SAMPLE. Revision. Extended-response questions revision. B x cm

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1 C H A P T E R 0 Etended-response questions revision This chapter contains etended-response questions. The can be attempted after the completion of the course or particular questions ma be chosen after the appropriate chapters have been completed. 1 a i Find the coordinates of the stationar point for the curve with equation: = ii Determine the nature of this stationar point. b In the triangle AC, the magnitude of AC = 90, A = 5cmand AC = 13 cm. The rectangle PQR A is such that its vertices P, Q and R lie on C, CA and A R Q respectivel. cm cm 60 5 i Given that P = cm and PQ = cm, show that = 1 ii Find the area of the rectangle, A cm,interms of. iii Find the maimum value of this area as varies. A theoretical model of the relationship between two variables and predicts the values in the following table: a An equation of the form = k( p)( q) is suggested, where k, p and q are constants and p < q. Use the information in the table to find p, q and k. 67 P C

2 Chapter 0 Etended-response questions revision 673 b A series of eperiments is carried out to test this model. The values of when = 0, 1 and 3 are found to be as predicted, but when = the value of is found to be. After further discussion a new model is proposed with an equation of the form = m( p) ( q) where p and q have the values alread calculated and m is a constant. i Find the value of m. ii Obtain the equation of this new model in the form, = a 3 + b + c + d iii Sketch the graph of against. State the coordinates of the stationar points and the nature of each of these points. 3 A curve C has equation = a,where a is a positive constant. a Sketch C, showing clearl the coordinate of the points of intersection with the aes. b Calculate the area of the finite region bounded b C and the -ais, giving our answer in terms of a. c The lines with equations = 1 3 a and = a intersect C at the points A and 3 respectivel. i Find, in terms of a, the -coordinates of A and. ii Calculate the area of the finite region bounded b C and the straight line A, giving our answer in terms of a. 4 a Find the equation of the straight line joining the points A(0, 1.5) and (3, 0). b Let = sin + cos A(0, 1.5) i Find d d. Q ii Solve the equation d θ d = 0 for, (3, 0) iii State the coordinates of the stationar point of = sin + cos for iv It can be shown that sin + cos can be written in the form r sin ( + ). Use the result of iii and the fact that when = 0, = tofind the values of r and. v Use addition of ordinates and the result of iv to sketch the graph of against for c The figure shows a map of a region of wetland. The units of the coordinates are kilometres, and the -ais points due north. A walker leaves her car somewhere on the straight road between A and. She walks in a straight line for distance of km to a monument at the origin O. While she is looking at it fog comes down, so that she cannot see her wa back to her car. She needs to work out the bearing on which she should walk. i Write down the coordinates of a point Q which is km from O on a bearing of. ii Show that for Q to be on the road between A and, must satisf the equation: sin + 4 cos = 3 iii Use the result of bivto solve the equation for, where is between 0 and 90. km

3 674 Essential Mathematical Methods 3&4CAS 5 The number of people unemploed in a particular population can be modelled b the function f (t) = 1000(t 10t + 44)e t 10 where t is the number of months after Januar 1997 and 0 t 35. a Use this function to find epression for: i the rate of increase of the number unemploed ii the rate of increase of this rate of increase b Find the values of t for which: i the number unemploed was increasing ii the rate of increase of the number unemploed was going down iii the number unemploed was increasing and the rate of increase of the number unemploed was going down 6 A square piece of card OAC,ofside 10 cm, 10 cm C is cut into four pieces b removing a square OXYZ of side cm, as shown, and then cutting out the triangle AY. 10 cm a i Find A cm, the sum of the areas OXYZ and Y AY,interms of. Z ii Find the domain of the function which cm determines this area. O cm X A iii Sketch the graph of the function, with domain determined in ii. iv State the minimum value of this area. b i Find the rule for the function of which represents the area of triangle AXY. ii Sketch the graph of this function for a suitable domain. c Find the ratio of the areas of the four pieces when the area of triangle AXY is a maimum. 7 The graph of = f ()isshown. a Sketch the graph of: i = f () ii = f () iii = f ( ) iv = f () v = f ( + ) b Eplain wh f does not have an inverse function. c i Sketch the graph of g:(, ) R, g() = f () 1 ii Sketch the graph of g 1. d i Given that g() = ( ) and g:(, ) R, calculate the gradient of the graph of = g () at the point (3, 9). ii Hence find the gradient of = g 1 () atthe point (9, 3).

