PENRITH HIGH SCHOOL MATHEMATICS EXTENSION 013 Assessor: Mr Ferguson General Instructions: HSC Trial Total marks 100 Reading time 5 minutes Working time 3 hours Write using black or blue pen. Black pen is preferred Board-approved calculators may be used. A table of standard integrals is provided at the back of this paper. Show all necessary working in Questions 11 16. Work on this question paper will not be marked. SECTION 1 Pages - 7 10 marks Attempt Questions 1 10 Allow about 15minutes for this section. SECTION Pages 8-13 90 marks Attempt Questions 11 16 Allow about hours 45 minutes for this section. Section1 Section Question Mark Question Mark Question Mark Total /100 1-3 11 /15 15motion /5 % 4-5 1 /15 15poly /7 6-7 13vol /7 15 harder ext /3 8 13graph /8 16complex /6 9 14conic /1 16 harder ext /9 10 14circle /3 Total /10 This paper MUST NOT be removed from the examination room Student Name:
Section I 10 marks Attempt Questions 1 10 Allow about 15 minutes for this section Use the multiple-choice answer sheet for Questions 1 10. 1 Consider the Argand diagram below. The inequality that represents the shaded area is: (A) 0 z (B) 1 z (C) 0 z 1 (D) 1 z 1 Let z = 1+ i and w= + i. The value of 5 iw is: (A) 1 i (B) 1+ i (C) 1 i (D) 1+ i
π 3 The locus of z if arg( z ) arg( z+ ) = is best shown as: 4 (A) (B) (C) (D) 3
4 The diagram shows the graph of the function y = f( x). The diagram that shows the graph of the function (A) (B) y = f( x) is: (C) (D) 4
5 The function y = f( x) is drawn below. y f( x ) x Which of the following is an incorrect statement? (A) y = f( x) has two asymptotes only (B) y = f( x) is continuous everywhere in its domain (C) y = f( x) has exactly one point of inflexion (D) y = f( x) is differentiable everywhere in its domain. 6 The values of the real numbers p and q that makes 1 i a root of the equation 3 z + pz + q = 0 are: (A) p = and q = 4 (B) p = and q = 4 (C) p = and q = 4 (D) p = and q = 4 5
r 7 Let Px ( ) be a polynomial of degree n > 0 such that Px ( ) = ( x α) Qx ( ), where r and α is a real number. Qx ( ) is a polynomial with real coefficients of degree q > 0. Which of the following is the incorrect statement? (A) n r+ q (B) Px ( ) changes sign around the root x = α (C) Let N r be the number of real roots of Pxand ( ) of Px. ( ) Then r Nr n and 0 Nc q. (D) Roots of Pxare ( ) conjugate one another. Nc the number of complex roots x y 8 The eccentricity of the ellipse with the equation + = 1is: 4 3 1 1 (A) (B) 4 (C) 3 4 (D) 9 16 9 A particle of mass m falls from rest under gravity and the resistance to its motion is mkv, where v is its speed and k is a positive constant. Which of the following is the correct expression for the velocity? (A) v g kt = ( 1 e ) k (B) g kt v= ( 1+ e ) k (C) g kt v= ( 1 e ) k (D) g kt v= ( 1+ e ) k 6
10 The region enclosed by the ellipse form a solid. y x + = is rotated about the y axis to 4 ( 1) 1 What is the correct expression for volume of this solid using the method of slicing? (A) V = π 1 y dy (B) V = π 1 y dy (C) V = π 4 y dy (D) V = π 4 y dy 7
Section II 90 marks Attempt Questions 11 16 Allow about hours and 45 minutes for this section Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. In Questions 11 16, your responses should include relevant mathematical reasoning and/or calculations. Question 11 (15 marks) Use a SEPARATE writing booklet. (a) The complex number z is given by z = 1+ 3i (i) Show that z = z (ii) Evaluate z and Arg( z ) (b) Calculate the product of the roots of the following equation in the form a + ib (3+ ) iz (1 ) iz+ (6 i) = 0 (c) Find the complex square roots of 7 6 i giving your answers in the form x + iy, 3 where x and y are real. (d) (i) Express z = 3 + i in modulus/argument form. 3 (ii) Show that z 7 + 64z = 0 (e) The points ABCD,,, on the Argand diagram represent the complex numbers abcd,,, 3 respectively. If a+ c= b+ d and a c= i( b d) find what type of quadrilateral ABCD is. 8
