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1 GPCC9 U 5 CSA Trials )! Yearl\Yr-U\cat-u.5 Qn) U5-a Write down the value of 6. )! Yearl\Yr-U\cat-u.5 Qn) U5-b If, find the value of f when u = 5 and v = 7 5. f u v )! Yearl\Yr-U\cat-u.5 Qn) U5-c Solve the equation ( ) = 9. )! Yearl\Yr-U\cat-u.5 Qn) U5-d Differentiate )! Yearl\Yr-U\cat-u.5 Qn5) U5-e Sketch the curve = e. State its range. 6)! Yearl\Yr-U\cat-u.5 Qn6) U5-f If, show that a. a 7)! Yearl\Yr-U\cat-u.5 Qn7) U5-a The definition of an odd function f() is given b the rule f( ) = f(). Show that the function f() = 5 is an odd function. 8)! Yearl\Yr-U\cat-u.5 Qn8) U5-b D A(, ) O B(, ) C(, ) In the diagram above, points A, B and C have coordinates (, ), (, ) and (, ) respectivel. Also AD BC and AD CD. Cop this diagram into our answer sheet. i. Show that the gradient of the line BC is equal to. ii. Show that the equation of the line AD is + 6 =. iii. Find the equation of line CD. iv. B solving simultaneousl the equations from (ii) and (iii), find the coordinates of point D. v. Find the area of the quadrilateral ABCD. 9)! Yearl\Yr-U\cat-u.5 Qn9) U5-a In a right angled triangle tan. Find sin, for. )! Yearl\Yr-U\cat-u.5 Qn) U5-b Differentiate the following functions: i. sin loge. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

2 GPCC9 ii. tan. )! Yearl\Yr-U\cat-u.5 Qn) U5-c Find: i. sin ( e ) d. ii. d, leaving our answer in eact form. )! Yearl\Yr-U\cat-u.5 Qn) U5-d Find the equation of the normal to the curve = e at the point on the curve where =. )! Yearl\Yr-U\cat-u.5 Qn) U5-a A quadratic equation with roots and has the form: ( + ) + =. Hence, or otherwise, form a quadratic equation whose roots are and. )! Yearl\Yr-U\cat-u.5 Qn) U5-b The first and the thirteenth terms of an arithmetic progression are 7 and respectivel. Calculate: i. the common difference. ii. the number of terms which have a sum of zero. 5)! Yearl\Yr-U\cat-u.5 Qn5) U5-c In the diagram below triangle QRP has a right angle at R. Also RQS = 5, SQP = 5 and QR = RS = cm. Cop the diagram into our writing booklet. Q 5 cm 5 R cm S P i. Using triangle QRS find the eact length of QS. ii. Using triangle QRP find the eact length of PR and hence the eact length of PS. iii. Use the Sine Rule in triangle QPS to prove that sin5. 6)! Yearl\Yr-U\cat-u.5 Qn6) U5-5a Consider the curve given b = i. Find the coordinates of the stationar points and determine their nature. ii. Find the coordinates of an point of infleion. iii. Sketch the curve, showing all of the above information. iv. Determine the values of for which d. d 7)! Yearl\Yr-U\cat-u.5 Qn7) U5-5b The diagram below shows the shading of a region bounded b the graph = and the lines = and =. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

3 GPCC9 = O i. Cop and complete the following table giving our answer correct to three decimal places. 5 5 = 7 ii. Use Simpson s Rule with five function values to approimate the shaded area to three decimal places. 8)! Yearl\Yr-U\cat-u.5 Qn8) U5-6a i. Factorise the epression a 7a +. ii. Hence, solve the following equation for : (log ) 7(log )+ =. 9)! Yearl\Yr-U\cat-u.5 Qn9) U5-6b B M C P cm A 6 cm D ABCD is a rectangle in which AB = cm and AD = 6 cm. M is the midpoint of BC and DP is perpendicular to AM. Draw a neat sketch on our answer sheet. Hence: i. Prove that triangles ABM and APD are similar. ii. Calculate the length of PD. iii. Using Pthagoras Theorem in triangle APD show that AP = 6 cm. iv. B finding the two areas of the triangles ABM and APD, prove that the area of the quadrilateral PMCD is 96 cm. )! Yearl\Yr-U\cat-u.5 Qn) U5-7a Nicole and Mariana pla against each other, in the third round of the Australian Open. In this tournament, the first plaer to win sets wins the match. The probabilit that Nicole wins an set is 7%. i. Find the probabilit that the game will last two sets onl. ii. Find the probabilit that Nicole wins the match. iii. Find the probabilit that Mariana wins the match. )! Yearl\Yr-U\cat-u.5 Qn) U5-7b The number N of bacteria in a culture at a time t seconds is given b the equation N = e t i. What is the number of bacteria initiall? ii. Determine the number of bacteria after seconds. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

4 GPCC9 iii. After what period of time will the number of bacteria have doubled? iv. At what rate is the number of bacteria increasing when t = seconds? )! Yearl\Yr-U\cat-u.5 Qn) U5-8a i. Sketch the graph of = cos, for. ii. Solve the trigonometric equation cos, for. iii. Hence, find the values of for which cos. )! Yearl\Yr-U\cat-u.5 Qn) U5-8b At time t seconds, the position cm of a point moving in the straight line XOX is given b = at + bt cm, where a and b are constants. The particle passes through the origin O with velocit 6 cm/s in the positive direction at time t = seconds, and after 8 seconds, it is again at O. i. Find the velocit of the particle at an time, in terms of a and b. ii. Find the values of the constants a and b. iii. Find the time when the object is at rest. iv. Find the position of the particle when it is at rest. )! Yearl\Yr-U\cat-u.5 Qn) U5-9a S P O cm r cm Q In the diagram above PQ and RS are arcs of concentric circles with centre O. POQ radians and OP cm. i. Find the area of the sector OPQ. ii. If OR is r cm, find the area of the shaded sector OSR in terms of r. 7 iii. If the shaded area is cm, find the length of PS. 6 5)! Yearl\Yr-U\cat-u.5 Qn5) U5-9b On Jul 5, Nadia invested $ in a bank account that paid interest at a rate of 6% p.a., compounded annuall. i. How much would be in the account after the pament of interest on Jul 5 if no additional deposits were made? ii. In fact Nadia added $ to her account on Jul each ear, beginning on Jul 6. After the pament of interest and her deposit on Jul 5, how much was in her account? iii. Nadia s friend Ana deposited $ in an account at another bank on Jul 5 and made no further deposit. On Jul 5, the balance of her account was $ What was the annual rate of compound interest paid on Ana s account? 6)! Yearl\Yr-U\cat-u.5 Qn6) U5-a i. Simplif logee a ii. Hence evaluate a a log e d 7)! Yearl\Yr-U\cat-u.5 Qn7) e. R EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

