Question 1.Marks 3x 2 (a) Find lim (b) Find the obtuse angle between the lines x − y − 1 = 0 and 2 x + y − 1 = 0. (c) Find the general solution to sin θ = (d) When the polynomial function f(x) is divided by x 2 − 16, the remainder is 3 x − 1 . What is the remainder when f(x) is divided by x − 4 ? (e) Solve for x: (f) Find a primitive of x→0 Question 2. tan 5 x . 3 2 2 . 1 − 2x ≥ 1. 1+ x 1 x2 − 9 2 2 3 1 . [START A NEW PAGE] (a) Given the function g ( x) = x + 2 and that g −1 ( x) is the inverse function of g(x), find g −1 (5). (b) (i) Show that: 2 tan x = sin 2 x. 1 + tan 2 x 2 1 π 4 (ii) Hence, or otherwise, find tan x ∫ 1 + tan 2 0 3 (c) Using the substitution u = 1 + x , evaluate ∫ 0 (d) x 2 dx. 5 x 2 + 10 x 1+ x 4 dx. Sketch the graph of the curve: y = 2 cos −1 ( x) − 1, showing all essential information. JRAHS Y12 ME1 Trial, 2008 3 Page 1 3 (c) The velocity v ms. The point Q divides CP.Question 3. 2008 x cos 2 dx. 2 2 Page 2 . 1 (ii) The tangent intersects the y-axis at C. Find the locus of all the Q points as parameter p varies. (i) Show that the equation of the tangent at P is y = px − 2 p 2 . internally. 2 p 2 ) be an arbitrary point on the parabola x 2 = 8 y with parameter p. Find the acceleration of the particle at any position. in the ratio 1 : 3. (a) (b) [START A NEW PAGE] Marks 12 Find the exact value of tan 2 cos −1 . 1005 and 1231 all have something in common. 2 Each is a four-digit number beginning with 1 that has exactly two identical digits How many such four-digit numbers exist? (e) Find ∫ JRAHS Y12 ME1 Trial. 13 2 Let point P(4 p. 2 (d) The numbers 1447.1 of a particle moving in a straight line at position x at time t seconds is given by: v = x 3 − x. 2008 Calculate the number of years that the $200 prize can be awarded. Page 3 1 . 2. From this fund he decided to donate a yearly prize of $200 to be awarded to the Dux of Agriculture in Year 12. C2 Copy or trace the diagram onto your writing booklet and prove that ADFG is a cyclic quadrilateral. numbered 1. (ii) JRAHS Y12 ME1 Trial. The prize money being withdrawn from this fund after the year’s interest had been added. BC produced meets circle C2 at E. 3. (c) (d) A bag contains eleven balls. … and 11. AB produced meets EF produced at G α E D Let ∠ DFE = α . (i) How many ways can the sum of the numbers on the balls drawn be odd? 2 (ii) What is the probability that the sum of the numbers on the balls drawn is odd? 1 When Farmer Browne retired he decided to invest $2 000 in a fund which paid interest of 8% pa. If six balls are drawn simultaneously at random.Question 4. (i) Show that the balance $ Bn remaining after n prizes have been awarded ( ) 3 will be: Bn = 500 5 − 1 ⋅ 08 n . [START A NEW PAGE] Marks 9 (a) 3 Find the term independent of x in the expansion of 2 x 2 − . x 2 G B (b) C A C1 3 F Two circles C1 and C2 intersect at C and D. compounded annually. (a) (b) [START A NEW PAGE] Marks Considering the expansion: (9 + 5 x )29 = p0 + p1 x + p 2 x 2 + K + p k x k + K + p 29 x 29 . the temperature of the water cubes is 60C. Find the time for the water cubes to reach − 10 0 C (correct to the nearest minute). find the largest coefficient in the expansion. 1 Page 4 . show that: (iii) After 5 minutes in the freezer. 9(k + 1) 2 2 An ice cube tray is filled with water which is at a temperature of 200 C and placed in a freezer that is at a constant temperature of − 15 0 C. or otherwise. 