Mathematics Extension 03 HSC ASSESSMENT TASK 3 (TRIAL HSC) General Instructions Reading time 5 minutes Working time 3 hours Write on one side of the paper (with lines) in the booklet provided Write using blue or black pen Board approved calculators may be used All necessary working should be shown in every question Each new question is to be started on a new page. Student Number Attempt all questions Class Teacher: (Please tick or highlight) Mr Fletcher Ms Beevers Ms Ziaziaris (To be used by the exam markers only.) Question No -0 3 4 5 6 Total Total Mark 0 5 5 5 5 5 5 00 00
Section I 0 marks Attempt Questions -0 Allow about 5 minutes for this section Use multiple choice answer sheet for questions -0. If z 5i, find the value of zz. (A) 5 (B) -5 (C) 5 (D) -5 4. For the equation xy x y 0, the equation of the tangent at (-,0) is given by, (A) y x (B) y x (C) Does not exist (D) y x 3. The locus of all complex numbers for z Re( z) is represented by which of the following diagrams. (A) (B) 4 (C) (D)
4. The graph of y f( x) is shown below. Y - X - The sketch of y f( x) is best represented by which of the following: (A) Y (B) Y 4 X X -4 - (C) Y (D) Y X X - 3
5. Find the coordinates of the foci for xy 8. (A) 4, 4, 4, 4 (B),,, (C) 8,8, 8, 8 (D) 4,4, 4, 4 6. The volume of the solid obtained by revolving the region bounded by e x y, the lines x0, x about the Y-axis can be evaluated with which of the following integrals. e Y y e x X y e and x (A) V x e e dx e x (B) V x e dx e x (C) V xe e dx 0 x (D) V xe dx 0 4
7. The square roots of ( 34 i) in the form of a bi is (A) ( i),( i) (B) ( i),( i) (C) ( i), ( i) (D) ( i), ( i) 8. The polynomial Pzhas ( ) real coefficients. Four of the roots of the equation Pz ( ) 0are z 0, z i, z iand z 3i. The minimum number of roots that the equation Pz ( ) 0 could have is (A) 4 (B) 5 (C) 6 (D) 8 9. Using a suitable substitution, the definite integral is equivalent to 4 tan xsec 0 xdx 4 (A) udu 0 4 (B) udu (C) (D) 0 3 udu 0 3 0 udu 5
0. Without evaluating the integrals which of the following will give an answer of zero. (A) (B) (C) (D) 4 3 cos 4 cos d x x 3 dx ( )( ) sin x 7 x cos xdx 4dx 6
Section II 90 marks Attempt Questions -6 Allow about hours and 45 minutes for this section Start each question on a NEW PAGE. Extra writing booklets are available. In Questions -6, your responses should include relevant mathematical reasoning and /or calculations. Question (5 marks) Start a NEW PAGE. (a) The graph of y f( x) is sketched below. Y 3 X - On separate axes sketch the graph of (i) y f( x) y f( x) (ii) (b) Consider the curve y cos x (i) Find dy dx, 0 x 4 and the gradient of the limiting tangent as x 0. (ii) Find any stationary points and determine their nature. 3 (iii) Sketch y cos x, 0 x 4 and then complete the sketch to show the graph of y cos x, 4 x 4, showing any intercepts on the coordinate axes and the coordinates of any stationary points. 3 (iv) u Use the substitution y cos x and the x axis between x, u 0 to find the area bounded by the curve x and 4 x. 3 4 7
Question (5 marks) Start a NEW PAGE. i (a) Express i in the form a ib. (b) (i) Express z 3 3 3i in modulus argument form. (ii)hence find the smallest positive integer n so that n z is real. (c) Solve the equation z ( z) 0, where z is a complex number. (d) Sketch the following loci on separate argand diagrams. (i) (ii) z z 6i. z i arg z (e) (i) Suppose z is any non-zero complex number. Explain why z z has modulus and argument twice the argument of z. z (ii) Find all complex numbers z so that z = i. Give your answer in the form a ib, where a and b are real. 8
