ascham2011-2u - Mathematics Trial Examination 2011 Tlme...

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Unformatted text preview: Mathematics Trial Examination 2011 Tlme Allowed: 3 hours plus 5 minutes' reading time _ Total marks - 120 Instructions 0 All questions may be attempted. o All questions are of equal value (12 marks). 0 All necessary working should be shown in every question. 0 Write using black or blue pen. 0 Board «approved calculators and templates may be used. 0 A table of standard integrals is provided at the back of this paper. 0 Start each question in a new booklet. o If you use a second booklet for a question, place it inside the first and indicate on your first booklet that it is the first of two booklets. 0 Write your name/number, teacher’s name and question number on each booklet. QUESTION 1 Find the exact value of cos(-5-—”—). a ) 6 b) Solve for x: i) Ix-3| < 4. ii) x2 2 6x. iii) 2—_—"‘1 :5. 10 5 iv) 21nx=ln(2x—l). c) Find the focus of the parabola (x —3)2 2 8y. d) Write down the equation of a circle with centre (-1, 2) and radius 5 units. ' QUESTION 2 a) Differentiate with respect to x: i) x2 cosx ii) 3x ~l x iii) In 2. b) Integrate with respect to x: i) 3 er .. 3): u . . ) x“ — 4 l c) Evaluate to 2 significant figures: sin —. J? 4 d) Evaluate I |x— 2[ dx . l (1) (2) (2) (2) (2) (2) (1) (2) (2) (1) (2) (2) (1) (2) QUES'HON 3 a) b) A is the point (6,3), 8 is (3,4) and C is (4,—3) Draw a set of axes and put in the above points. Your diagram need not be to scale. i) Show that AB and AC are perpendicular. ii) Without using a protractor, find the size of angle AC3 to the nearest minute iii) If D is the point (7, — 4), show that ABCD is a parallelogram. iv) Find the area of the parallelogram ABCD. v) If AC and BD intersect at E, find the coordinates of E. The gradient function of a curve is 65in 2x. Find the equation of the curve if it passes through the point (~33). QUESTION 4 a) b) d) :4 . Evaluate 2(4r - 7). r:4 On a set of axes, shade the region represented by 2x —— y + 6 2 0 . A circle has a circumference of 12” cm. If an angle of Z;— is subtended at the centre of the circle, find the exact area of the minor segment. The third term of a geometric series is 36 and the eighth term is i . Find the first term of the series. 27 (3) (3) (2) (1) (1) (2) (3) (2) (3) (4) QUESTION 5 a) For the curve y = x3 — 3x2 —9x + 2, i) find any stationary points and determine their nature, (4) ii) find any points of inflexion, (2) iii) sketch the graph showing all significant points, (2) iv) for what values of x is the curve decreasing with downward concavity? (1) b) Not to scale In the diagram above, P5 = 1.2, SQ = 0.8, PT = 0.8 and TR = 2.2. Draw the diagram in your answer booklet. i) Prove that A PST is similar to A PRQ . (2) ii) If ST = 1.7, find the length of QR. (1) QUESTION 6 a) Use Simpson’s Rule with 5 function values, to approximate to 3 decimal places: I log((x —- 3) dx . (3) b) A substance has a half-life of 8 days. Its rate of decay is given by the differential equation d—M = —kM . dt i) Show that M = M0 e“ is a solution to the differential equation above, where M is the mass of the substance present at time t days. (1) ii) Show that k = “1% . (2) iii) How long will it take for 75% of the substance to have decayed? (2) c) The equation x2 + (m —3)x+(m _— l) = 0 has one root that is half the other. Find the values of m and the values of the two roots. (4) QUESTION 7 a) i) Draw the graph of y = tan; for -7r 3 x 3 3n. (2) ii) Find the equation of the normal to the graph y = tang- when x = (3) iii) As accurately as possible, sketch this normal on to the same set of axes as the graph in i), showing the coordinates of the points of intersection with the axes. (2) b) The position of a particle on a number line is given by x = 25in t +t‘f3- where t is measured in seconds and x in metres. i) When is the particle at rest for 0 S t 3 Zn (2) ii) How far has the particle travelled in the first nseconds? Give an exact answer in simplest form. (3) QUESHON 8 3) ii) iii) iv) b) The graph of velocity vs time for a particle is shown below. Not to scale When is the particle at rest? When is the particle moving to the right? When is the acceleration of the particle equal to zero? is the particle speeding up or slowing down when t = 6. Explain. i) By solving an equation, show that the graphs y = e" and y = e“ —2 intersect when x = ln2. ii) Draw the two graphs in part i) on the same set of axes, showing intercepts on axes. iii) Find the exact area bounded by the graphs y = c" and y 2 e3" —2 and the line x = 0. (1) (1) (1) (1) (2) (2) (4) QUES‘HON 9 a) The curve y = —x3 + 6x —8 is given below. i) Find the points of intersection with the x axis (1) ii) The equation of the parabola can be rearranged as x = 3 i 1 ~— y . Find the exact volume created when the area between the curve the x axis, x = 3 and x = 4 is rotated about the y axis. (4) V “at. AA_ b) Charlotte decides she would like to buy a car. She opens a savings account on 1 January 2010 by investing $2000 in the Bell Bank. She continues to invest $2000 at the start of each year thereafter. She receives interest at the rate of 6% per annum compounded annually. i) Show that the value of her account at the end of 2013 is $9274.19. (3) ii) 0n 1 January 2014, the interest rate goes up to 8% per annum and she receives this new interest rate on all money in her account. How much does she have in her account at the end of 2020 if she continues her original plan of investing $2000 at the start of each year? (4) QUESTION 10 a) For what values of m will the curve y = +2; —4x+ 7 be decreasing for all x? (3) b) Towns P, Q and R are located 6km due west, 6km due east and 10 km due south respectively, of a town 5. A road is to run due north from R to a point A which is x km from R. From A, a branch road is to run to P and another branch road to Q. Copy the diagram below into your answer booklet and fill in the data. i) Show that the total length L of roads AP, AQ and AR is given by L=x+2 x3-20x+l36 (2) ii) Find the exact value of x such that L is a minimum. (5) iii) Show that the minimum total length for the three roads AP, AQ and AR is (10+6J3) km. (2) END OFEXAM Ix‘dx =——l—x“', n¢—1;x¢0,ifn<0 x a: 1 OJ je dx =——e , 0:0 0 1 . Icosaxdx -—smax, a¢0 a 1 Ismaxdx ———cosax, a¢0 a 2 l Isec axdx =—tanax, a¢0 a l Isecaxtanaxdx =—secax, a¢0 a I 21 2dx =ltan‘li, 0:0 0 +x 0 a . '-1x sm —, a>0, —a<x<a a 1 ——-dx J 1 ’9 \ =ln(+ x““02y x>a>0 \ 1 ("' _ =ll’l‘+ 1324-02] NOTEzlnx :logex, x>0 ...
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