HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 2/3 UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time)

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1 HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS / UNIT (COMMON) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions. ALL questions are of equal value. All necessary working should be shown in every question. may be deducted for careless or badly arranged work. Standard integrals are printed on page. Board-approved calculators may be used. Answer each question in a SEPARATE Writing Booklet. You may ask for etra Writing Booklets if you need them. 584

2 QUESTION. Use a SEPARATE Writing Booklet. (a) Epress as a recurring decimal. (b) Simplify 5 8. (c) A coin is tossed three times. What is the probability that heads appears every time? (d) Find a primitive of + 7. (e) Find the eact value of sin π π sin. 4 + (f) By rationalising the denominators, epress + in simplest form. + (g) A merchant buys tea from a wholesaler and then sells it at a profit of 7 5%. If the merchant sells a packet of tea for $ 08, what price does he pay to the wholesaler per packet of tea?

3 QUESTION. Use a SEPARATE Writing Booklet. (a) Differentiate the following functions: 6 ( + 4) 5 ( ) sin + tan (iii). (b) Evaluate the following integrals: 4 d 4 e d. 0 (c) Find d. +

4 4 QUESTION. Use a SEPARATE Writing Booklet. y C NOT TO SCALE O A θ B The diagram shows points A(, 0), B(4, ) and C(, 6) in the Cartesian plane. Angle ABC is θ. Copy or trace this diagram into your Writing Booklet. (a) Show that A and C lie on the line + y =. (b) Show that the gradient of AB is. (c) Show that the length of AB is 0 units. (d) Show that AB and AC are perpendicular. (e) Find tanθ. (f) Find the equation of the circle with centre A that passes through B. (g) The point D is not shown on the diagram. The point D lies on the line + y = between A and C, and AD = AB. Find the coordinates of D. (h) On your diagram, shade the region satisfying the inequality + y.

5 5 QUESTION 4. Use a SEPARATE Writing Booklet. (a) The following table lists the values of a function for three values of f() Use these three function values to estimate Simpson s rule f( ) d by: the trapezoidal rule. (b) The third term of an arithmetic series is and the sith term is 7. Find the common difference. Find the sum of the first ten terms. (c) The first term of a geometric series is 6 and the fourth term is 4. Find the common ratio. Find the limiting sum of the series. (d) The equation of a parabola is = 8 y+. ( ) Find the coordinates of the verte of the parabola. Find the equation of the directri of the parabola.

6 6 QUESTION 5. Use a SEPARATE Writing Booklet. (a) D 8 E C A B The diagram shows a regular pentagon ABCDE. Each of the sides AB, BC, CD, DE and EA is of length metres. Each of the angles ABC, BCD, CDE, DEA and EAB is 08. Two diagonals, AD and BD, have been drawn. Copy or trace the diagram into your Writing Booklet. (iii) (iv) State why triangle BCD is isosceles, and hence find CBD. Show that triangles BCD and DEA are congruent. Find the size of ADB. Find an epression for the area of the pentagon in terms of and trigonometric ratios. (b) The population P of a city is growing at a rate that is proportional to the current population. The population at time t years is given by where A and k are constants. P = Ae kt, The population at time t = 0 was and at time t = was Find the value of A. Find the value of k. (iii) At what time will the population reach ? 4

7 7 QUESTION 6. Use a SEPARATE Writing Booklet. (a) A particle P moves along a straight line for 8 seconds, starting at the fied point S at time t = 0. At time t seconds, P is (t) metres to the right of S. The graph of (t) is shown in the diagram. 5 O 6 t At approimately what times is the velocity of the particle equal to 0? At approimately what time is the acceleration of the particle equal to 0? (iii) (iv) At approimately what time is the distance from S greatest? At approimately what time is the particle moving with the greatest velocity? (b) The function f( )= e + has first derivative f ( )= e e and second derivative f ( )= 4e 4e. 7 (iii) Find the value of for which y = f () has a stationary point. Find the values of for which f () is increasing. Find the value of for which y = f () has a point of inflection and determine where the graph of y = f () is concave upwards. (iv) Sketch the curve y = f () for 4. (v) Describe the behaviour of the graph for very large positive values of.