5 GPCC9 U5-b A(, ) P(, ) = O The diagram above shows the graph of the parabola =. The point A(, ) is outside the parabola while the point P(, ) is on the parabola as shown in the above diagram. i. If D is the distance between the two points A and P, show that D ( ) ( ). ii. Show that the value of D in the equation in part i. is a minimum when =. iii. Show that the minimum distance between A and P is 5 units. )! Yearl\Yr-U\cat-u. Qn) -a If 5 = 5, find correct to significant figures. )! Yearl\Yr-U\cat-u. Qn) -b a Epress + in the form, where a and b are integers. b )! Yearl\Yr-U\cat-u. Qn) -c Solve tan =, for 6, giving the answers to the nearest degree. )! Yearl\Yr-U\cat-u. Qn) -d a b Simplif. a b 5)! Yearl\Yr-U\cat-u. Qn5) -e Solve 8 =, leaving the answer as a fraction. 6)! Yearl\Yr-U\cat-u. Qn6) -f Find the integers a and b such that a b. 7)! Yearl\Yr-U\cat-u. Qn7) -a Write down the derivatives of: i. ( +) 7 ii. e iii. tan )! Yearl\Yr-U\cat-u. Qn8) -b i. Write down the primitive function of e. ii. Find the eact value of d. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

6 GPCC9 iii. d Given that = sin and = when =, find in terms of. d 9)! Yearl\Yr-U\cat-u. Qn9) -a For what values of a, will a a be positive definitive? )! Yearl\Yr-U\cat-u. Qn) -b k Find the values of k if ( ) d 6. )! Yearl\Yr-U\cat-u. Qn) -c The points A, B, and C have co-ordinates (, 5), (6, ) and (5, 7) respectivel. Plot these points on a number plane. Hence: i. Show that the length of AB is 5. ii. Show that the triangle ABC is isosceles b finding the length of BC. iii. Find the equation of the line AB. iv. BA is produced to meet the line = 7 at P; show that P has co-ordinates (, 7). v. Find the area of triangle PAC. )! Yearl\Yr-U\cat-u. Qn) -a In the diagram below, AB = 9 cm, BC = 6 cm, AD = 8 cm, AC = cm and ABC = DAC. A 9 cm B 8 cm cm 6 cm D i. Prove ABC CAD, giving clear reasons. ii. Hence, find the value of side CD. )! Yearl\Yr-U\cat-u. Qn) -b A parabola whose equation is = a, where a is a constant, has the line = + as a tangent. i. B equating the two given equations, find a quadratic equation in terms of and a. ii. B using the discriminant of the quadratic equation found, find the value of a. iii. Find the coordinates of the point of contact between the tangent and the parabola. iv. Sketch the parabola and the tangent line, showing the co-ordinates of the point of contact. )! Yearl\Yr-U\cat-u. Qn) -5a Below is the graph of = sin, for 6 6. C EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98- One solution of the equation sin = 5 is =. Find the other solutions of this equation for )! Yearl\Yr-U\cat-u. Qn5)

7 GPCC9-5b i. On the same graph, sketch the curves = sin and = sin, for. ii. Give three solutions to the equation sin + sin =, for. 6)! Yearl\Yr-U\cat-u. Qn6) -5c A radioactive substance decas at a rate proportional to the mass present. The rate of change is given b dm km, where k is a positive constant and M grams is the mass present at an time, t hours. dt i. Show that M = Me kt is a solution to this equation. ii. If grams of this substance decas to 8 grams in hours, find:. The value of k, correct to decimal places.. The mass present after a further hours, to the nearest gram. iii. The half-life time is the time taken for grams of this substance to deca to 5 grams. What is the half-life time of this substance? Give our answer to the nearest hour. 7)! Yearl\Yr-U\cat-u. Qn7) -6a i. The curve = + a has a stationar point at =. Find the value of a, and hence: ii. Find the coordinates of all stationar points. iii. Determine the nature of the stationar points. iv. Sketch the curve and then determine for what values of the curve is increasing. 8)! Yearl\Yr-U\cat-u. Qn8) -6b If the K th term of an arithmetic series is L, and the L th term is K: i. Show that L = a + (K )d. ii. Find another epression for K. iii. B solving the two equations founded, show that d =. iv. Hence, find the first term of this series in terms of L and K. 9)! Yearl\Yr-U\cat-u. Qn9) -7a The diagram above shows the area bounded b the graph, (for > ), the - ais and the lines = and =. i. Find the shaded area. Leave our answer as a fraction. ii. Find the volume of the solid formed when the shaded area is rotated about the ais. Leave our answer in eact form. )! Yearl\Yr-U\cat-u. Qn) -7b i. Sketch the graph of f() = e for all values of in the domain and state its range. ii. The curve f() = e is rotated about the -ais to give a solid. Show that the volume V of the solid formed, from = to = 5, is given b V ( ln ) 5 d. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

8 GPCC9 iii. Use Simpson s rule with 5 function values to find the volume of this solid, correct to significant figures. )! Yearl\Yr-U\cat-u. Qn) -8a A certain soccer team has a probabilit of 6 of winning a match and a probabilit of of drawing a match. i. If this soccer team plas two matches, draw a tree diagram to show all possible outcomes. ii. Find the probabilit of this soccer team winning at least one match out of the two matches. iii. Find the probabilit of this soccer team not winning either of the two matches. )! Yearl\Yr-U\cat-u. Qn) -8b A(5, ) B(, 5) O The figure above shows a sector of a circle OAB, centre O, with its arc joining the points A(5, ) and B(, 5). Cop this figure into our answer booklet. i. Find the value, in degrees, of one radian, correct to the nearest minute. ii. Show that the size of AOB is 78 radians, correct to decimal places. iii. Calculate the perimeter of sector OAB, correct to decimal places. )! Yearl\Yr-U\cat-u. Qn) -9a A bo is made from a 5 cm b cm rectangle of cardboard b cutting out four equal squares of side cm from each corner as shown below: cm cm 5 cm The edges are turned up to make an open bo. i. Show that the volume V of this bo is given b the equation: V = + (cm ). ii. Find the value of, correct to one decimal place, that gives this bo its greatest volume. iii. Hence, find the maimum volume of this bo, correct to decimal places. )! Yearl\Yr-U\cat-u. Qn) -9b Jordan has to pa annual instalments for his superannuation at the beginning of each ear according to the formula: EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