4 Chapter 0 Etended-response questions revision The diagram shows part of the graph of = cos and the graphs of two quadratic functions, denoted b Q and R,which approimate to the cosine function around = 0 and = respectivel. It is known that the equation of Q is = 1 1 a i Find an estimate of cos 0.1 b using the approimation = 1 1 ii Find an approimation for the solution of the equation cos = 0.98, b solving the quadratic equation 1 1 ( = 0.98 ) b i The graph Q can be transformed into R b areflection in the -ais, followed b a translation. Use this fact to find an equation for the graph R. ii Estimate the value of cos 3 using this approimation. 9 In the figure, ACD is a rectangle A with A = 30 cm, AD = 10 cm. The portions shaded are cut awa, leaving the parallelogram S PQRS Q = SD = cm and cm R = PD = 3 cm. D 3 cm P a Find the area, S cm,ofthe parallelogram in terms of. b Find the allowable values of. c Find the value of for which S is a maimum. d Sketch the graph of S against for a suitable domain. 10 In the figure, OA is a quadrant of a circle of radius Q 1 unit. OA is produced to a varing point P. From P a tangent to the quadrant is drawn touching it at T and meeting another tangent Q at Q. Let OPQ =. a i Find the length OP as a function of. ii Find the length Q as a function of. cos b Show that the area S of trapezium OPQ is sin. c Show that ds 4 cos = d 4 sin d Find the minimum value of S and the distance AP when S is a minimum. π Q π π R 3π R 3 cm θ O A P 11 Adogis at A on the edge of a certain circular lake of diameter a metres and she wishes to reach her owner whoisatthe diametricall opposite point. The dog can swim at 1 m/s and run at 1 m/s. A θ T O a metres Q C cm

5 676 Essential Mathematical Methods 3&4CAS a b c If she swims in a direction making an angle of with A and then runs round the edge of the lake to, show that the time taken, T s, is given b: T = a( + cos ) On the one set of aes sketch the graphs: = 00 and = 400 cos for 0 Using addition of ordinates sketch the graph of: = 00( + cos ) (Find the maimum value of for 0 d b finding and solving the equation d d d = 0.) Sketch the graph of T = a( + cos ) for 0 and state the minimum value of T. 1 a i Show that if f () = ( 1)g() and f () = ( 1)h()where g() and h() are polnomials, then ( 1) must be a factor of g(). ii Let F() = 3 k (3 k) (k ) Show that F(1) = F (1) = 0 iii Using the results of i and ii solve the equation F() = 0 b The parabola with equation = a + b + c and the cubic = 3 touch at P(1, 1) (and have the same gradient at this point). The curves also = 3 meet at Q. i Find b and c in terms of a. P ii If the coordinates of Q are (h, k), find h (1, 1) in terms of a (use the result of a). = a + b + c iii If Q has coordinates (, 8) find the values of a, b and c. Q iv If Q has coordinates ( 3, 7) find the values of a, b and c. 13 P is the point with coordinates (t, 0), where 0 < t < 7. The line PA is parallel to the -ais. a Let Z be the length of A.Find Z in terms of t. b Sketch the graph of Z against t. c State the maimum value of Z and the value of t for which it occurs. 0 t A P = (4 ) = 14 A stud of the numbers of male and female children in families in a certain population is being carried out. a A simple model is that each child in an famil is equall likel to be male or female, and the se of each child is independent of the se of an previous children in the

