Catholic Schools Trial Examination 2004 Mathematics

Transcription

1 0 Catholic Trial HSC Examination Mathematics Page Catholic Schools Trial Examination 0 Mathematics a If x 5 = 5000, find x correct to significant figures. b Express in the form b a, where a and b are integers. c Solve tan α =, for 0 o α 60 o, giving the answers to the nearest degree. d Simplify - a b a + b e Solve: 8 x =, leaving the answer as a fraction. f Find the integers a and b such that a Write down the derivative of: (i) (x + 4) 7 b (i) xe x tan = a + b Write down the primitive of e x + x x 4 + Find the exact value of dx. x dy Given that = x sin x and y = when x = 0, find y in terms of x. dx a For what values of a will ax + + a be positive definite? b k Find the values of k if ( x + ) dx = 6. c The points A, B and C have coordinates (, 5), (6, 0) and (5, 7) respectively. Plot these points on a number plane. Hence: (i) Show that the length of AB is 5. Show that the triangle ABC is isosceles by finding the length of BC. Find the equation of the line AB. (iv) BA is produced to meet the line y = 7 at P. Show that P has coordinates (-, 7). (v) Find the area of triangle PAC.

2 0 Catholic Trial HSC Examination Mathematics Page 4a In the diagram, AB = 9cm, BC = 6cm, AD = 8cm, AC = cm and ABC = DAC. 5a (i) Prove that ABC CAD, giving clear reasons. Hence, find the value of side CD. 4b A parabola whose equation is y = ax, where a is a constant, has the line y = x + as a tangent. (i) By equating the two given equations, find a quadratic equation in terms of x and a. By using the discriminant of the quadratic equation found, find the value of a. Find the coordinates of the point of contact between the tangent and the parabola. (iv) Sketch the parabola and the tangent line, showing the coordinates of the point of contact. 5a Here is the graph of y = sin x, for 60 o x 60 o. One solution of the equation sin x = 0.5 is x = -0 o. Find the other solutions of this equation for 0 o x 60 o. 5b (i) On the same graph, sketch the curves y = sin x and y = - sin x for 0 x π. Give three solutions to the equation sin x + sin x, for 0 x π. 5c A radioactive substance decays at a rate proportional to the mass present. The rate dm of change is given by = -km, where k is a positive constant and M grams is the dt mass present at any time t hours. (i) Show that M = M o e -kt is a solution to this equation. If 00 grams of this substance decays to 80 grams in 0 hours, find: (α) The value of k, correct to decimal places. (β) The mass present after a further 0 hours, to the nearest gram. The half-life is the time taken for 00 grams of this substance to decay to 50 grams. What is the half-life of this substance? Give your answer to the nearest hour.

3 0 Catholic Trial HSC Examination Mathematics Page 6a (i) The curve y = x + ax + 7x 5 has a stationary point at x =. Find the value of a, and hence: Find the coordinates of all stationary points. Determine the nature of the stationary points. (iv) Sketch the curve and then determine for what values of x is the curve increasing. 6b If the K th term of an arithmetic series is L and the L th term is K: (i) Show that L = a + (K )d. Find another expression for K. By solving the two equations found, show that d = -. (iv) Hence, find the first term of this series in terms of L and K. 7a The diagram shows the area bounded by the graph y =, (for x > 0), x the x axis and the lines x = and x =. 5a (i) Find the shaded area. Leave your answer as a fraction. Find the volume of the solid formed when the shaded area is rotated about the x axis. Leave your answer in exact form. 7b (i) Sketch the graph of f(x) = e x for all values of x in the domain and state its range. The curve f(x) = e x is rotated about the y axis to give a solid. Show that the volume V y of the solid formed, from y = to y = 5, is given by 5 V y = p (ln y) dy. Use Simpson s rule with 5 function values to find the volume of this solid, correct to significant figures. 8a A certain soccer team has a probability of 0.6 of winning a match and a probability of 0. of drawing a match. (i) If this soccer team plays two matches, draw a tree diagram to show all possible outcomes. Find the probability of this soccer team winning at least one match out of the two matches. Find the probability of this soccer team not winning either of the two matches.

4 0 Catholic Trial HSC Examination Mathematics Page 4 8b The figure shows a sector of a circle OAB, centre O, with its arc joining the points A(5, ) and B(, 5). Copy this figure into your answer booklet. 5a (i) Find the value, in degrees, of one radian to the nearest minute. Show that the size of AOB is 0.78 radians, correct to decimal places. Calculate the perimeter of sector OAB, correct to decimal places. 9a A box is made from a 50cm by 0cm rectangle of cardboard by cutting out four equal squares of side x cm from each corner as shown. The edges are turned up to make an open box. 5a (i) Show that the volume V of this box is given by the equation V = 4x 40x + 000x (cm ). Find the value of x, correct to one decimal place, that gives this box its greatest volume. Hence, find the maximum volume of this box, correct to decimal places. 9b Jordan has to pay annual instalments for his superannuation at the beginning of r each year according to the formula: M n = + M n-, n, where r(%) is the 00 annual rate of interest paid by the fund and M n is the instalment at the beginning of the n th year. If the interest rate is % p.a., compounded yearly, and Jordan s first instalment is $500, find: (i) How much is his second instalment? Find the amount Jordan has to pay into the fund at the beginning of the 0 th year. Find the total value of his investment after 0 years. 0 dv The acceleration of a particle at any time t seconds is given by = k, where k is a dt a constant.

5 0 Catholic Trial HSC Examination Mathematics Page 5 0 b (i) How that v = kt + c, for some constant c. The displacement x metres, at any time t seconds, is shown in the table: t (sec) 0 Show that x = t t + x (metres) 9 Find when the particle comes to rest. The diagram shows a triangle ABC, and CD is perpendicular to AB. It is given that AB = a, AC = b, ACD = β and BCD = α. (i) By using triangles ACD and BCD, show that h = b cos β = a cos α. Show that the area of triangle ACD is equal to ab sin β cos α. Find another expression for the area of triangle BCD in terms of a, b, α and β. (iv) Show that the area of triangle ABC is equal to ab sin(α + β). (v) Hence, but not otherwise, deduce that: sin(α + β) = sin α cos β + cos α sin β. 9 A a b. c. 7 o or 5 o d. 0 x e x (x + ) sec tan b a + b x e b.(i) e. 5 f. a =, b = a.(i) (x + 4) 6 + x + c + ln y = x + cos x + 4 a. a > 5 b. k = or -5 c. x + y 6 = 0 (v) 6 units 4a. 6cm 4b.(i) ax x = 0 a = - (-0.5, -) 5a.(i) 0, 50 o, -0 o, -0 o 5b. 0, π, π 5c.(α) (β) 7 grams 6 hours 6a.(i) a = -5 (, -) and ( 7, -.8) max(, - ) and min( 7, -.8) (iv) x < or x > 7 6b. K = a + (L )d (iv) a = L + K 7a.(i) units 6π units 7b.(i) y > 0 units 8 8a b.(i) 57 o units 9a. x = cm 9b.(i) $560 $406.8 $ a. seconds 0b. ab sin β cos α