = 5 = 5 = ± 9 The function f : R, f = + will have an inverse function for = R =, = [, =, ] = R + 70 Let f : R R, f = and let g: [ 5, R, g = +. Then the domain of the inverse function of h = f + g is [ 5, [ 5, R 0, 5] [ 5, R [0, 5 ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

13 8 tended-response questions 5 8 tended-response questions n arch is constructed as shown. F Z H G = 7 m The height of the arch is 9 metres Z = 9 m. The width of the arch is 0 metres = 0 m. The equation of the curve is of the form = a + b, taking aes as shown. a Find the values of a and b. b man of height.8 m stands at = 7 m. How far above his head is the point on the arch? That is, find the distance. c horizontal bar FG is placed across the arch as shown. The height, H, of the bar above the ground is. m. Find the length of the bar. a The epression + a 7 8 leaves a remainder of 7 when divided b + 5. etermine the value of a. b Solve the equation = c i Given that the epression 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. s a pendulum swings, its horizontal position, cm, measured from the central position, varies from cm at to cm at. The position is given b the rule = sinπt a Sketch the graph of against t for t [0, ]. b Find the horizontal position of the pendulum for: i t = 0 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 and back to. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

14 hapter 8: of hapters 7 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. The vertical position, m, of the P point P on the rope at time t seconds is given b the rule.5 m =.5 cosπt +.5 a Find when:.5 m i t = 0 ii t = iii t = b How long does it take for one rotation of the rope? c Sketch the graph of against t. d Find the first time that the point P on the rope reaches a height of m above the ground. 5 The population of a countr is found to be growing continuousl at an annual rate of.9% after Januar 950. The population t ears after Januar 950 is given b the formula pt = 50 0 e kt a Find the value of k. b Find the population on Januar 950. c Find the population on Januar 000. d fter how man ears would the population be 00 0? large urn was filled with water. It was turned on, and the water was heated until its temperature reached 95. 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. ommencing at p.m. and finishing at midnight, Jenn measured the temperature of the water ever hour on the hour for the net 0 hours and recorded the results. t p.m., Jenn recorded the temperature of the water as 55. She found that the temperature, T, of the water could be described b the equation T = e kt + 5, for 0 t 0 where t is the number of hours after p.m. a Find the values of and k. b Find the temperature of the water at midnight. c t what time did Jenn first record a temperature less than? d Sketch the graph of T against t. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

15 8 tended-response questions 7 7 football is kicked so that it leaves the plaer s foot with a velocit of V m/s. The total horizontal distance travelled b the football, m, is given b = V sinα 0 where α is the angle of projection. a Find the horizontal distance travelled b the ball if V = 5 m/s and α = 5. b For V = 0, sketch the graph of against α for 0 α 90. c If the ball goes 0 m and the initial velocit is 0 m/s, find the angle of projection. 8 The diagram shows a conical glass fibre. The circular cross-sectional area at end is 0.0 mm. The cross-sectional area diminishes b a factor of per metre length of the fibre. The total length is 5 m. a Write down a rule for the cross-sectional area of the fibre at a distance m from. b What is the cross-sectional area of the fibre at a point one-third of its length from? c The fibre is constructed such that the strength increases in the direction to. t a distance of m from, the strength is given b the rule S = 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 of m from. d piece of glass fibre that will have to carr loads of up to units is needed. How much of the 5 m fibre could be used with confidence for this purpose? 9 a The graph is of one complete ccle of πt = h k cos R i How man units long is P? ii press Q and R in terms of h and k. Q P t b For a certain cit in the northern hemisphere, the number of hours of dalight on the st da of each month is given b the table: ec Jan Feb Mar pr Ma Jun Jul ug Sep ct Nov ec Using suitable scales, plot these points and draw a curve through them. all ecember month 0, Januar month, etc., and treat all months as of equal length. c Find the values of h and k so that our graph is approimatel that of πt = h k cos ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

16 8 hapter 8: of hapters 7 0 n an overnight interstate train, an electrical fault affected the illumination in two carriages, and. efore the fault occurred, the illumination in carriage was I units and that in carriage was 0. I units. ver time the train stopped, the illumination in carriage reduced b 7% and that 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? a The curve = a in the figure intersects the -ais at and. Point is the verte of the curve and a is a positive constant. i Find the coordinates of and in terms of a. ii Find the area of triangle in terms of a. b The graph shown has rule = a a + a where a > 0. i Use a calculator to sketch the graph for a =,,. ii Find the values of a for which 7 a + a = 0. iii Find the values of a for which 7 a + a < 0. iv Find the value of a for which 7 a + a =. Pa, a S Q 5a, 7 a + a v Find the value of a for which 7 a + a =. vi Plot the graphs = a a + a for the values of a obtained in iv and v. c 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 500. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

17 8 tended-response questions 9 The deficit of a government department in Ningteak, a small monarch east of frica, is continuall assessed over a period of 8 ears. The following graph shows the deficit over these 8 ears. eficit in millions of Ningteak dollars 0,.8,., Time t The graph is read as follows: The deficit at the beginning of the 8-ear period is $.8 million. t the end of the third ear the deficit is $.5 million, and this is the smallest deficit for the period 0 t 8. a Find the rule for in terms of t, assuming that it is of the form = at + bt + c. b Use this model to predict the deficit at the end of 8 ears. The rate of rainfall, R mm per hour, was recorded during a ver rain da in North Queensland. The recorded data are given in the table. ssume a quadratic rule of the form R = at + bt + c Time Rainfall a.m. 7.5 mm per hour 8 a.m. 9.0 mm per hour 0 a.m. 8.0 mm per hour is applicable for 0 t, where t = 0 is a.m. Use the quadratic model to predict the rate of rainfall at noon. t what time was the rate of rainfall greatest? population of insects is determined b a rule of the form c n = + ae, t 0 bt where n is the number of insects alive at time t das. a onsider the population for c = 5790, a = and b = 0.0. i Find the equation of the horizontal asmptote b considering values of n as t becomes large. ii Find n when t = 0. iii Sketch the graph of the function. iv Find the eact value of t for which n = 000. b i Use our calculator to find values of a, t 0 00 b and c such that the population growth n ields the table on the right. ii Sketch the graph for this population. ambridge Senior Maths /V ISN vans et al. 0 ambridge Universit Press Photocoping is restricted under law and this material must not be transferred to another part.

SAMPLE. Revision. Revision of Chapters 1 7. The implied (largest possible) domain for the function with the rule y = is: 2 x. 1 a

SAMPLE. Revision. Revision of Chapters 1 7. The implied (largest possible) domain for the function with the rule y = is: 2 x. 1 a 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