6 Chapter 0 Etended-response questions revision 677 b famil. Using this model calculate the probabilit that, in a randoml chosen famil of four children: i there will be two males and two females ii there will be eactl one female given that there is at least one female An alternative model is that the first child in an famil is equall likel to be male or female, but that, for an subsequent children, the probabilit that the will be of the same se as the previous child is 3. Using this model calculate the probabilit that in a 5 randoml chosen famil of four children: i all four will be of the same se ii no two consecutive children will be of the same se iii there will be two males and two females 15 In the figure ACD is a rectangle. OA = OD = a, A = b The equation of the parabola OC is = k a Epress k in terms of a and b. b If D cuts the parabola at T, find: T i the equation of the straight line D D O A ii the coordinates of T c Show that the area bounded b the parabola and the line C is 4 ab square units. 3 d Let S 1 be the area of the region bounded b the line segment T and the curve OT. Let S be the area of the region bounded b curve CT and the line segments C and T. Find the ratio S 1 : S. 16 A certain tpe of brass washer is manufactured as follows. A length of brass rod is cut, cross-sectionall, into pieces of mean thickness 0.5 cm and with standard deviation These brass slices are then put through a machine that punches out a circular hole of mean diameter 0.5 cm through the middle of the slice, with a standard deviation of 0.05 cm. The thickness of the washers and the diameters of the holes are known to be normall distributed and do not depend on each other. a Find the probabilit that a washer, selected at random, will: i have a thickness of less than 0.53 cm ii have a thickness of less than 0.47 cm iii have a hole punched with a diameter greater than 0.56 iv have a hole punched with a diameter less than 0.44 cm b If the brass washers are acceptable onl if the are between 0.47 cm and 0.53 cm in thickness with a hole of diameter between 0.44 cm and 0.56 cm, find: i the percentage of washers that are rejected ii the epected number of washers of acceptable thickness in a batch of 1000 washers iii the epected number of washers of acceptable thickness that will be rejected in a batch of 1000 washers C

7 678 Essential Mathematical Methods 3&4CAS 17 A ditch is to be dug to connect the D 90 m points A and in the figure. The earth on the same side of AE as is hard and on the opposite side A is soft. The cost of digging hard earth is $00 per metre and soft earth is $100 per metre. Where should C be chosen? (C is the point where a turn is made.) 18 The diagram shows a sketch of the graph of = e. The points A and have coordinates (n, 0)and (n + 1, 0) respectivel, and the points C and D on the curve are such that AD and C are parallel to the -ais. 1 O D A C 4 m E C = e a i Find the equation of the tangent to = e at the point D. ii Find the intercept of the tangent with the -ais. b i Find the area of the region ACD. ii The line D divides the region into two parts. Find the ratio of the two areas of these parts. 19 A closed capsule is to be constructed as shown in the diagram. It consists of a circular clinder of height h cm which has flat base of radius r cm. It is surmounted b a hemispherical cap. h cm a i Show that the volume, V cm 3,ofthe capsule is given b V = r (3h + r) 3 r cm ii Show that the surface area of the capsule, S cm,is given b S = r(h + 3r) b i If V = a 3,where a is a positive constant, find h in terms of a and r. ii Hence find S in terms of a and r. c i using addition of ordinates sketch the graph of S against r for a suitable domain. ii Find the coordinates of the turning point b finding ds and solving the equation dr ds = 0 for r and determining the corresponding value of S. dr 0 A manufacturer sells clinders whose diameters are normall distributed with mean 3 cm and standard deviation 0.00 cm. The selling price is $s per clinder and the cost of manufacture of each clinder is $1. A clinder is returned and the purchase mone is refunded if the diameter of the clinder is found to differ b more than d cm. A returned clinder is regarded as a total loss to the manufacturer. The probabilit that a clinder is returned is 0.5.