1 (i) Use the Binomial theorem to write the expression for p k . where k is the rate constant of proportionality. 2008 W (t ) = 35e − kt − 15. dt 2 (i) Show that: (ii) Hence. r pk +1 pk = 5(29 − k ) . so that W(t) satisfies the rate equation: d [W (t )] = −k [W (t ) + 15]. JRAHS Y12 ME1 Trial. (ii) Show that: (ii) Hence. Find the rate of cooling at this time (correct to 1 decimal place) 2 2. The cooling rate of the water is proportional to the difference between the temperature of the water W (t ) 0 C and the freezer temperature at time t. dt [ ] d W (t )e kt = −15ke kt . 29 [you may leave your answer in the form: 3 a 5 b ].Question 5. 2 1. (a) [START A NEW PAGE] Marks A ball is projected from a point O on horizontal ground in a room of length 2R metres with an initial speed of U ms. 2. … . − 3. 1! f 2 ( x) = and state the zeros of f 2 ( x). and the vertical component of motion is &y& = − g . − 2. (ii) 2! 2 ( x + 1)( x + 2) Hence complete the proof by mathematical induction that the zeros of the polynomial function f n (x) are − 1. R= g (iii) Suppose that the room has a height of 3 ⋅ 5 metres and the angle of projection π is fixed for 0 < α < but the speed of projection U varies.2. 3.1 at an angle of projection of α. 3. for n = 1. is given by: x = Ut cos α . 2. There is no air resistance and the acceleration due to gravity is g ms. that is prove that: f n ( x) = JRAHS Y12 ME1 Trial. (i) Assuming after t seconds the ball’s horizontal distance x metres. 2. Page 5 3 . 2 (b) Prove that: (α) the maximum range will occur when U 2 = 7 g cos ec 2α . … f n ( x) = 1 + + 1! 2! n! where for n = 1: (i) f1 ( x) = 1 + Show that for n = 2: x = x + 1 which has a zero at − 1. 2008 1 1 n! ( x + 1)( x + 2)( x + 3) K ( x + n). 3. show that the vertical displacement y of the ball is given by: y = Ut sin α − 1 2 2 gt 2 . for n = 1.Question 6. 2 (ii) Hence show that the range R metres for this ball is given by: U 2 sin 2α . 2 ( β) 1 the maximum range would be 14 cot α . Given the polynomial function: x x( x +1) x( x +1)K( x + n −1) +K+ . … . K and − n for n = 1. 2 Show that: (ii) Hence show that the volume of water V cubic units displaced by the sphere is given by: V ( h) = (c) (H − h )R . (a) y -r 2 The shaded area of thickness w is rotated about the x-axis to form the volume of a ‘cap’. 3 (b) An inverted cone ABC of height H units with a base radius of R units is filled with water. [START A NEW PAGE] Marks Given the semi-circle equation: y = r 2 − x 2 . OP = r and ME = h. (iii) Hence. or otherwise find the radius of the sphere that displaced the maximum volume of water under the above conditions.Question 7. Not to scale O M B C h P Given: MB = MC = R. w r−w r O x Show that the volume of the solid of revolution V is given by: π V = (3r − w)w 2 . as shown below. 1 (ii) 2n 2 Hence show that: 2n 2n 2n 2n 2n 2n + 3 + K + (2n − 1) = 2 + 4 + K + 2n . 1 3 2 n − 1 2 4 2n THE END J JRAHS Y12 ME1 Trial. MA = H . (i) L−R π 3(L − R ) [3RHh 2 1 ] − (L + 2 R )h 3 . A sphere of radius r units is inserted into the inverted cone so as to touch the inner walls of the cone at P & Q to a depth of h units. AC = L. 4 (i) Write down the binomial expansion of (1 − x ) in ascending powers of x. Q H E A r= where L = H 2 + R 2 . 2008 K L C Page 6 .