9 GPCC9 r M n ( ) M n,n, where r(%) is the annual rate of interest paid b the fund and Mn is the instalment at the beginning of the n th ear. If the interest rate is % p.a., compounded earl, and Jordan s first instalment is $5, find: i. How much is his second instalment? ii. Find the amount Jordan has to pa into the fund at the beginning of the th ear. iii. Find the total value of his investment after ears. 5)! Yearl\Yr-U\cat-u. Qn5) -a The acceleration of a particle at an time t seconds is given b dv dt k, where k is a constant. i. Show that v = kt + c, for some constant c. ii. The displacement metres, at an time t seconds, is shown in the table below: t (sec) (metres) 9 iii. Show that = t t +. Find when the particle comes to rest. 6)! Yearl\Yr-U\cat-u. Qn6) -b C b h a A D B The diagram above shows a triangle ABC, and CD is perpendicular to AB. It is given that BC = a, AC = b, ACD = and BCD =. i. B using triangles ACD and BCD, show that h = bcos = acos. ii. Show that the area of a triangle ACD is equal to ab sin cos. iii. Find another epression for the area of triangle BCD in terms of a, b, and. iv. Show that the area of triangle ABC is equal to ab sin( ). v. Hence, but not otherwise, deduce that: sin( + ) = sin cos + cos sin. )! Yearl\Yr-U\cat-u. Qn) U-a 5 Evaluate correct to one decimal place. 5 )! Yearl\Yr-U\cat-u. Qn) U-b Factorise full 8 6. )! Yearl\Yr-U\cat-u. Qn) U-c Graph on a number line the solution to + 5. )! Yearl\Yr-U\cat-u. Qn) U-d EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

10 GPCC9 Rationalise the denominator of 5)! Yearl\Yr-U\cat-u. Qn5) U-e 5. Find the eact value of tan cos ec. 6)! Yearl\Yr-U\cat-u. Qn6) U-f Sketch the curve = e, clearl showing where the curve cuts the -ais. 7)! Yearl\Yr-U\cat-u. Qn7) U-a Consider the quadratic function (k + ) + =. For what value of k does the quadratic function have real roots? 8)! Yearl\Yr-U\cat-u. Qn8) U-b L A S R L C Line L has equation + = and intersects the -ais at point A. Line L has equation = and intersects the -ais at point C. Line L and line L intersect at point R. The horizontal line through A intersects the vertical line through R, at S. Find the coordinates of point A and C. Show that R has coordinates (, ). State the equation of the line SR. Find the gradient of line L. Find the distance AR. Show that triangle ARC is a right-angled isosceles triangle. Find the equation of the circle with centre R, passing through the points A and C. 9)! Yearl\Yr-U\cat-u. Qn9) U-a Differentiate with respect to : i. e tan. )! Yearl\Yr-U\cat-u. Qn) U-b e Find e d. )! Yearl\Yr-U\cat-u. Qn) U-c Evaluate ( cos ) d. )! Yearl\Yr-U\cat-u. Qn) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

11 GPCC9 U-d In the diagram, PQRS is a parallelogram. QR is produced to U so that QR = RU. P Q S T R U Giving clear reasons, show that the triangles PST and URT are congruent. Hence, or otherwise, show that T is the midpoint of SR. )! Yearl\Yr-U\cat-u. Qn) U-a Evaluate k 5. k )! Yearl\Yr-U\cat-u. Qn) U-b The third term of a geometric series is and the seventh term is. Find the th term of this series. 5)! Yearl\Yr-U\cat-u. Qn5) U-c Consider the function f() =. Sketch the function f(). 6 Evaluate f ( ) d. 6)! Yearl\Yr-U\cat-u. Qn6) U-d Solve for the equation m n. 7)! Yearl\Yr-U\cat-u. Qn7) U-e Sketch a graph of = cos for. 8)! Yearl\Yr-U\cat-u. Qn8) U-f dp d P The population (P) of a coastal town is increasing at a decreasing rate. Comment on and for dt dt this function. 9)! Yearl\Yr-U\cat-u. Qn9) U-5a In a bag are marbles. The bag consists of 7 red marbles, 9 gold marbles and blue marbles. One marble is drawn from the bag and not replaced, and then a second marble is drawn. With the aid of a tree diagram, or otherwise, find the probabilit of choosing: two gold marbles marbles of different colour. )! Yearl\Yr-U\cat-u. Qn) U-5b Consider the curve given b = 6. Find the coordinates of the two stationar points. Determine the nature of the stationar points. Show that there eists a point of infleion when =. Sketch the curve for the domain 6. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

12 GPCC9 )! Yearl\Yr-U\cat-u. Qn) U-5c Given that = e d d, show that. d d )! Yearl\Yr-U\cat-u. Qn) U-6a Find the equation of the normal to the curve = sin at the point where )! Yearl\Yr-U\cat-u. Qn) U-6b The table shows the values of a function f() for five values of.. f() 5 8 Use Simpson s rule with these five values to find an approimation to )! Yearl\Yr-U\cat-u. Qn) U-6c = B = + f ) ( d. A i. The curve = and the line = + intersect at the points A and B as shown in the diagram above. Find the coordinates of the points A and B. ii. Find the area bounded b the curve = and the line = +. 5)! Yearl\Yr-U\cat-u. Qn5) U-6d Find the volume generated when the curve cot is rotated about the -ais between and. Leave our answer in eact form. 6)! Yearl\Yr-U\cat-u. Qn6) U-7a d Find given that log e ( ). d 7 7)! Yearl\Yr-U\cat-u. Qn7) U-7b A particle moves in a straight line so that its velocit, v metres per second, at time t is given b v. The particle is initiall metre to the right of the origin. t Find an epression for the position, of the particle at time t. Eplain wh the velocit of the particle is never metres per second. Find the acceleration of the particle when t = seconds. 8)! Yearl\Yr-U\cat-u. Qn8) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