8 Chapter 0 Etended-response questions revision 679 a Find d. b The profit, Q, per clinder is a random variable. Give the possible values of Q in terms of s, and the probabilities of these values. c Epress the mean and standard deviation of Q in terms of s. 1 The length of a certain species of worm has a normal distribution with mean 0 cm and standard deviation 1.5 cm. a Find the probabilit that a randoml selected worm has a length greater than cm. b If the lengths of the worms are measured to the nearest centimetre, find the probabilit that a randoml selected worm has a length of 0 cm. c If five worms are randoml selected, find the probabilit that eactl two will have their lengths measured as 0 cm (measurement to the nearest centimetre). The number of tonnes of coal, P, produced b miners per shift is given b the rule: P = (56 ) where a Find dp d. b i Sketch the graph of P against for ii State the maimum value of P. c Write down an epression in terms of for the average production per man in the shift. Denote the average production per man b A (tonnes). i Sketch the graph of A against for ii State the maimum value of A and the value of for which it occurs. 3 Consider the famil of quadratic functions with rule f () = (k + ) + (6k 4) + where k is an arbitrar constant. a Sketch the graph of f when: i k = 0 ii k = iii k = 4 b Find the coordinates of the turning point of the graph of = f () interms of k. Ifthe coordinates of the turning point are (a, b) find: i {k: a > 0} ii {k: a = 0} iii {k: b > 0} iv {k: b < 0} c Forwhat values of k is the turning point a local maimum? d using the discriminant state the values of k for which: i f ()isaperfect square ii there are no solutions for the equation f () = 0 4 a Find the solution to the equation e = e b For = e e : i Find d. d ii d Solve the equation d = 0 (cont d.)

9 680 Essential Mathematical Methods 3&4CAS c iii State the coordinates of the turning points of = e e iv Sketch the graph of = e e for 0. State the set of values of k for which the equation e e = k has two distinct positive solutions. 5 a Sketch, on a single clear diagram, the graph of: i = ii = ( + a) iii = b( + a) iv = b( + a) + c where a, b, and c are positive constants with b > 1. b Show that = 3 ( + 1) + for all values ecept = 1. c Hence state precisel a sequence of transformations b which the graph of = ma be obtained from the graph of = d Evaluate d. e Sketch the graphs of = 1 and = 3 + onthe one set of aes and indicate ( + 1) the region for which the area has been determined in d. 6 A real estate agent has a block of land to sell. An coordinate grid is placed with the origin at O, asshown in the diagram. A(0, 50) (5, 50) The block of land is OACE. OA, A, CE and EO are straight line segments. 5 C(50, 5) Points and C lie on a parabola with equation of the form = a c O E(5, 0) 50 a Find the equation of line segments: i A ii EC b Find the values of a and c and hence the equation of the parabola through points and C. c Find the area of: i rectangle OEA ii region EC (with boundaries as defined above) iii the block of land 7 In the diagram PQRST is a thin metal plate. PQST is a rectangle with PQ = cmand QRS is an isosceles triangle with QR = RS = 4 cm. a Show that the area of the metal plate A cm is given b: A = 16(cos + cos sin ), 0 < < Q cm P R 4 cm 4 cm θ θ S cm T

10 Chapter 0 Etended-response questions revision 681 b c d Show that da d = 16(1 sin sin ) Solve the equation da d = 0 for 0 < < b first solving the equation 16(1 a a ) = 0 for a. Hence sketch that graph of A against for 0 < < and state the maimum value of A. 8 The length of an engine part must be between 4.81 cm and 5.0 cm. In mass production it is found that 0.8% are too short and 3% are too long. Assume these lengths are normall distributed. a Find the mean and standard deviation of this distribution. b Each part costs $4.00 to produce; those that turn out too long are shortened at an etra cost of $.00 and those that turn out to be too short have to be rejected. Find the epected total cost of producing 100 parts that meet the specifications. 9 The temperature, T C, of water in a kettle at time t (minutes) is given b the formula T = + Ae kt where is the temperature of the room in which the kettle sits. a At :3 p.m., the water in a kettle boils at 100 Cinaroom of constant temperature 1 C. After 10 minutes the temperature of the water in the kettle is 84 C. Use this information to find the values of k and A, giving our answer correct to two decimal places. b At what time will the temperature of the water in the kettle be 70 C? c Sketch the graph of T against t for t > 0. d Find the average rate of change of temperature for the interval of time [0, 10]. e Find the instantaneous rate of change of temperature when: i t = 6 ii T = Large batches of similar components are delivered to a compan. A sample of five articles is taken at random from each batch and tested. If at least four of the five articles are found to be good, the batch is accepted. Otherwise the batch is rejected. a b c d If the fraction of defectives in the batch is 1, find the probabilit of a batch being accepted. If the fraction of defectives in the batch is p, show that the probabilit of the batch being accepted is given b a function of the form A(p) = (1 p) 4 (1 + bp), 0 p 1 and find the value of b. Sketch the graph of A against p for 0 p 1. (Using a calculator would be appropriate.) Find correct to two decimal places: i the value of p for which A(p) = 0.95 ii the value of p for which A(p) = 0.05 (cont d.)