Question 1 (15 marks) Use a SEPARATE writing booklet. (a) Find 3 x dx 1 (Hint: let + x 0 u = x ) 3 (b) By using a suitable trigonometric substitution show that dx 1 = sin x + c 1 x where c is some constant x + 6 A Bx + C (c) (i) Find real numbers AB, and C such that dx + ( x+ 1)( x + 9) x+ 1 x + 9 x + 6 (ii) Hence, find dx ( x+ 1)( x + 9) (d) Using the substitution π x dx t = tan, evaluate 5 3cos 3 + x 0 (e) For n 0 let, 3 π n In = x sin x dx 0 n 1 π n Show that for n, In = n nn ( 1) I 9
Question 13 (15 marks) Use a SEPARATE writing booklet. (a) The area bounded by the curve y x x = 1, the x axis, x = and x = 10 is 4 rotated about the y axis to form a solid. By using the method of cylindrical shells calculate the volume of the solid. (b) The base of a solid is the segment of the parabola x = 4y cut off by the 3 line y =. Each cross section perpendicular to the y axis is a right angled isosceles triangle with the hypotenuse in the base of the solid. Find the volume of the solid. (c) x e 1 Consider the function f( x) = x e + 1 (i) Show that f( x ) is an odd function 1 (ii) Show that the function is always increasing (iii) Find f (0) 1 (iv) Discuss the behaviour of f( x ) as x ± (v) Sketch the graph of y = f( x) 10
Question 14 (15 marks) Use a SEPARATE writing booklet. x y (a) For the ellipse + = 1 4 3 (i) Find the eccentricity 1 (ii) Find the coordinates of the foci S and S 1 (iii) Find the coordinates of the directices. 1 (iv) Sketch the curve showing foci and directices. 1 (v) P is an arbitrary point on this ellipse Prove that the sum of the distances SP and SP is independent of P (b) (i) Find the slope of the tangent to the ellipse x a y + = at the point b 1 Pa ( cos θ, bsin θ ) xcosθ ysinθ (ii) Hence show that the equation of this tangent is + = 1. a b (iii) If the point Pa ( cos θ, bsin θ) is on the ellipse in quadrant one, find the minimum area of the triangle made by this tangent and the coordinate axes. (c) Two fixed circles intersect at AB. P is a variable point on one circle. Copy or trace the diagram into your writing booklet. (i) Let APB = θ, explain why θ is a constant. 1 (ii) Prove that MN is of a constant length. 11
Question 15 (15 marks) Use a SEPARATE writing booklet. (a) A body of unit mass falls under gravity through a resisted medium. The body falls from 1 rest. The resistance to its motion is v Newtons where v metres per second is the 100 speed of the body when it has fallen a distance x metres. (i) 1 Show that the equation of motion of the body is x= g v, where g is the 100 1 magnitude of the acceleration due to gravity. (Note: Draw a diagram!) (ii) Show that the terminal speed, V T, is given by VT = 10 g 1 (iii) Show that V = VT 1 e x 50 3 (b) 3 x 6x + 9x+ k = 0 has two equal roots. (i) Show that k = 4 is a possible value for k (ii) Solve 3 x x x 6 + 9 4= 0 (c) αβ,, and γ are the roots of Find 3 x px qx r + + + = 0 (d) (i) (ii) (i) (ii) α + β + γ 1 3 3 3 α + β + γ Prove that x + x+ 1 0 for all real x 1 Hence or otherwise, prove that a + ab + b 0 1
Question 16 (15 marks) Use a SEPARATE writing booklet. (a) Let (i) (ii) (iii) π π w= cos + isin 7 7 Show that k w is a solution of 7 z 1= 0, where k is an integer. 3 4 5 6 Prove that w+ w + w + w + w + w = 1 π 4π 6π 1 Hence show that cos + cos + cos = 7 7 7 (b) The Bernoulli polynomials Bn ( x ) are defined by B ( ) 1 0 x = and for n=1,, 3,, dbn dx ( x) = nb ( ), 1 x and n Thus 1 0 B ( ) 0 n x dx = 1 B1 ( x) = x 1 B ( x) = x x+ 6 3 3 1 B3 ( x) = x x + x (i) 1 Show that B4 ( x) = x ( x 1) 30 (ii) Show that, for n, Bn(1) Bn(0) = 0 1 (iii) Show by mathematical induction, that for n 1 3 n 1 Bn( x+ 1) Bn( x) = n x, (iv) Hence show that for n 1and any positive integer k k n 1 n m = Bn k+ Bn m= 0 ( 1) (0) (v) Hence deduce that 135 4 m = 913496308 1 m= 0 13
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