13 GPCC9 U-7c i. Show that (cosec A )sin A = cos A. ii. Hence, or otherwise, solve ( cos ec A ) sin A for A. 9)! Yearl\Yr-U\cat-u. Qn9) U-8a Christina borrows $8 from a finance compan to bu a house. She pas interest at 6% per annum, calculated quarterl on the balance still owing. The loan is to be repaid at the end of ears with equal quarterl repaments of $P. Let An = the amount owing after the nth repament. Show that after the first quarterl repament of $P Cristina owes an amount equivalent to A = $87 $P. Find an epression for the amount still owing after repaments of $P. Find the value of $P to the nearest cent. )! Yearl\Yr-U\cat-u. Qn) U-8b Water is draining from a storage tank at a rate which is proportional to the volume of water contained in the tank. When full, the storage tank holds litres of water. On inspection, the tank was found to be full but minutes later it was found to contain onl 8 litres of water. How much water (to the nearest litre) will the tank hold after hour? How long, in hours and minutes, will it take to reach litre of water in the tank? )! Yearl\Yr-U\cat-u. Qn) U-8c Consider the series sin + sin + sin 6 +. Show that a limiting sum eists. Find the limiting sum epressing the answer in simplest form. )! Yearl\Yr-U\cat-u. Qn) U-9a Arcsec Landscaping Compan are designing a garden bed for a local park in the shape of a sector with radius r and sector angle. The have a total of 75 metres of garden edging materials to use as the perimeter of the garden bed. r r r Show that the area A of the garden bed is given b A ( 75 r). Find the greatest area of garden bed which can be made using 75 metres of edging material. After inspecting the location for the garden bed the designers calculate that the sector angle for the garden must be less than. Can the still create the garden bed with maimum area found in (ii)? Justif our answer. )! Yearl\Yr-U\cat-u. Qn) U-9b A train is travelling at a constant velocit of 8 kilometres per hour as it passes through the railwa station at a town. At the same time, a second train commences its journe from rest at the railwa station. The second train accelerates uniforml for 5 minutes until it reaches kilometres per hour and maintains this velocit for a further 5 minutes. At this time each of the trains then begin to slow down at a constant rate, arriving at the net station at the same time. B illustrating graphicall the relationship between velocit and time, calculate the time taken for the trains to travel between the two stations. How far apart are the stations? EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

14 GPCC9 )! Yearl\Yr-U\cat-u. Qn) U-a d i. Show that ( n ) n. d ii. Hence, or otherwise, find n d. iii. The graph shows the curve = n, ( > ) which meets the line = 5 at Q. Using our answers from (i) and (ii), or otherwise, find the area of the shaded region. = 5 = n Q 5)! Yearl\Yr-U\cat-u. Qn5) U-b Consider the function f() = e cos for. Find the values where the stationar points occur. Determine the nature of the stationar points. Sketch the curve showing the coordinates of the stationar points in eact form and the intercepts with the aes. Find the number of solutions to the equation e cos, in the domain. Justif our answer. )! Yearl\Yr-U\cat-u. Qn) U-a Evaluate if ( ) and ( ). Give our answer in fractional form. 5 5 )! Yearl\Yr-U\cat-u. Qn) U-b Epress as a fraction in its simplest form. )! Yearl\Yr-U\cat-u. Qn) U-c Factorise 5. )! Yearl\Yr-U\cat-u. Qn) U-d Solve + = leaving our answer in simplest surd from. 5)! Yearl\Yr-U\cat-u. Qn5) U-e i. Solve =. ii. 6)! Yearl\Yr-U\cat-u. Qn6) U-a Differentiate with respect to : i. ii. Hence, or otherwise, write down the value of log. 5 e EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

15 GPCC9 iii. sin 7)! Yearl\Yr-U\cat-u. Qn7) U-b B A( 5, 5) C(, ) D(, ) The diagram shows the points A( 5, 5), C(, ) and D(, ). B is a point on the ais. i. Find the gradient of AC. ii. Find the midpoint of AC. iii. Show that the equation of the perpendicular bisector of AC is + =. iv. Find the coordinates of B given that B lies on + =. v. Show that the point D(, ) lies on + =. vi. Show that ABCD is a rhombus. 8)! Yearl\Yr-U\cat-u. Qn8) U-a Find 5 d 9)! Yearl\Yr-U\cat-u. Qn9) U-b Evaluate 8 sec d )! Yearl\Yr-U\cat-u. Qn) U-c Find the equation of the normal to the curve )! Yearl\Yr-U\cat-u. Qn) U-d B O log e cm at the point (e, e). A 8 cm C 6 cm D In the diagram AB is cm, AD is 8 cm and CD is 6 cm. BAD is and CBD is. i. Use the cosine rule to find the length of BD. ii. Hence, find the size of BCD to the nearest degree. )! Yearl\Yr-U\cat-u. Qn) U-a The second term of a geometric series is and the fifth term is i. Find the common ratio and the first term of the series. ii. Find the limiting sum of the series. iii. Hence, find the difference between the limiting sum and the sum of the first terms giving our answer in scientific notation correct to significant figures. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

16 GPCC9 )! Yearl\Yr-U\cat-u. Qn) U-b For the quadratic equation + k + k =, i. find the value of the discriminant in terms of k, ii. eplain wh the roots of this quadratic equation are real for all values of k. )! Yearl\Yr-U\cat-u. Qn) U-c B C A H D E ABCDEFGH is regular octagon. G F i. Eplain clearl wh ABC is 5. ii. Calculate the size of GAH. iii. Using (i), or otherwise, calculate the size of CGF. iv. Hence, calculate the size of AGC. 5)! Yearl\Yr-U\cat-u. Qn5) U-5a O A B C D The diagram shows two concentric circles with centre O. The radius of the larger circle is 8 cm. i. Calculate the area of sector COD. ii. The area of the sector AOB is 8 cm. Calculate the radius of this sector AOB. iii. Calculate the area of triangle COB. 6)! Yearl\Yr-U\cat-u. Qn6) U-5b Let f() = +. i. Cop the following table and suppl the missing values. ii. 6 8 f() Use these si values of the function and the trapezoidal rule to find the approimate vale of ( ) d. 7)! Yearl\Yr-U\cat-u. Qn7) U-5c The population P of a town is growing at a rate proportional to the town s current population. The population at time t ears is given b P = Ae kt, where A and k are constants. The population ears ago was people and toda the population of the town is 5 people. i. Find the value of A. ii. Find the value of k. iii. Find the population that will be present ears from now. 8)! Yearl\Yr-U\cat-u. Qn8) U-6a EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

17 GPCC9 Consider the curve given b = d i. Find. d ii. Find the coordinates of the two stationar points. iii. Determine the nature of the stationar points. iv. Sketch the curve for the domain 5. v. B drawing an appropriate line on our graph, or otherwise, solve =. 9)! Yearl\Yr-U\cat-u. Qn9) U-6b Calculate the eact volume generated when the region enclosed b the curve = + e for, is rotated about the ais. )! Yearl\Yr-U\cat-u. Qn) U-7aS A bag contains 5 blue balls, red balls, ellow balls and green ball. Three balls are selected at random without replacement from the bag. Calculate the probabilit that i. the three balls drawn are blue, ii. the three balls drawn are of the same colour, iii. eactl two of the balls drawn are blue. )! Yearl\Yr-U\cat-u. Qn) U-7b A particle is projected verticall upwards from a point metres above horizontal ground. The displacement at time t seconds is given b = 5t 9t, t. i. Find an epression for the velocit of the particle. ii. Find when the particle comes to rest. iii. Hence, find the greatest height of the particle above the ground. iv. Find the length of time for which the particle is at least 6 metres above the ground. )! Yearl\Yr-U\cat-u. Qn) U-8a A function = f () is continuous for all values. After finding the first and second derivatives a student discovers the following, for all values of. When <, f () < and f () > When =, f () = and f () = When >, f () < and f () <. Draw a neat sketch of = f (), showing all the important characteristics of the function given that f () =. )! Yearl\Yr-U\cat-u. Qn) U-8b The graphs of the function = and =. = = i. Describe, using inequalities, the shaded region. ii. B solving simultaneousl, show that the points of intersection are at = and =. iii. Calculate the area of the shaded region. )! Yearl\Yr-U\cat-u. Qn) U-8c EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

