Trending ▼   ResFinder  

NSW HSC 2005 : MATHEMATICS EXTENSION-2

20 pages, 66 questions, 0 questions with responses, 0 total responses,    0    0
nsw_hsc
  
+Fave Message
 Home > nsw_hsc >

Instantly get Model Answers to questions on this ResPaper. Try now!
NEW ResPaper Exclusive!

Formatting page ...

2005 H I G H E R S C H O O L C E R T I F I C AT E E X A M I N AT I O N Mathematics Extension 2 General Instructions Reading time 5 minutes Working time 3 hours Write using black or blue pen Board-approved calculators may be used A table of standard integrals is provided at the back of this paper All necessary working should be shown in every question 412 Total marks 120 Attempt Questions 1 8 All questions are of equal value BLANK PAGE 2 Total marks 120 Attempt Questions 1 8 All questions are of equal value Answer each question in a SEPARATE writing booklet. Extra writing booklets are available. Marks Question 1 (15 marks) Use a SEPARATE writing booklet. (a) (b) cos Find 5 d . sin 2 (i) Find real numbers a and b such that (ii) 5x a + b 5x dx . Hence find 2 x x 6 x2 x 6 x 3 x+2 . 2 1 e (c) Use integration by parts to evaluate x 7 loge x dx . 1 3 (d) dx Using the table of standard integrals, or otherwise, find . 4x2 1 2 (e) Let t = tan . 2 ( ) 1 . 2 dt 1 = 1 + t2 . d 2 (i) Show that (ii) Show that sin = (iii) Use the substitution t = tan 2t 1 + t2 to find cosec d . 2 3 2 Marks Question 2 (15 marks) Use a SEPARATE writing booklet. (a) Let z = 3 + i and w = 1 i . Find, in the form x + iy , (i) 1 (ii) zw 1 (iii) (b) 2 z + iw 6. w 1 Let = 1 i 3 . (i) 2 (ii) Express 5 in modulus-argument form. 2 (iii) (c) Express in modulus-argument form. Hence express 5 in the form x + iy . 1 Sketch the region on the Argand diagram where the inequalities z z < 2 and z 1 1 hold simultaneously. Question 2 continues on page 5 4 3 Marks Question 2 (continued) (d) Let l be the line in the complex plane that passes through the origin and makes an angle with the positive real axis, where 0 < < . 2 Q l P O The point P represents the complex number z1, where 0 < arg(z1) < . The point P is reflected in the line l to produce the point Q, which represents the complex number z2. Hence z1 = z2 . (i) Explain why arg(z1) + arg(z2) = 2 . (ii) Deduce that z1 z2 = z1 (cos 2 + i sin 2 ) . (iii) 2 and let R be the point that represents the complex 4 number z1 z2 . Let = Describe the locus of R as z1 varies. End of Question 2 5 2 1 1 Marks Question 3 (15 marks) Use a SEPARATE writing booklet. (a) The diagram shows the graph of y = (x) . y 4 O 1 2 x 4 Draw separate one-third page sketches of the graphs of the following: (i) 1 (ii) y = ( x) 1 (iii) y = ( x) 2 (iv) (b) y = (x + 3) y = ( x ). 2 Sketch the graph of y = x + 8x , clearly indicating any asymptotes and any x2 9 points where the graph meets the axes. Question 3 continues on page 7 6 4 Marks Question 3 (continued) (c) Find the equation of the normal to the curve x3 4xy + y3 = 1 at (2, 1) . (d) 2 N P mv 2 r mg The diagram shows the forces acting on a point P which is moving on a frictionless banked circular track. The point P has mass m and is moving in a horizontal circle of radius r with uniform speed v. The track is inclined at an angle to the horizontal. The point experiences a normal reaction force N from the track and a vertical force of magnitude mg due to gravity, so that the net mv 2 force on the particle is a force of magnitude directed towards the centre of r the horizontal circle. By resolving N in the horizontal and vertical directions, show that N = m g2 + v4 r2 . End of Question 3 7 3 Marks Question 4 (15 marks) Use a SEPARATE writing booklet. (a) y y = e x 2 N x 2 The shaded region between the curve y = e x , the x-axis, and the lines x = 0 and x = N , where N > 0, is rotated about the y-axis to form a solid of revolution. (i) 3 (ii) (b) Use the method of cylindrical shells to find the volume of this solid in terms of N. What is the limiting value of this volume as N ? 1 Suppose , , and are the four roots of the polynomial equation x 4 + px 3 + qx 2 + rx + s = 0. (i) Find the values of + + + and + + + in terms of p, q, r and s . 2 (ii) Show that 2 + 2 + 2 + 2 = p 2 2 q . 