11 68 Essential Mathematical Methods 3&4CAS e i Find A (p), 0 p 1. ii Sketch the graph of A (p)against p. iii Forwhat value of p is A (p) aminimum? iv Describe what the result of iii means. 31 A liquid is contained in a tank which is cuboid with square cross-section as shown in the diagram. The depth of liquid, h cm, in the tank at time t minutes is given b the function with the rule: h(t) = ( t) cm 0.8 cm a State the depth of the liquid at time t = 0. b State the practical domain for the function h. c State the rule for the volume, V cm 3,ofwater in the tank at time t. d Eplain briefl wh an inverse function h 1 eists and find its rule and domain. e Draw graphs of both h and h 1 on the one set of aes. 3 A machine produces ball-bearings with a mean diameter of 3 mm. It is found that 6.3% of the production is being rejected as below the lower tolerance limit of.9 mm and a further 6.3% is being rejected as being above the upper tolerance level of 3.1 mm. Assume that the diameters are normall distributed. a Calculate the standard deviation of the distribution. b A sample of eight ball bearings is taken. Find the probabilit that: i at least one is rejected ii two are rejected c The setting of the machine now wanders so that the standard deviation remains the same, but the mean changes to i Calculate the total percentage of the production that will now fall outside the given tolerance levels. ii Find c such that the probabilit that the diameter lies in the interval (3.05 c, c)is There is a probabilit of 0.8 that a boarding student will miss breakfast if he oversleeps. There is a probabilit of 0.3 that the student will miss breakfast even when he does not oversleep. The student has a probabilit of 0.4 of oversleeping. a On a random da what is the probabilit of: i the student oversleeping and missing breakfast? ii the student not oversleeping and missing breakfast? iii the student not missing breakfast? b Given that the student misses breakfast find the probabilit that he overslept. c It is found that 10 students in the boarding house have identical probabilities for sleeping in and missing breakfast to the student mentioned above. Find the probabilit that: i eactl two of the 10 students miss breakfast ii at least one of the 10 students misses breakfast iii at least eight of the students don t miss breakfast h cm

12 Chapter 0 Etended-response questions revision a On the one set of aes sketch the graphs of = 1 and = e for > 0. b Using addition of ordinates sketch the graph of = 1 + e for > 0. (Do not attempt at this stage to find the coordinates of the turning points.) c Find d d, for = 1 + e d i Show that d d = 0 log e =, > 0 ii Eplain wh this means that the local minimum of = 1 + e lies in the interval (0, 1). iii Using a careful sketch graph or a calculator show that the point of intersection of the graphs of = log e and = is at (0.70, 0.70) (correct to two decimal places). iv Hence find the coordinates of the local minimum of = 1 + e (correct to one decimal place). 35 A section of a creek bank can be modelled b the function f :[0, ( 50] ) R, f () = a + b sin 50 (e, g) where is measured in metres. a i Find the values of a, b, d, e, and g. (0, 7.5) = f() ii The other bank of the river can be modelled b the function = f () (d, 0) Sketch the graph of this new function. b Find the coordinates of the points on the first bank which have -coordinate 10. c A particular river has a less severe bend than this creek. It is found that a section of the bank of the river can be modelled b the function: ( ) g:[0, 50] R, g() = f 5 d Sketch the graph of this function. Clearl indicate the coordinates of the turning points. Over the ears the river bank moves. The shape of the bends are maintained but there is translation of 10 metres in the positive direction of the -ais. i Give the rule that describes this section of the river bank, after the translation (relative to the original aes). ii Sketch the graph of this new function. 36 The continuous random variable X has probabilit densit function f with rule given b: if 0 0 f () = k(5 ) if < 5 0 if > 5