18 GPCC9 On a factor production line a tap opens and closes to fill containers with liquid. As the tap opens, the 6t rate of flow increases for the first seconds according to the relation R, where R is measured in 5 L /sec. The rate of flow then remains constant until the tap begins to close. As the tap closes, the rate of flow decreases at a constant rate for seconds, after which time the tap is full closed. i. Show that, while the tap is full open, the volume in the container at an time is given b 6 V ( t 5). 5 ii. For how man seconds must the tap remain full open in order to eactl fill a L container with no spillage. 5)! Yearl\Yr-U\cat-u. Qn5) U-9a ABCD is a rectangle and AE BD. AE = 5 cm and DE = cm. A B E D C i. Cop the diagram and prove that triangles AED and BCD are similar. ii. Hence, show that AD = BD DE. iii. Find the area of ABCD. 6)! Yearl\Yr-U\cat-u. Qn6) U-9b A closed water tank in the shape of a right clinder is to be constructed with a surface area of 5 cm. The height of the clinder is h cm and the base radius is r cm. 7 i. Show that height of the water tank in terms of r is given b h r. r ii. Show that the volume V that can be contained in the tank is given b V = 7r r. iii. Find the radius r cm which will give the clinder its greatest possible volume. Justif our answer. 7)! Yearl\Yr-U\cat-u. Qn7) U-a i. Show that is a solution of sin = cos. 8 ii. On the same set of aes, sketch the graphs of the functions = sin and = cos for. iii. Hence, find graphicall the number of solutions of tan = for. iv. Use our graphs to solve tan for. 8)! Yearl\Yr-U\cat-u. Qn8) U-b Mr and Mrs Matthews decide to borrow $5 to bu a house. Interest is calculated monthl on the balance still owing, at a rate of 6 6% per annum. The loan is to be repaid at the end of 5 ears with equal monthl repaments of $M. Let $An be the amount owing after the nth repament. i. Derive an epression for A6. ii. Find the value of M. iii. Hence, calculate the amount still owing after 5 ears of pament at this rate. iv. At the end of five ears, the interest rate is increased to 7 % per annum and Mr and Mrs Matthews change their paments to $8 per month. How man more months are needed to pa off the remainder of the loan? )! Yearl\Yr-U\cat-u. Qn) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

19 GPCC9 U-a Factorise completel ab a b +. )! Yearl\Yr-U\cat-u. Qn) U-b Simplif + 5. )! Yearl\Yr-U\cat-u. Qn) U-c Find integers a and b such that )! Yearl\Yr-U\cat-u. Qn) U-d a b. Find the value of cos, correct to decimal places. 8 5)! Yearl\Yr-U\cat-u. Qn5) U-e Solve tanθ for 6. 6)! Yearl\Yr-U\cat-u. Qn6) U-f i. Write down the discriminant of + k. ii. For what values of k does + k = have unequal real roots? 7)! Yearl\Yr-U\cat-u. Qn7) U-a A(, 6) O C(, ) B(5, ) The diagram shows the origin O and the points A(, 6), B(5, ) and C(, ). The line AB and BC are perpendicular. Cop or trace this diagram onto our writing sheet. a. Show that A and B lie on the line + = 9. b. Show that the length of AB is 5 units. c. Find the perpendicular distance from O to AB. d. Find the area of triangle AOB. e. Show that C has coordinates (, ). f. Does the line AC pass through the origin? Eplain. g. The point D is not shown on the diagram. The point D lies on the ais and ABCD is a rectangle. Find the coordinates of D. h. On our diagram, shade the region satisfing the inequalit )! Yearl\Yr-U\cat-u. Qn8) U-h On our diagram, shade the region satisfing the inequalit )! Yearl\Yr-U\cat-u. Qn9) U-a Find: i. sec d ii. ( ) d e. )! Yearl\Yr-U\cat-u. Qn) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

20 GPCC9 U-b Evaluate d. )! Yearl\Yr-U\cat-u. Qn) U-c R C A B S In the diagram, AC = BC, RCA and CBS are straight lines, ABS = and BCR =. Cop the diagram onto our writing sheet. Find the value of giving reasons. )! Yearl\Yr-U\cat-u. Qn) U-d Differentiate the following i. sin ii.. )! Yearl\Yr-U\cat-u. Qn) U-a The nth term of an arithmetic series is given b Tn = n +. i. What is the th term of this series? ii. What is the sum of the first terms of this series? )! Yearl\Yr-U\cat-u. Qn) U-b Two cards are chosen at random and without replacement from the seven cards above. What is the probabilit that i. both cards show a ii. the sum of the two numbers on the cards chosen is greater than? 5)! Yearl\Yr-U\cat-u. Qn5) U-c Use the sine rule to find the value of where is obtuse. 6)! Yearl\Yr-U\cat-u. Qn6) U-d 5 8 The geometric series a + ar + ar + has a second term of and has a limiting sum of. i. Show that a = r. ii. Solve a pair of simultaneous equations to find r. 7)! Yearl\Yr-U\cat-u. Qn7) U-5a Consider the curve given b =. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

21 GPCC9 i. d Find. d ii. Find the coordinates of two stationar points. iii. Determine the nature of the stationar points. iv. Sketch the curve for. 8)! Yearl\Yr-U\cat-u. Qn8) U-5b = B C A D O = = 7 In the diagram, the shaded region OABCD is bounded b = + the lines = 7, = and the and aes. i. Show that B has coordinates (, 5). ii. Use Simpson s rule with 5 function values to estimate the area of the shaded region. 9)! Yearl\Yr-U\cat-u. Qn9) U-6a = The diagram shows the shape of a vessel obtained b rotating about the ais, the part of the parabola = 5 between = and = 5. Show that the volume of the vessel is 5 units. )! Yearl\Yr-U\cat-u. Qn) U-6b The number N of bacteria in a colon is growing at a rate that is proportional to the current number. The number at time t hours is given b N = Ne kt where N and k are positive constants. i. If the size of the colon doubles ever half hour, find the value of k. ii. If the colon now contains 6 million bacteria, how long ago did the colon contain million iii. bacteria? Show that the numbers of bacteria present at consecutive integer hours form a geometric sequence. )! Yearl\Yr-U\cat-u. Qn) U-6c EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

