STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE MATHEMATICS. Time allowed Three hours (Plus 5 minutes reading time)

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1 STRATHFIELD GIRLS HIGH SCHOOL TRIAL HIGHER SCHOOL CERTIFICATE 00 MATHEMATICS 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. Marks may be deducted for careless or badly arranged work. Standard integrals are supplied. Board-approved calculators may be used. Each question attempted is to be started on a new page clearly marked Question, Question, etc. You may ask for etra paper if you need it. Diagrams are. Students are advised that this is a Trial Eamination only and cannot in any way guarantee the content of the HSC Eamination

2 QUESTION Marks a) Epress 0.8 as a fraction in simplest form (b) Solve and graph the solutions on a number line (c) Evaluate 8 correct to decimal places (d) Find the eact value of sin 60 tan 0 cos 60 (e) Find a primitive function for cos

3 QUESTION (Start a new page) Marks (a) Differentiate with respect to 5 i) log e i cos e (b) A 65 B ABCD is a rhombus which has 4 AB 65mm and BX 60mm 60 as shown D X C i) Eplain why i BXA 90 Find the length of AX State the length of BD giving reasons. (d) In an eperiment the following values of f ( ) were recorded:.5 f().5.7 Use the trapezoidal rule to find an approimation for f ( ) d giving your answer to significant figures.

4 QUESTION (start a new page) Marks B y. C(,5) 6 A (a) i) Find the gradient of the line AB. Show that the equation of the line AB is y 6 0 i Find the length of AB. iv) Find the perpendicular distance of C from AB. v) Hence find the area of the triangle ABC. vi) There is a point D such that ABCD is a parallelogram. Give the coordinates of the point D. (b) The area under the curve y e from =0 to = is rotated 4 about the -ais. Find its volume to decimal place.

5 QUESTION 4 (Start a new page) Marks (a) Evaluate 0 sec d (b) The second term of a geometric series is 4 and the fifth term is i) Find the common ratio, r Find the first term, and hence, the sum to infinity (c) In an arithmetic sequence the tenth term is 58 and the sum of the first 0 terms is 55. i) Find the first term Find the common difference (d) The graph shows the gradient function y ' f y - ' f ( ) Copy this graph onto your answer page and below it sketch y f ( ) showing all essential features. (e) The number, N, of koalas in a certain area was studied over a period of time, t. At the beginning of the study there were 450 koalas. dn dt i) During this period, 0. What does this say about the number of koalas over this time? d N During this period 0. dt Sketch a graph showing the number of koalas against time.

6 QUESTION 5 (Start a new page) Marks (a) If and are the roots of the equation 7 6 0, find the value of i) and (b) Find all the values for if sin for0. (c) 0 B 8 A C 6 4 D i) Prove that DCA is similar to BCD. Hence find the value of. 7 (d) If cos and sin is negative find tan as a fraction 5

7 QUESTION 6 (Start a new page) Marks (a) Si books labeled A, B, C, D, E and F are placed at random on a shelf. What is the probability that book A or book E is placed third from the left? (b) A drawer contains 5 red socks, white socks and black sock. Ms Felicietti takes out socks and puts them on in the dark 7 i) Construct a probability tree showing this information i iv) Calculate the probability that the first sock taken out is red Calculate the probability that Ms Felicietti s socks are the same colour Calculate the probability that Ms Felicietti is wearing odd socks. (c) i) Sketch the curve y cos( ) in the domain 4 showing the main features of the graph. Use your sketch to determine the number of solutions to the equation cos( ).

8 QUESTION 7 (Start a new page) Marks (a) For the curve y 4 find 9 i) where the curve cuts the -ais any stationary points and determine their nature i any points of infleion iv) hence, sketch the curve for the domain v) for what range of values is the curve concave up and decreasing. (b) For what values of k does the quadratic equation, k 8 k 0 have real roots? QUESTION 8 (Start a new page) Mark (a) The region in the first quadrant enclosed by the curve y the y-ais and the line y and y 9 is shown below. Find the area of the region. y 9 0 (b) Consider the sketch of y sin and eplain why 0 sin d 0 y 0 -

9 QUESTION 8 (Continued) Mark (c) Solve log log 8 (d) $, 880 is paid into a superannuation fund at the beginning of each year. Interest is paid at the end of each year at the rate of 8 %p.a. compounded annually. Find the amount in the fund at the end of 5 years. (e) y P - For the shaded area shown i) Show that (, ) is one point of intersection P of and y. y Hence, calculate the shaded area.

10 QUESTION 9 (Start a new page) Mark (a) For the parabola y find 4 i) the focus the directri i the equation of the tangent to the parabola at (b) If log. 5 and log. evaluate i) log log (c) The diagram shows a right square pyramid. The height (h) 5 plus the side length (s) equals 0 cm. i) Show that the volume of the pyramid is given by V s ( 0 s ) Find the value of s which maimizes the volume of the pyramid. h s i Calculate this maimum volume.

11 QUESTION 0 (Start a new page) Mark (a) Simplify sin sin (b) A garden bed is in the shape of a circle with a segment removed a shown. The circle has centre O and radius 5metres. The length of the straight edge AB is 5 metres. Find the eact area of the garden bed. 5 A O. B (c) A farmer buys a new tractor for $ The interest rate is % p.a. compounded monthly. If the farmer wishes to repay the loan in half yearly instalments, calculate the amount of each instalment if he is to repay the loan in 0 years. 5 END OF EXAM