13 684 Essential Mathematical Methods 3&4CAS a Find the value of k. b Find: i E(X) ii the median of X iii, the standard deviation of X, correct to two decimal places c Find Pr(X < ( )) where = E(X) 37 The lifetime, X in das, of a large batch of computer components has a probabilit densit function given b 0 if 0 f () = k(a ) if 0 < a 0 if > a where k and a are positive constants. Find: a k in terms of a b the mean,, and the variance,,ofxin terms of a c Pr(X > + ), giving our answer to two three decimal places d the value of a if the median lifetime is 1000 das 38 The diagram shows a sketch graph of: = 3 = 10 log e ( + 3), > 3 0 M a Find the -coordinate of the local minimum at M. b Show that the gradient of the curve is alwas less than c Find the equation of the line through M with a gradient of d i Hence show that the value of the -ais intercept at P is greater than 10 log e 10. ii Find, correct to three decimal places, the value of the intercept at P. 39 The random variable X has the probabilit densit function given b 0 if < 0 f () = k n if if > 1 where n and k are constants and n > 0 Find in terms of n: a k b E(X) c Var(X) d the median of X e the mode of X P

14 Chapter 0 Etended-response questions revision A particle is moving along a path with equation = + 4 a Find d d. b Find the coordinates of the local minimum of the curve. c Is the function with this rule even? d As,itisevident that and that as,. Sketch the graph of = + 4 showing the asmptotes. e Find the equation of the normal to the curve at the point with coordinates (1, 5) and sketch the graph of this normal with the graph of d. f When the particle is at the point with coordinates (5, 7), is increasing at a rate of 10 units per second. At what rate is increasing? ( d g Show that 1 log d e ) = + 4 h Use this result to find the area of the region contained between the curve and the lines =, = 5 and the -ais. 41 The volume, V cm 3,ofright circular cone of height h cm and 60 radius r cm is given b V = 1 3 r h and the curved surface cm area, S cm,ofthe cone is given b S = r r + h Sand falls onto a horizontal floor at a rate of 0 cm 3 /s. The sand falls in a pile so that it is in the shape of a right circular cone with vertical angle 60. The height of the sand t seconds after the sand starts to fall is cm. a Find the radius of the base of the base of the pile of sand at time t seconds in terms of. b Find V in terms of. c Find dv d in terms of. d Find d in terms of. dt e Find dt d in terms of and, given that = 0when t = 0, find an epression for 3 in terms of t. f Find ds when t = 60. dt 4 The boplot is a displa used to describe the distribution of a data set. Located on the boplot are the minimum, the lower quartile, the median, the upper quartile and the maimum. oplots also show outliers. These are values which are more than 1.5 interquartile ranges below the lower quartile or above the upper quartile. a Suppose that a random variable Z is normall distributed with a mean of 0 and a standard deviation of 1. i Find the value of the median, i.e. m such that Pr(Z m) = 0.5 ii Find the value of the lower quartile, i.e. q 1 such that Pr(Z q 1 ) = 0.5 iii Find the value of the upper quartile, i.e. q 3 such that Pr(Z q 3 ) = 0.75 iv Hence find the interquartile range (IQR) for this distribution. (cont d.)

15 686 Essential Mathematical Methods 3&4CAS b v Find Pr(q IQR < Z < q IQR). vi What percentage of data values would ou epect to be designated as outliers for this distribution? Suppose that a random variable X is normall distributed with a mean of and a standard deviation of. i Find the value of the median, i.e. m such the Pr(X m) = 0.5 ii Find the value of the lower quartile, i.e. q 1 such that Pr(X q 1 ) = 0.5 iii Find the value of the upper quartile, i.e. q 3 such that Pr(X q 3 ) = 0.75 iv Hence find the interquartile range (IQR) for this distribution. v Find Pr(q IQR < X < q IQR). vi What percentage of data values would ou epect to be designated as outliers for this distribution?