22 GPCC9 5 6 t A particle moves along a straight line for 6 seconds. The particle s velocit v at time t seconds is shown on the graph above. i. When is the particle at rest? ii. What does the shaded region represent? iii. Is this particle further from its initial position at t = or at t = 5? Eplain briefl. )! Yearl\Yr-U\cat-u. Qn) U-7a i. Show that is a solution of cos = cos. ii. On the same set of aes, sketch the graphs of = cos and = cos for, showing the coordinate of all points of intersection. iii. Find the eact area of the region bounded b the curves = cos and = cos over the interval. )! Yearl\Yr-U\cat-u. Qn) U-7b N M S Q In the figure, MNQ is the sector of a circle, MQN =, NQ = cm and MS = SQ = cm. i. Calculate the eact length of NS. ii. Find the perimeter of the shaded region MSN. )! Yearl\Yr-U\cat-u. Qn) U-7c i. Without using calculus, sketch the graph of = e. ii. On the same sketch, find graphicall the number of solutions of the equation e =. 5)! Yearl\Yr-U\cat-u. Qn5) U-8a Differentiate = log. 6)! Yearl\Yr-U\cat-u. Qn6) U-8b The diagram shows a rectangle ABCD inscribed in the region bounded b the parabola and the line = 6. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

23 GPCC9 A B = 6 D O i. Show that the area of ABCD is given b A. ii. Find the dimensions of the rectangle so that its area is a maimum. 7)! Yearl\Yr-U\cat-u. Qn7) U-8c Liquid petroleum is pumped out of a 5 litre storage container through a valve such that the volume dv flow rate of petroleum in litres per second is given b 9t dt ( t ). i. Show that if the tank was initiall full, then V = 5 96t. ii. How long before the tank is onl % full? 8)! Yearl\Yr-U\cat-u. Qn8) U-9a Mia would like to save $6 for a deposit on her first home. She decides to invest her net monthl salar of $ in a bank account that pas interest at a rate of 6% per annum compounded monthl. Mia intends to withdraw $E at the end of each month from this account for living epenses, immediatel after the interest has been paid. i. Show that the amount of mone in the account following the second withdrawal of $E is given b $(5 + 5) $E(5 + ). ii. Calculate the value of E if Mia is to reach her goal after ears. 9)! Yearl\Yr-U\cat-u. Qn9) U-9b A particle is moving along the ais. Its position at time t is given b 6t e 5 ( t ). i. Find the initial position of the particle. ii. Show that the particle is alwas moving to the right. iii. What happens to the acceleration eventuall? )! Yearl\Yr-U\cat-u. Qn) U-9c C t = f () The diagram shows the graph of the derivative of the curve = f(). i. The curve = f() has a stationar point of infleion at =. Justif this statement b reference to the graph. ii. Draw a possible graph of = f() if f() =. )! Yearl\Yr-U\cat-u. Qn) U-a EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

24 GPCC9 D 7 B A 6 M In the diagram, ABC is an isosceles triangle where ABC = BCA = 7, AB = AC = and BC =. Angle BCA is bisected b CD and angle BAC is bisected b AM which is also the perpendicular bisector of BC. Cop the diagram onto our writing sheet. i. Show that AD =. ii. Show that triangles ABC and CBD are similar. iii. B using (ii), find the eact value of. iv. Hence find the eact value of sin 8. )! Yearl\Yr-U\cat-u. Qn) U-b 6 C U A O I E The spinner shown above is used in a game. Once spun, it is equall likel to stop at an one of the letters A, E, I, O or U. i. If the spinner is spun twice, find the probabilit that is stops on the same letter twice. ii. How man times must the spinner be spun for it to be 99% certain that it will stop on the letter E at least once? )! Yearl\Yr-U\cat-u. Qn) U-a Find the value of 6 5 )! Yearl\Yr-U\cat-u. Qn) U-b 6 Solve the equation. 5 )! Yearl\Yr-U\cat-u. Qn) U-c X correct to two decimal places. Z 8 5 Y Not to scale U V W The diagram shows XY parallel to UW, XYU = 5, UZV = 8 and ZVW =. Find the value of. Give reasons. )! Yearl\Yr-U\cat-u. Qn) U-d Find a primitive function for +. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

25 GPCC9 5)! Yearl\Yr-U\cat-u. Qn5) U-e when A function g() is defined as: g( ). when Evaluate g( ) + g(). 6)! Yearl\Yr-U\cat-u. Qn6) U-f Graph the solution of + on the number line. 7)! Yearl\Yr-U\cat-u. Qn7) U- B(,) C(5,) O A(, ) Not to scale The diagram shows ABC with vertices A(, ), B(, ) and C(5, ). Cop the diagram onto our answer sheet. a. E is the midpoint of AC. Show that the coordinates of E are (, ). b. Show that the gradient of AC is. c. A line L is drawn through B, perpendicular to AC. Show the equation of line L is =. d. Show that E lies on line L. e. On our diagram, draw line L and plot point E. Prove BEC is congruent to BEA. f. AC is a diameter of a circle. i. Calculate the radius of the circle. ii. Hence find the equation of the circle. 8)! Yearl\Yr-U\cat-u. Qn8) U-a Differentiate the following epressions with respect to : i. ( + ) ii. cos e iii. 9)! Yearl\Yr-U\cat-u. Qn9) U-b Evaluate the following definite intervals: / i. sin d ii. e d )! Yearl\Yr-U\cat-u. Qn) U-c EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

26 GPCC9 Find d. )! Yearl\Yr-U\cat-u. Qn) U-a The third term and the tenth term of an arithmetic series are 7 and respectivel. Find the: i. first term and the common difference. ii. sum of the first ten terms of the series. )! Yearl\Yr-U\cat-u. Qn) U-b A bo contains five blue, three ellow and eight red beads. Two beads are selected at random from the bo without replacement. Find the probabilit that: i. both beads are blue. ii. at most one of the beads are blue. )! Yearl\Yr-U\cat-u. Qn) U-c = 9 (,8) O The diagram shows the graph of the parabola = 9. A tangent is drawn to the parabola at the point (, 8). i. Show that the equation of the tangent at (, 8) is + =. ii. Eplain how ou know the tangent crosses the ais at (5, ). iii. Calculate the area bounded b the parabola, the tangent and the ais. )! Yearl\Yr-U\cat-u. Qn) U-5a Not to B scale 9 8 cm 7 cm A D C In the diagram, AB = 9 8 cm, BC = 7 cm, BCD = 69, BDA = 6 and BAD = 9. i Find the length of BD correct to the nearest mm. ii Calculate the area of ABC correct to the nearest cm. 5)! Yearl\Yr-U\cat-u. Qn5) U-5b Consider the curve = + 6. i. Find the coordinates of an turning points and determine their nature. ii. Find the coordinate of an points of infleion. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