8 8 QUESTION 7. Use a SEPARATE Writing Booklet. (a) Write down the discriminant of + + k. For what values of k does + + k = 0 have real roots? (b) Consider the function y = + sin + cos. 5 Find the equation of the tangent to the graph of the function at = 5 π. 6 Find the maimum and minimum values of + sin + cos in the interval 0 π. (c) r 5 θ The diagram shows a sector of a circle with radius r cm. The angle at the centre is θ radians and the perimeter of the sector is 8 cm. Find an epression for r in terms of θ. Show that A, the area of the sector in cm, is given by θ A = ( θ + ). π (iii) If 0 θ, find the maimum area and the value of θ for which this occurs.

9 9 QUESTION 8. Use a SEPARATE Writing Booklet. (a) Sand is tipped from a truck onto a pile. The rate, R kg/s, at which the sand is flowing is given by the epression R = 00t t, for 0 t T, where t is the time in seconds after the sand begins to flow. Find the rate of flow at time t = 8. 7 (iii) (iv) (v) What is the largest value of T for which the epression for R is physically reasonable? Find the maimum rate of flow of sand. When the sand starts to flow, the pile already contains 00 kg of sand. Find an epression for the amount of sand in the pile at time t. Calculate the total weight of sand that was tipped from the truck in the first 8 seconds. (b) y 5 y = log O The diagram shows the graph of y = log between = and = 8. The shaded region, bounded by y = log, the line y =, and the and y aes, is rotated about the y ais to form a solid. Show that the volume of the solid is given by y ln 4 V = π e dy. 0 Hence find the volume of the solid.

10 0 QUESTION 9. Use a SEPARATE Writing Booklet. (a) Solve ln (7 ) = ln. (b) y 4 y = sin π O π 6 π y = cos The diagram shows the graphs of the functions y = cos and y = sin between = π and = π. The two graphs intersect at = 6 π and = π. Calculate the area of the shaded region. (c) D C 6 M X A B In the diagram, ABCD is a rectangle and AB = AD. The point M is the midpoint of AD. The line BM meets AC at X. Show that the triangles AXM and BXC are similar. Show that CX = AC. (iii) Show that 9(CX) = 5(AB ).

11 QUESTION 0. Use a SEPARATE Writing Booklet. (a) A game is played in which two coloured dice are thrown once. The si faces of the blue die are numbered 4, 6, 8, 9, 0 and. The si faces of the pink die are numbered,, 5, 7, and. The player wins if the number on the pink die is larger than the number on the blue die. 5 By drawing up a table of possible outcomes, or otherwise, calculate the probability of the player winning a game. Calculate the probability that the player wins at least once in two successive games. (b) A fish farmer began business on January 998 with a stock of fish. He had a contract to supply fish at a price of $0 per fish to a retailer in December each year. In the period between January and the harvest in December each year, the number of fish increases by 0%. Find the number of fish just after the second harvest in December Show that F n, the number of fish just after the nth harvest, is given by F n = ( ) n. (iii) (iv) When will the farmer have sold all his fish, and what will his total income be? Each December the retailer offers to buy the farmer s business by paying $5 per fish for his entire stock. When should the farmer sell to maimise his total income? End of paper

12 STANDARD INTEGRALS n d n+ =, n ; 0, if n< 0 n + d = ln, > 0 e a d a e a =, a 0 cosa d = sin a, a 0 a sin a d = cos a, a 0 a sec a d = tan a, a 0 a sec a tan a d = sec a, a a 0 a d = a tan, 0 + a a a d = sin, a> 0, a< < a a ( ) > > d = ln + a, a a ( ) d = ln + + a + a NOTE : ln = log, > 0 e 0 Board of Studies NSW 998