Mathematics Extension 2

Transcription

1 009 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions o Reading Time- 5 minutes o Working Time hours o Write using a blue or black pen o Approved calculators may be used o A table of standard integrals is provided at the back of this paper. o All necessary working should be shown for every question. o Use a separate answer booklet for each question Total marks - 0 o Attempt Questions -8 o All questions are of equal value

2 Question 5 marks Use a separate writing booklet (a) Find cos d sin (b) Use completing the square to find 4 d (c) Use integration by parts to find tan d 0 (d) Find real numbers A and B so that Hence find 4 9 A B ( )( 4) d ( )( 4) (e) Use the substitution sin to find d 0 (4 ) 4

3 Question 5 marks Use a separate writing booklet (a) Let z i (i) Find z a ib in the form 5 (ii) Find in the form a ib z (b) Find all pairs of real numbers and y such that ( iy) 4 i. (c) Let i (i) Epress in modulus-argument form. (ii) Hence evaluate 0 in the form a ib (d) On separate Argand diagrams, sketch the region satisfying the following inequalities: (i) z (ii) arg( z) 4 4 (iii) 0 Re( z) and Im( z) (e) y D C B O A On the Argand diagram shown, ABCD is a square. The points A and B represent comple numbers and respectively. Hence AB represents the comple number. Find, in terms of and, (i) the comple number represented by AD (ii) the comple number represented by BD (iii) the comple number that is represented by the point C

4 Question 5 marks. Use a separate writing booklet (a) Find the gradient of the tangent to the curve with equation y y 5 at the point (4,) on the curve. (b) y (, ) y f( ) y The diagram shows the graph of y f( ) Make separate, one-third page sketches of the following: (i) y = f( ) (ii) y f( ) (iii) y f( ) (iv) y f( ) f (v) ( y ) (c) Consider the graph of the function y. 9 (i) Write down the equations of the vertical and horizontal asymptotes. (ii) Make a neat sketch of the graph of y clearly showing the 9 asymptotes and any intercepts with the coordinate aes.

5 Question 4 5 marks. Use a separate writing booklet (a) y y ( 4) y 4 O y ( ) y y The region bound by 4 and 4 is rotated about the -ais to form a solid. (i) Use the method of cylindrical shells to form an integral whose value will give the volume of the solid (ii) Evaluate this integral to find the volume of the solid. (b) y C y 4 A B O The base of a solid is the region bound by y 4 and the -ais. Vertical cross-sections parallel to the -ais are equilateral triangles. A typical cross-section ABC is shown. (i) Show that the area of an equilateral triangle with side s units is given by A s 4 (ii) Form an integral whose value will give the volume of the solid and evaluate this integral to find the volume. 4

6 Question 4 (Continued) (c) y 4 A O h B y A solid is formed by rotating the circle y about the line 4. Any cross-section perpendicular to the line 4 will be an annulus. (i) Consider the annulus formed when the chord AB, at y h, is rotated about the line 4. Show that the area of this annulus is given by A = 6 h (ii) Form an integral whose value will give the volume of the solid and evaluate it to find the volume of the solid.

7 Question 5 5 marks. Use a separate writing booklet 4 (a) The polynomial P( ) has a triple zero. Find all the zeros of P ( ). (b) The cubic equation 4 0 has roots, and. (i) Find the cubic polynomial with integer coefficients whose roots are, and. (ii) Find the value (iii) Find the value of (c) It is given that is a root of 0 = 0 where and are real. i a b a b (i) Find all the roots of a b 0 = 0. (ii) Hence epress 0 as a product of linear and quadratic a b factors with real coefficients. (d) Let z cos i sin (i) Find the roots of z modulus-argument form. (ii) Hence show that cos cos 5 5 5, epressing the comple roots in

8 Question 6 5 marks. Use a separate writing booklet y (a) Consider the ellipse whose equation is. 9 4 (i) Find the eccentricity of the ellipse. (ii) Write down the coordinates of the focii and the equations of the directrices. (iii) Make a neat sketch of the ellipse, clearly showing the focii and the directrices. (iv) Let P be any point on the ellipse. Show that PS PS ' 6, where S and S ' are the focii of the ellipse. (b) y T(0,5) 5 P( 5 p, ) p R 5 Q( 5q, ) q O 5 5 P( 5 p, ), p 0 and Q(5q, ), q0 are two points on the hyperbola y 5. p q (i) Show that the equation of the chord PQ is pqy 5( p q) (ii) Show that the equation of the tangent at is 0 P p y p ( iii) The tangents at P and Q intersect at R. Show that the coordinates 0 pq 0 of R are (, ) pq pq) (iv) PQ passes through the point T(0,5). Find the equation of the locus of R.

9 Question 7 5 marks. Use a separate writing booklet (a) A r A body of mass m kg is attached by a light, inetensible string of lenth metres o to the verte of a smooth, inverted cone whose semi-vertical angle is. The body remains in contact with the surface of the cone and rotates as a conical pendulum with angular velocity radians per second and radius r metres.. The forces acting on the body are tension in the string ( T), the normal reaction of the cone ( N) and the gravitational force mg. (i) Copy the diagram and show the forces that are acting on the body. (ii) By resolving forces vertically and horizontally, show that Tcos Nsin mg and Tsin Ncos mr T mgcos mr sin (iii) Show that and find a similar epression for N. (iv) The angular velocity is increased until the body is about to lose contact with the cone. Find an epression for this value of in terms of g, r and.

10 Question 7 (continued) (b) A body of mass m kg is projected vertically under the influence of gravity in a medium whose resistance is mkv where v is its velocity and k is a constant. Its initial velocity is U metres per second. (i) Make a neat sketch showing the forces acting on the body as it is moving up. (ii) Show that its acceleration is given by g kv ( ) g ku (iii) Show that the displacement of the body is given by log e k g kv (iv) Find the maimum height reached by the body. (v) The body now starts falling back towards the ground. Make a neat sketch showing the forces acting on the body and hence write an equation for the acceleration of the body. (vi) Determine the terminal velocity of the body as it falls.

11 Question 8 5 marks. Use a separate writing booklet (a) Use mathematical induction to prove that sin( n ) n ( ) sin where n an integer (b) Two circles intersect at A and B and a common tangent touches them at P and Q as shown. P B Q R A S A chord PR is drawn parallel to QA. RA is produced to meet the other circle at S. Copy the diagram onto your own paper. (i) Eplain why PQA = QSA. (ii) Prove that PQSR is a cyclic quadrilateral (iii) Hence prove that PA is parallel to QS

12 Question 8 (continued) n (c) Let I = n cos d for n = 0,,,... 0 n n 0 0 (i) By writing cos d as cos cos d and using integration n - by parts, show that In In- for n =,,... n (ii) Hence evaluate 6 cos d 0 4 END OF EXAMINATION