2 (iii) Apply the result in part (ii) to show that x4 3x3 + 5x2 + 7x 8 = 0 cannot have four real roots. 1 (iv) By evaluating the polynomial at x = 0 and x = 1, deduce that the polynomial equation x4 3x3 + 5x2 + 7x 8 = 0 has exactly two real roots. 2 Question 4 continues on page 9 8 Marks Question 4 (continued) (c) y P(x1, y1) x B(0, b) The point P (x1, y1 ) lies on the ellipse x2 a2 + y2 b2 = 1 , where a > b > 0. The equation of the normal to the ellipse at P is a2y1x b2x1y = (a2 b2 ) x1y1 . (i) The normal at P passes through the point B (0, b) . Show that y1 = (ii) b3 or y1 = b . a2 b2 Show that if y1 = b3 a b 2 2 2 , the eccentricity of the ellipse is at least End of Question 4 9 1 2 . 2 BLANK PAGE 10 Marks Question 5 (15 marks) Use a SEPARATE writing booklet. (a) (i) 1 A c b d B C a The triangle ABC is right-angled at A and has sides with lengths a, b and c, as shown in the diagram. The perpendicular distance from A to BC is d. By considering areas, or otherwise, show that b2 c2 = d 2 (b2 + c2 ) . (ii) 2 T h A B P C South East The points A, B and C lie on a horizontal surface. The point B is due south of A. The point C is due east of A. There is a vertical tower, AT, of height h at A. The point P lies on BC and is chosen so that AP is perpendicular to BC. Let , and denote the angles of elevation to the top of the tower from B, C and P respectively. Using the result in part (i), or otherwise, show that tan 2 = tan 2 + tan 2 . Question 5 continues on page 12 11 Marks Question 5 (continued) (b) Mary and Ferdinand are competing against each other in a competition in which the winner is the first to score five goals. The outcome is recorded by listing, in order, the initial of the person who scores each goal. For example, one possible outcome would be recorded as MFFMMFMM . (i) 1 (ii) (c) Explain why there are five different ways in which the outcome could be recorded if Ferdinand scores only one goal in the competition. In how many different ways could the outcome of this competition be recorded? 2 Let a > 0 and let (x) be an increasing function such that (0) = 0 and (a) = b . a (i) b 1 Explain why ( x ) dx = ab ( x ) dx . 0 0 1 2 (ii) x Hence, or otherwise, find the value of sin 1 dx . 4 0 Question 5 continues on page 13 12 3 Marks Question 5 (continued) (d) The base of a right cylinder is the circle in the xy-plane with centre O and radius 3. A wedge is obtained by cutting this cylinder with the plane through the y-axis inclined at 60 to the xy-plane, as shown in the diagram. B C y 3 A 60 O x 3 ( 3 D x, 9 x 2 x ) A rectangular slice ABCD is taken perpendicular to the base of the wedge at a distance x from the y-axis. (i) Show that the area of ABCD is given by 2 x 27 3 x 2 . 2 (ii) Find the volume of the wedge. 3 End of Question 5 13 Marks Question 6 (15 marks) Use a SEPARATE writing booklet. x (a) For each integer n 0 , let In ( x ) = t ne t dt . 0 (i) 4 Prove by induction that x2 x3 xn In ( x ) = n! 1 e x 1 + x + + + + . 2! 3! n! (ii) Show that 1 1 1. 0 t ne t dt n +1 0 (iii) Hence show that 1 11 1 1 0 1 e 1 1 + + + + . 1! 2! n! (n + 1)! (iv) Hence find the limiting value of 1 + 11 1 + + + as n . 1! 2! n! Question 6 continues on page 15 14 1 Marks Question 6 (continued) (b) Let n be an integer greater than 2. Suppose is an nth root of unity and 1. (i) By expanding the left-hand side, show that 2 (1 + 2 + 3 2 + 4 3 + + n n 1) ( 1) = n . (ii) Using the identity 1 z2 1 = z 1 z z 1 , or otherwise, prove that 1 1 cos i sin , = cos 2 + i sin 2 1 2i sin provided that sin 0. 1 2 2 + i sin , find the real part of . n n 1 (iii) Hence, if = cos (iv) Deduce that 1 + 2 cos (v) By expressing the left-hand side of the equation in part (iv) in terms of cos 2 4 6 8 5 + 3 cos + 4 cos + 5 cos = . 5 5 5 5 2 2 and cos , find the exact value, in surd form, of cos . 5 5 5 End of Question 6 15 1 1 3 Marks Question 7 (15 marks) Use a SEPARATE writing booklet. (a) A u q N B p T t r M s P The points A, B and P lie on a circle. The chord AB produced and the tangent at P intersect at the point T, as shown in the diagram. The point N is the foot of the perpendicular to AB through P, and the point M is the foot of the perpendicular to PT through B . Copy or trace this diagram into your writing booklet. (i) Explain why BNPM is a cyclic quadrilateral. 