27 GPCC9 iii. Sketch the curve for the domain. iv. What is the maimum value of + 6 in the domain? 6)! Yearl\Yr-U\cat-u. Qn6) U-6a Find all the values of for which ln = ln. 7)! Yearl\Yr-U\cat-u. Qn7) U-6b D C A X B Y Not to scale In the diagram, ABCD is a parallelogram. X is a point on AB. DX and CB are both produced to Y. i. Cop this diagram onto our answer sheet. ii. Prove that ADX is similar to CYD. iii. Hence find the length of XY given AX = 8 cm, DC = cm and DX = cm. 8)! Yearl\Yr-U\cat-u. Qn8) U-6c Not to scale Q(, 5) = = O P(5, ) The diagram shows the graphs of the curve 5 and the line 5 5 =. The curve and the line intersect at the points P(5, ) and Q(, 5). The region bounded b the curve and the line is rotated about the ais. Find the volume of the solid generated. 9)! Yearl\Yr-U\cat-u. Qn9) U-7a An eperimental vaccine was injected into a cat. The amount, M millilitres, of vaccine present in the bloodstream of the cat, t hours later was given b M = e t +. i. How much vaccine was initiall injected into the cat? ii. At what rate was the amount of vaccine decreasing at the end of hours? iii. Show that there will alwas be more than millilitres of vaccine present in the cat s bloodstream. iv. Sketch the curve of M = e t + to show how the amount of vaccine present in the cat s bloodstream changes over time. )! Yearl\Yr-U\cat-u. Qn) U-7b A particle moves in a straight line. At times t seconds, its displacement, metres from a fied point O on the line is given b = cos t, EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

28 GPCC9 i. What is the initial displacement of the particle? ii. Sketch the graph of as a function of t. iii. Find an epression for the velocit of the particle at an time t. iv. What is the velocit of the particle at time t? 6 v. At what time does the particle first reach its maimum speed? )! Yearl\Yr-U\cat-u. Qn) U-8a When Jack left school, he borrowed $5 to bu his first car. The interest rate on the loan was 8% p.a. and Jack planned to pa back the loan in 6 equal monthl instalments of $M. i. Show that immediatel after making his first monthl instalment, Jack owed $[5 5 M]. ii. Show that immediatel after making his third monthl instalment, Jack owed $[5 5 M( )]. iii. Calculate the value of M. )! Yearl\Yr-U\cat-u. Qn) U-8b The table shows the value of a function f() for five values. 8 Approimate the value of )! Yearl\Yr-U\cat-u. Qn) U-8c 6 8 f() f ( ) d using the five function values and Simpson s rule. i. On the same number plane, sketch the curves = sin and = ii. Hence find the number of real solutions to the equation sin = )! Yearl\Yr-U\cat-u. Qn) U-9a Not to scale A B tan in the domain. tan for. O The diagram shows a sector OAC with area of 9 cm and OA = 5 cm. i. Find the size of in radians. ii. Find the perimeter of the segment ABC. Give our answer correct to the nearest cm. 5)! Yearl\Yr-U\cat-u. Qn5) U-9b C EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

29 GPCC9 The water s edge is a straight line ABC which runs East-West. A lighthouse is 6 km due north of A. km due east of A is the general store. To get to the general store as quickl as possible, the lighthouse keeper rows to a point B, km from A, and then jogs to the general store. The lighthouse keeper s rowing speed is 6 km/h and his jogging speed is km/h. 6 i. Show that it takes the lighthouse keeper hours to row from the lighthouse to B. 6 ii. Show that the total time taken for the lighthouse keeper to reach the general store is given b: 6 T hours. 6 iii. Hence show that when = km, the time it takes from the lighthouse keeper to travel from the lighthouse to the general store is a minimum. iv. Hence find the quickest time it takes the lighthouse keeper to go to the general store from the lighthouse. Give our answer correct to the nearest minute. 6)! Yearl\Yr-U\cat-u. Qn6) U-a P A O N B Not to scale The diagram shows a circle with centre O and diameter AB. P is a point on the circumference of the circle. PN is drawn perpendicular to AB and AP is perpendicular to PB. Let POB =. i. Eplain wh APO is isosceles. ii. Eplain wh OAP = OPA =. PN iii. Show that sin. AB iv. Use APN and PAB to show that sin cos = sin. 7)! Yearl\Yr-U\cat-u. Qn7) U-b Kellie and Lachlan pla a game where the each take turns at throwing two ordinar dice. The winner is the first person to throw a double. Kellie throws first. i. Show that the probabilit that Lachlan wins the game on his first throw is 6 5. ii. iii. [[End Of Qns]] Show that the probabilit Lachlan wins the game on his first or second throw is given b Calculate the probabilit that Lachlan wins the game. EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

30 GPCC9 «) 6 «) f = 5 «) = 6 or = 8 «) 5 (, ) «5), Range: > «6) Proof «7) Proof 8 «8) i) ii) Proof iii) + + = iv) D(, ) 5 5 v) 7 square units «9) 5 sin «) i) cos log e ii) sec «) i) cos(e ) + c ii) ln «) = «) + = «) i) 5 ii) 9 «5) i) QS = ii) PR =, PS = iii) Proof «6) i) (, 8) is a maimum turning point, (, ) is a minimum turning point ii) (, 6) (, 8) (, 6) (, ) iii) iv) < < «7) i) 5 5 = ii) 7 85 «8) i) (a )(a ) ii) = or = 8 «9) i) Proof ii) PD = 8 cm iii) iv) Proof [Answers] «) i) ii) iii) «) i) ii) 7 iii) seconds iv) 6 7 bacteria/second «) i) ii), iii) «) i) v = at + b ii) a =, b = 6 iii) t = seconds iv) 8 cm from the origin «) i) cm ii) r cm iii) cm 6 «5) i) $ 9 7 ii) $ iii) 5% «6) i) a ii) a «7) Proof «) «) «) = 7 or 5 b «) a b 5 «5) «6) a = and b = «7) i) ( + ) 6 ii) e ( + ) iii) 5 sec 5 tan 5 5 e «8) i) c ii) ln iii) = + cos + «9) a > 5 «) k = or 5 «) i) ii) Proof iii) + = 6 iv) Proof v) 6 units «) i) Proof ii) 6 cm EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