Question 3 (5 marks) Start a NEW PAGE. (a) x 6x5 dx 3 (b) d 4 sin (c) Use the substitution 6 u x to find dx 3 3 x x (d) (i) Show that x x x x (ii) If n n n n I x dx for n 0 n show that In In for n. n 0 n (iii) Deduce that I n n! n!! 9
Question 4 (5 marks) Start a NEW PAGE. (a) A(0,) and B(3, ) lie on the curve y x. The shaded region in the diagram is bounded by the lines y, x3 and arc AB. A slice perpendicular to the line x 4 has been taken and the region is rotated about x 4. Y B(3,) A(0,) X Ay ( ) 5 y (ii) Find the volume of the solid generated. (i) Show that the area of a slice is given by (b) (i) Sketch the ellipses x 5y 00 and 5x y 00 on the same diagram, showing their intercepts with the coordinate axes. You DO NOT need to show their foci or directrices. (iii) A child s spinning top is made by revolving the total area enclosed by each of these ellipses (ie. total area, not just common area ), around the vertical axis. Let the two ellipses intersect at x ain the first quadrant. By using cylindrical shells, find an expression for the volume in terms of a. (Complete simplification is not necessary). 4 Question 4 continues overleaf 0
(c) -a x Y X a A solid is formed with a square cross section at rightangles to the X-Y plane as shown in the diagram. The equation of the base is 4 ( ) y xy (i) 4 Show that the volume of a thin slice is approximately equal to 4 x x, where x is the thickness of the slice. (ii) Hence calculate the volume of the solid bounded by a x a. (iii) What is the limiting value of the volume of the solid as a approaches infinity.
Question 5 (5 marks) Start a NEW PAGE. (a) The polynomial 4 3 Px ( ) x 7x 9x 7x C has a triple zero. (i) Determine the value of the triple zero. (ii) Hence, find the value of C. (iii) Factorise Px ( ) (b) The equation (i) (ii) x x 3 0 has roots, and. 3 Find the polynomial equation that has roots, and. Express with integral powers. Find the value of 4 4 4. (c) The quartic polynomial 4 3 f ( x) x px qx rx s such that the sum of and equals the sum of and. Let C. Let P. Let Q. has four zeroes,, and, (i) Find pqr,, and s in terms of C, P and Q. 3 (ii) Show that the coefficients of f( x) satisfy the condition 3 p 8r 4pq. (iii) It is given that the polynomial 4 3 gx ( ) x8x 79x 8x 440 has the property that the sum of two of the zeroes equals the sum of the other two zeroes. Using the identities of part (i) or otherwise, find all four zeroes of gx ( ). 3
Question 6 (5 marks) Start a NEW PAGE. x (a) Tangents to the ellipse a M is the midpoint of PQ. y b at the points P x, y and, Q x y intersect at T. (i) xx yy Given the tangent to the ellipse at P has equation, a b write down the equation of the tangent to the ellipse at Q. (ii) xx yy xx yy Show that the line passes through T and M. a b a b (iii) Deduce that the points OT,, Mare collinear. (iv) Show that the product of the gradients of PQ and TM is a constant. xx yy (v) If PTQ is a right angle, show that 0. 3 4 4 a b Question 6 continues overleaf 3
Y x a y b (b) O -a a ae X x ae y a e x y (i) The tangent at Pa ( sec, btan ) on the hyperbola has equation a b xsec ytan 0. (DO NOT PROVE THIS). a b Show that if the tangent at P is also tangent to the circle with centre ae,0 and radius a e, then sec e. 4 (ii) Deduce that the points of contact P, Q on the hyperbola of the common tangents to the circle and hyperbola are the extremities of a latus rectum of the hyperbola, and state the co-ordinates of P and Q. END OF PAPER 4
NORTH SYDNEY BOYS HIGH SCHOOL 03 HSC ASSESSMENT TASK 3 (TRIAL HSC) MATHEMATICS EXTENSION MULTIPLE CHOICE ANSWER SHEET ANSWER QUESTIONS -0 on this sheet NAME/NUMBER... TEACHER S NAME.... A B C D. A B C D 3. A B C D 4. A B C D 5. A B C D 6. A B C D 7. A B C D 8. A B C D 9. A B C D 0. A B C D 5