Catholic Schools Trial Examinations 2007 Mathematics. as a single fraction in its simplest form. 2

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1 0 Catholic Trial HSC Eaminations Mathematics Page Catholic Schools Trial Eaminations 0 Mathematics a The radius of Uranus is approimately m. Write the number in scientific notation, correct to two significant figures. b Given that = 5 + evaluate +. Leave your answer in eact form. c Epress ( a ) 4 - ( a ) as a single fraction in its simplest form. d Find a primitive of 5 + cos, e Find the values of for which. f Solve the pair of simultaneous equations. + y = 6 6 y = -8 a In the diagram, ABCD is a quadrilateral. The coordinates of A, B, C and D are (, 4), (6, -), (0, -5) and (, 0) respectively. Calculate the length of BC. Leave your answer in eact form. Show that the gradient of AB is - 5. (iii) Find the -coordinate of D such that AB is parallel to DC. (iv) Show that the equation of the line AD is y + 6 = 0. (v) (vi) Find the perpendicular distance from C to AD. Leave answer in eact form. Hence, or otherwise, find the area of the quadrilateral ABCD. b The diagram shows a quadrilateral JKLM, in which JK is parallel to ML, JM = JK, KM = KL and KLM = 4 o. Copy or trace this (iii) diagram into your writing booklet. Find the size of LKM. Give reasons for your answer. Hence, or otherwise, find the size of MJN. Give reasons for your answer. a Differentiate with respect to :

2 0 Catholic Trial HSC Eaminations Mathematics Page y = (e + ) 4 f() = tan b Find: d 5 e d. Leave your answer in eact form. 0 c Find the value of p in eact form. d Given that sin θ = and tan θ < 0, find the eact value of cos θ. 5 4a An infinite geometric series has a limiting sum of 4. If the first term is 5, find the common ratio. 4b In the diagram, CD is parallel to AB and DE is parallel to CA. AC = 5cm, AB = cm, CD = 8cm, BE = cm. Prove that ABC is similar to triangle DCE. Hence find the length of BC. 4c Find the equation of the normal to y = e cos π at the point where =. 4d Solve tan θ + = 0 for 0 θ π. 4e α and β are the roots of = 0. Find the value of 4αβ + 4α β. 5a Consider the function f() = 8. Find the coordinates of the stationary points of the curve y = f() and determine their nature. Sketch the curve, clearly labeling any stationary points and the y-intercept. (iii) For what values of is the curve decreasing?

3 0 Catholic Trial HSC Eaminations Mathematics Page 5b Two urns contain green and red marbles. Urn A contains 6 green and red marbles while Urn B contains 4 green and 7 red marbles. A person selects either of the urns at random and chooses two marbles without replacement from their selected urn. Determine the probability that the person chooses green marbles from Urn B. Determine the probability that the person selected at least one green marble from Urn B. 5c The diagram shows the cubic y = and the line l with equation y = 0 for 0 intersecting at the point P, in the first quadrant. Q is the -intercept of the line l with coordinates (0, 0). Show that the coordinates of the point P are (, 6). 6a By considering the sum of two areas, find the shaded area. 4 Evaluate w 4. Leave your answer in eact form. w = 0 6b With the drought ever worsening, James and Theodore design a counting generator that can simulate the number of rain drops per minute that fall over a river during a storm. The rain drops falling per minute forms the series with the nth term given the formula R n = n + n. Verify that 5 is a term of this series. Find the total amount of rain drops which fall over the river in the first twenty five minutes. (iii) If the surface area of the river is 50m find the average rainfall per cm over the first twenty five minutes. 6b Copy and complete the following table into your writing booklet, correct to decimal places where necessary, for the curve y = log e. Use Simpson s rule with 5 function values to find an approimation to 4 log e d.

