Year Trial HSC Examination - Mathematics (U) 008 Question Solve for t: 7 4t >. Simplify: x 5 x +. 6 (c) Solve sinθ = for 0 θ. (d) Differentiate with respect to x: 6 y =. x y = x ln x. 5 (e) Evaluate + dx. x Question y A D P B O C x In the diagram AB = BC and CD is perpendicular to AB. CD intersects the y-axis at P. Find the length of AB. Hence show that the co-ordinates of point C are (, 0 ). (iii) Show that the equation of CD is x + 4y = 6. (iv) Show that the co-ordinates of P are (, ) 0. (v) Show that the length of CP is. (vi) Prove that ADP is congruent to COP (vii) Hence calculate the area of the quadrilateral DPOB. JRAHS 008 Trial HSC Mathematics ( Unit)
Question Find the equation of the tangent to the curve = x 4x y at the point ( ) T,. C D A ABC is a sector with cm B BAC = and AC = AB = cm. Calculate the area of sector ABC. Calculate the area of the shaded region. (c) The equation of a parabola is given as y = x x. Rewrite the equation in the form ( x x ) = b( y + ) o y o where x, y and b are constants. o o 8 Hence write down the focal length of the parabola and the co-ordinates of its vertex and focus. Question 4 The price of gold, $P, was studied over a period of t years. dp Throughout the period of study > 0. What does this say about the price of gold? dt It was also observed that the rate of change of the price of gold decreased over the period of study. d P What does this statement imply about? dt A pool is being drained at a rate of 40t 9600 litres/minute. How long does it take before the draining of the pool stops? If the pool initially held 90000 litres of water, find the volume of water in the pool after 5 minutes. (c) In a hat are 5 red and green jellybeans. Jason reaches into the hat and randomly selects two jellybeans. Find the probability that: both selected jellybeans are red. the chosen jellybeans are different colours. (d) The tangent to the curve y = 4 x at a point P has a slope of. Find the co-ordinates of point P. JRAHS 008 Trial HSC Mathematics ( Unit)
Question 5 Pattern # Pattern # Pattern # The above patterns are made using small sticks. Pattern # requires 6 sticks, pattern # requires sticks and pattern # requires 6 sticks. Write a formula for the number of sticks, U n, needed to construct pattern number n. What is the largest pattern that can be constructed from 00 sticks? (iii) How many sticks would be needed to construct all the patterns from pattern # to pattern #0? Show that the curves y = + x and y x = 7 meet at the point (, ) 4. Sketch, on the same set of axes, the curves y = + x and y = 7 x and shade the region bounded by these curves and the y-axis. (iii) Find the volume of the solid formed when the area bounded by the y = + x, y = 7 x and the y-axis is rotated one revolution about the y-axis. 4 Question 6 Show that the curves y = sin x and y = cos x meet at the point with abscissa x = 6. Sketch, on the same set of axes, the curves y = sin x and y = cos x in the domain 0 x and shade the region bounded only by these curves. (iii) Find the area of the shaded region described in. ABCD is a rectangle with (Diagram not drawn to scale) AB = cm, BC = 9 cm and AM perpendicular to BD. Copy the diagram onto your answer sheet and find the length of BD. Prove that ABM is similar to BDC. (iii) Hence find the length of BM. (Give reasons) JRAHS 008 Trial HSC Mathematics ( Unit)
Question 7 In an experiment, the amount of crystals, x grams, that dissolved in a solution after t minutes was given by: x = 00 kt ( e ) After minutes it was found that 50 grams of crystals had dissolved, hence find the value of k. At what rate were the crystals dissolving after 5 minutes? Give your answer to the nearest gram/minute. Two particles A and B start from the origin at the same time and move along a straight line so that their velocities in m/s at any time t seconds are given by: v = t A + and v B = 8 t Show that the two particles never move with the same acceleration. Write a formula for the position of particle A at time t seconds. (iii) After leaving the origin, find when the two particles are again at the same position. Question 8 Given that f ( x) = x 0 x, show that f ( x) ( 8 x) 5x =. 0 x State the domain of the function y = x 0 x. (iii) Find all the stationary points on the curve y = x 0 x and determine their nature. 5 (iv) Sketch the curve y = x 0 x position of all stationary points. indicating its intercepts with the co-ordinate axes and the JRAHS 008 Trial HSC Mathematics ( Unit) 4
Question 9 Prime land along a foreshore is to be reclaimed and developed as part of a housing estate. A plan of the land to be reclaimed is shown below: Use Simpson s rule with five function values to find the approximate area, expressed in hectares, of the land to be reclaimed. y x The point P ( x, y) moves so that its distance from the point (,0) distance from the point N ( 0,). Show that the equation of the locus of the point ( x y) M is always twice its P, is x + x + y 8y + 9 = 0. Give a geometric description of the locus. (c) Graph y = f ( x) if ( x) Evaluate f ( ) + f ( ) (iii) Solve for > 0 4x for x < 0 f = x + for x 0. a : f ( a) f ( a) =. JRAHS 008 Trial HSC Mathematics ( Unit) 5
Question 0 Abigale decides to save for a car. She creates a savings account into which she deposits $00 at the beginning of each week. The account pays interest at a rate of 0.6% per week calculated on the value of the account at the end of each week. n Show that Abigale s account is worth 6600(.006 ) weeks. dollars at the end of n Find the value of Abigale s account at the end of 5 weeks. Give your answer correct to the nearest dollar. (iii) Find the number of weeks needed for Abigale s account to accumulate $0000. (Give your answer to nearest week) When a ship is travelling at a speed of v km/hr, its rate of consumption of fuel in tonnes per hour is given by 5 + 0.004v. Show that on a voyage of 5000km at a speed of v km/hr the formula for the total fuel used, T tonnes, is given by: 65000 T = + 0v v Hence find the speed for the greatest fuel economy and the amount of fuel used at this speed. (Justify your answer.) 4 [ This is the END of the examination paper < JRAHS 008 Trial HSC Mathematics ( Unit) 6
North 45 0 PQ = 500 m 0 5 diagram not A tourist at point P sees two towers at X and Y on bearings of 40 T and 05 T respectively. The tourist then walks 500 metres due north point Q and now records the bearings as 5 T and 00 T respectively. Show that 50 XP =. sin 5 Show that 50 YP =. sin5 (iii) Find the distance from tower X to tower Y correct to the nearest 0 metres. Curve Sketching Consider the curve given by the equation: ( 6) y = 0.4x x Find the co-ordinates of the stationary points and determine their nature. 4 Sketch the curve ( ) y = 0.4x x 6 in the domain x 0, clearly showing the intercepts with the co-ordinate axes, the stationary points and end points. (iii) What is the maximum value of ( ) 0.4x x 6 in the domain x 0? JRAHS 008 Trial HSC Mathematics ( Unit) 7