1 (ii) Prove that MN is parallel to PA. 3 Let TB = p, BN = q, TM = r, MP = s, MB = t and NA = u. (iii) sr Show that u < p . 2 (iv) Deduce that s < u . 1 Question 7 continues on page 17 16 Marks Question 7 (continued) (b) The acceleration of any body towards the centre of a star due to the force of gravity is proportional to x 2, where x is the distance of the body from the centre k of the star. That is, x = , where k is a positive constant. x2 (i) A satellite is orbiting a star with constant speed, V, at a fixed distance R 2 from the centre of the star. Its period of revolution is T. Use the fact that the satellite is moving in uniform circular motion to show that k = (ii) 4 2 R3 T2 . The satellite is stopped suddenly. It then falls in a straight line towards the centre of the star under the influence of gravity. 3 Show that the satellite s velocity, v, satisfies the equation v2 = 8 2 R2 R x , T2 x where x is the distance of the satellite from the centre of the star. (iii) Show that, if the mass of the star were concentrated at a single point, T the satellite would reach this point after time . 42 x x dx = R sin 1 You may assume that R R x End of Question 7 17 x( R x ) . 3 Marks Question 8 (15 marks) Use a SEPARATE writing booklet. (a) Suppose that a and b are positive real numbers, and let ( x ) = for x > 0. (i) Show that the minimum value of (x) occurs when x = (ii) a+b+x 1 3( abx ) 3 a+b . 2 Suppose that c is a positive real number. 3 2 2 a+b+c a + b + c a+b Show that and deduce that 3 2 ab 3 3 abc You may assume that 3 3 abc . a+b ab . 2 (iii) Suppose that the cubic equation x3 px2 + qx r = 0 has three positive real roots. Use part (ii) to prove that p3 27r . 1 (iv) Deduce that the cubic equation x3 2x2 + x 1 = 0 has exactly one real root. 2 Question 8 continues on page 19 18 Marks Question 8 (continued) y (b) C A U P( a sec , b tan ) T x O V Q D B x2 y2 = 1 . The line a2 b2 through P perpendicular to the x-axis meets the asymptotes of the hyperbola, xy xy = 0 and + = 0 , at A (a sec , b sec ) and B (a sec , b sec ) ab ab respectively. A second line through P, with gradient tan , meets the hyperbola xy at Q and meets the asymptotes at C and D as shown. The asymptote = 0 ab makes an angle with the x-axis at the origin, as shown. The point P (a sec , b tan ) is a point on the hyperbola (i) Show that AP PB = b2 . (ii) Show that CP = (iii) Hence deduce that CP PD depends only on the value of and not on the position of P. 1 (iv) Let CP = p, QD = q and PQ = r . 2 1 AP cos PB cos and show that PD = . sin( ) sin( + ) 2 Show that p = q . (v) A tangent to the hyperbola is drawn at T parallel to CD. This tangent meets the asymptotes at U and V as shown. Show that T is the midpoint of UV. End of paper 19 1 STANDARD INTEGRALS n x dx = 1 x dx = ln x, x > 0 ax e dx = 1 ax e , a 0 a cos ax dx = 1 sin ax, a 0 a sin ax dx 1 = cos ax, a 0 a 2 sec ax dx = 1 tan ax, a 0 a sec ax tan ax dx = 1 sec ax, a 0 a 1 dx 2 a + x2 = 1 x tan 1 , a 0 a a 1 dx 2 a x2 x = sin 1 , a > 0, a < x < a a 1 dx 2 x a2 = ln x + x 2 a 2 , x > a > 0 1 dx 2 x + a2 = ln x + x 2 + a 2 1 n +1 x , n 1; x 0, if n < 0 n +1 ( ) ( ) NOTE : ln x = loge x, x>0 20 Board of Studies NSW 2005

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

Formatting page ...

 

  Print intermediate debugging step

Show debugging info


 

Additional Info : New South Wales Higher School Certificate Mathematics Extension-2 2005
Tags : new south wales higher school certificate mathematics (extension-2) 2005, nsw hsc online mathematics extension-2, nsw hsc maths, nsw hsc mathematics extension-2 syllabus, nsw hsc maths model exam papers, mathematics sample papers, mathematics course, nsw hsc maths solved paper., australia new south wales, nsw hsc online, nsw hsc past papers, nsw hsc papers, nsw hsc syllabus, nsw board of studies, higher school certificate new south wales, nsw australia, hsc syllabus, nsw hsc exams, nsw hsc question papers, nsw hsc solved question papers, nsw hsc previous exam papers, nsw university.  

© 2010 - 2025 ResPaper. Terms of ServiceContact Us Advertise with us

 

nsw_hsc chat