31 GPCC9 «) i) a = ii) a = iii) (, ) = +.5 (, ) iv) = «) =, 5,, «5) i) = sin ii) =, π and π «6) i) Proof ii) ) k = 57 ) 7 grams iii) 6 hours = sin «7) i) a = 5 ii) (, ) and ( 7, 8) iii) (, ) is a local maimum and ( 7, 8) is a local (, ) 5 7 (,. 8) minimum iv) «8) i) Proof ii) K = a + (L )d iii) Proof iv) a = L + K «9) i) units 6 ii) units 8 «) i) Proof iii) units 6 f() = e L «) i) ii) 8 iii) 6 «) i) 578 ii) Proof iii) 6 units «) i) Proof ii) = iii) cm «) i) $56 ii) $6 8 iii) $6 6 W D L «5) i) ii) Proof iii) t = second W D L W D L W «6) i) ii) Proof iii) ab cos sin iii) iv) Proof «) 6 «) 6( )( + + ) «) 5 «) 7 «5) (,) «6) «7) k 6 and k «8) i) A(, ), C(, ) ii) R(, ) iii) = iv) v) units vi) Proof vii) ( ) ) ( ) ( D ii) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

32 GPCC9 «9) i) ii) e ( ) sec sec tan iii) ( ) «) ln( e ) c «) 6 «) Proof «) «) 56 «5) i) log m «6) log n ii) «7) dp «8) since the function is increasing. dt d P since the function is increasing at a dt decreasing rate. 8 7 «9) i) ii) 95 9 «) i) (, ) and (, ) ii) (, ) is a minimum turning point and (, ) is a maimum turning point (, ) (, ) 6 «5) ( ln ) cubic units «6) 7 «7) i) = t ln( + t) + ii) t can never be zero v will never be iii) ms 9 5 «8) i) Proof ii) A, 6 6 «9) i) Proof ii) A = $8 (5) $P( ) ii) $ «) i) 76 L ii) hours and 8 minutes «) i) Proof ii) tan «) i) Proof ii) m (to d.p) iii) No. The maimum area found in part (ii) created with a radius of 975 m would not be possible with an angle less than. is required to be radians (5 to the nearest degree). km/h 8 6 st train nd train time «) i) t = 55 minutes ii) 5 kilometres «) i) Proof ii) ( ln ) + constant iii) 8 units 7 «5) i), ii) Minimum turning point at 7. Maimum turning point at. iii) 7π 7π e (, ) (, 6) iii) Proof iv) (, ) (6, ) «) Proof «) + = «) 86 «) i) At A, = and at B, = 5 ii) square units 6 e (, ) iv) One «) 5 7 «) «) 5( )( + + ) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

33 GPCC9 «) 5 «5) i) = 5 ii) 5 6 «6) i) 5 ii) ( sin cos) e iii) sin «7) i) ii) (, ) iii) Proof iv) B(, ) v) vi) Proof «8) «9) ln( 5) c «) + e = «) i) 9 cm (to s.f) ii) 9 «) i) r = 75, a = 6 ii) 6 iii) 6 «) i) k k + ii) k k + = (k ) for all k «) i) Proof ii) 5 iii) 675 iv) 5 «5) i) 5 cm (to nearest cm ) ii) 59 cm (to s.f) iii) cm (to nearest cm ) «6) i) 6 8 f() ii) «7) i) A = ii) (to s.f) iii) 5 «8) i) ( + ) ii) (, ) and (, ) iii) (, ) is a relative maimum and (, ) is a relative minimum (, ) 5 5 (, ) 5 5 = iv) ( 5, ) (, ) v) = 5 and = «9) (7 e e ) cubic units 7 7 «) i) ii) iii) «) i) v = 5 98t ii) t = 5 seconds iii) 65 metres iv) seconds «) «) i), ii) Proof iii) 9 square units «) i) Proof ii) 9 seconds «5) i) ii) Proof ii) 75 cm «6) i) ii) Proof iii) r = cm «7) i) Proof ii) = f() π iii) Two solutions iv) «8) i) A = cos = sin 6 ( 55 ) M 55 ii) $775 iii) $9 676 iv) 69 months «) (a )(b ) «) 7 «) a =, b = «) 9 «5) 5, 9 «6) i) 9 8k ii) k 8 9 «7) a) b) Proof c) d) 8 square units 5 e) Proof f) Yes g) D( 6, ) h) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

34 GPCC9 π «) i) 5 ii) 5 = e «8) «9) i) tan c ii) e c «) loge «) = «) i) cos + sin ii) «) i) ii) 7 «) i) ii) «5) «6) i) Proof ii) r «7) i) 6 6 ii) (, 7) and (, ) iii) (, 7) is a maimum turning point and (, ) is a minimum turning point (, 7) iv) (, ) «8) i) Proof ii) 8 (, 9) (, ) square units «9) Proof «) i) k = 869 (to 5 d.p) ii) 89 hours (to d.p) iii) Proof «) i) t = or t = 5 ii) The distance travelled during the third second iii) The particle changes direction after it comes to rest at t = and begins to move back towards its initial position. Hence the particle is further from its initial position at t =. «) i) Proof (, ) = cos «) i) ii) solutions «5) log e = «6) i) Proof ii) «7) i) Proof ii) 5 seconds «8) i) Proof ii) (to d.p) «9) i) units to the right of the origin ii) Proof iii) It approaches zero «) i) From the graph f() =, hence = f() has a stationar point at =. Also, from the graph, f() < for, hence = f() is decreasing everwhere but at =. This means = is a stationar point of infleion = f() (, ) ii) 5 «) i) ii) Proof iii) iv) «) i) 5 ii) «) «) 7 «) = 6 «) c «5) 5 ii) iii) square units (, ) = cos «6) EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-

35 GPCC9 C B E A «7) a) b) c) d) e) Proof f) i) units ii) ( ) + ( ) = 8 «8) i) 8( + ) ii) 6 sin + cos e e iii) «9) i) ( ) ii) ( e 9 e ) «) ln ( ) c «) i) a =, d = 5 ii) 95 «) i) ii) «) i) = ii) iv) m/s v) t sec «) i) ii) Proof iii) $8 9 «) 67 (to d.p) = tan t iii) sin t «) i) ii) Proof iii) 6 units «) i) 67 mm ii) 5 cm «5) i) Local minimum at (, ), local maimum at (, ) ii) (,) (, ) (, ) = sin «) i) ii) solutions «) i) ii) 66 cm 5 «5) i) ii) iii) Proof iv) hour 8 minutes «6) Proof 5 «7) i) ii) Proof iii) 5 (, ) iii) (, 8) «6) = «7) ii) Proof iii) 5 cm «8) 65 units «9) i) ml ii) e 6 ml/hr iii) Proof M iv) iv) t EDUDATA SOFTWARE PTY LTD:995- BOARD OF STUDIES NSW 98- CSSA NSW 98-