4 0 Catholic Trial HSC Eaminations Mathematics Page 4 6c A set of three concentric circles with centre O and radii.5cm, 9cm and.5cm is drawn opposite. The outer arc length of AB is.5cm. Calculate the size of the refle angle AOB. Give answer in radians. Calculate the area of the shaded region. 7a The population P of Sydney rose from million at the beginning of 980 to.7 million at the beginning of 990. Assuming natural growth, P = 0 6 e kt, where t is the time in years since the beginning of 980 and k is a constant. dp Show that = kp satisfies the equation 0 6 e kt. dt Find the value of k, correct to three decimal places. (iii) Find the population of Sydney at the beginning of 0, correct to three significant figures. (iv) The population of Sydney s sister city Guangzhou in China at the beginning of 0 was 8.5 million. In what year will the population of Sydney reach the population of its sister city Guangzhou? 7b Another sister city of Sydney is San Francisco. Sydney City Council decodes to build an art gallery in San Francisco to allow local Sydney artists to ehibit their work. The loan required to build the art gallery is $P with interest charged at an introductory rate of 6% p.a. for the first three months. The loan is to be initially repaid in equal monthly repayments of $4000 over years and interest is charged monthly before each repayment. Let $A n be the amount owing by Sydney City Council at the end of the nth repayment. Find an epression for A. Show that A = P(.005) 4000( ). At the end of three months interest rates rise to 9% p.a. and the loan is to be repaid in total in equal monthly repayments of $4800 for the net.75 years. (iii) If the loan s interest rate is fied at 9% p.a. for the remainder of the loan, find the value of $P. sin 8a Evaluate lim. 0 8b The displacement of a particle is given by: = t 4log e (t ) + 5, t > where is in metres and t is in seconds.

5 0 Catholic Trial HSC Eaminations Mathematics Page 5 Find the eact displacement of the particle when t = 4. Find an epression for v and hence find when the particle comes to rest. (iii) Show that the acceleration remains positive for t >. (iv) Find the distance traveled by the particle between the times the particle comes to rest and t = 4. 8c The shaded region bounded by the graph y = 9a Consider the function g() = for <, the line y = and the y-ais is rotated about the y-ais to form a solid of revolution.. Show that g() is an even function. State the domain of y = g(). 9b Consider the trigonometric function y = cos. State the amplitude of y = cos. Show that the volume of the solid is given by: V y = π + dy. y y Find the volume V y of the solid formed. Draw a neat and accurate graph of y = cos for 0 π. (iii) On the same diagram accurately draw the graph of y = +. Hence determine the number of solutions to the equation + cos = 0 over the domain 0 π. 9c Helen is training to complete a mini triathlon. The course she practices on consists of three legs which starts at S and finishes at F. The first leg is a straight line swim from S to a point X. The second leg is a bike ride from X to Y along a straight road OY and the final leg is a jog from Y to F around a circular path. The perpendicular distance from S to O is km while the distance OY is 4km. Helen can swim at 6km/h, bike ride at km/h and jog at 8km/h. If the distance OX = a km, show that the time T that it takes Helen to

6 0 Catholic Trial HSC Eaminations Mathematics Page 6 0 a 4 a 4 (8 ) complete the three legs is given by T = + a + + π hours. 4 Find the value of a that will allow Helen to minimize the time taken to complete the three legs of her practice course. A rural water dam is to be emptied by means of a control valve. The valve operates so that the volume of the water, V litres, remaining in the dam varies with the time, dv t minutes, according to the equation = -bt, where b is a constant. dt Initially the dam contains litres of water. Show that after t minutes V = bt. 0 b If b = 0.4, at what rate will the dam be emptying when V = litres? Show that BD = 5 cm. Hence by using triangle DBE, prove that the perpendicular distance of B from π the line XY is 5 sin ( + θ). 4 (iii) By using triangles DAG and BFA, find an epression for the length of FG. (iv) Hence, prove that sin θ + cos θ = sin ( 4 π + θ). A a ( a + ) m b. 5 c. d. 5 + sin + c e. - 5 f. =, y = 5 a. b.(iii) = - (v) (vi) 4u b. 96 o (iii) 84 o a. 4e (e + ) tan sec b. loge ( 4 6 5) + c - c. 9 cm d. - 4a. 4b. 8 tan e π 5π π c. y = - 4d., 4e. -4 5a. ma(-, ), min(, -7) (iii)- < < b. 5c. 7unit 6a b (iii) 6.8 drops/cm c..77, 5.77, 9.888, 5.46, d..8 rads cm 7a (iii) 4.8 million (iv) 00 7b. A = P(.005) (iii)$ a. 8b. 4ln t = (iv) 4ln 8c. 0.57units 9a. all reals ecept =, - 9b. (iii) 9c. 0a L/min 0b.(iii) FG = 5(sinθ + cosθ).