MATHEMATICS EXTENSION 2

Transcription

1 Sydney Grammar School Mathematics Department Trial Eaminations 008 FORM VI MATHEMATICS EXTENSION Eamination date Tuesday 5th August 008 Time allowed hours (plus 5 minutes reading time) Instructions All eight questions may be attempted. All eight questions are of equal value. All necessary working must be shown. Marks may not be awarded for careless or badly arranged work. Approved calculators and templates may be used. A list of standard integrals is provided at the end of the eamination paper. Collection Write your candidate number clearly on each booklet. Hand in the eight questions in a single well-ordered pile. Hand in a booklet for each question, even if it has not been attempted. If you use a second booklet for a question, place it inside the first. Keep the printed eamination paper and bring it to your net Mathematics lesson. Checklist SGS booklets: 8 per boy. Candidature: 74 boys. A total of 750 booklets should be sufficient. Eaminer BDD

2 SGS Trial Form VI Mathematics Etension... Page QUESTION ONE (5 marks) Use a separate writing booklet. Marks (a) By completing the square, find d. 5 4 (b) Use integration by parts to find e d. (c) Use log laws to assist in finding ln d. (d) Use the substitution u = tan to find sec 4 d. (e) Use the substitution = cos θ to find 4 d. 9 (f) Use a careful substitution and a symmetry argument to find π π (π ) sin d. Eam continues net page...

3 SGS Trial Form VI Mathematics Etension... Page QUESTION TWO (5 marks) Use a separate writing booklet. Marks (a) (i) Epress i and + i in modulus argument form. (ii) Hence use de Moivre s theorem to evaluate ( i ) 0 +(+i ) 0. (b) Shade the region of the comple plane described by z < and Re(z). You need not find the coordinates of any points of intersection. (c) The three comple numbers +i, i and 5 + 4i are represented by the points A, B and C respectively in the comple plane. (i) Show that ABC is isosceles. (ii) Find the midpoint M of BC. (d) Im C B 8 O θ 5 A Re The diagram above shows a rhombus OABC in the first quadrant of the Argand diagram, with the origin O as one verte and another verte A lying on the real ais. The longer diagonal OB is 8 units, and each side is 5 units. Let AOB = θ and let z = cos θ + i sin θ. (i) Eplain why OC is represented by the comple number 5z. (ii) Show that z satisfies the quadratic equation 5z =8z 5. (iii) Solve this quadratic equation and then find the comple numbers representing the vertices B and C. Eam continues overleaf...

4 SGS Trial Form VI Mathematics Etension... Page 4 QUESTION THREE (5 marks) Use a separate writing booklet. Marks (a) y y= + y= metal base = A glass is designed by rotating the region bounded by the curves y = and y = + and the line = about the y-ais, as in the diagram above. The resulting volume is to be grafted onto a cylindrical metal base whose volume need not be included in the following calculations. (i) Find the point where the curves intersect in the first quadrant. (ii) Use the method of cylindrical shells to determine the volume of glass in the solid. (b) Draw a neat half-page diagram of the ellipse 4 + y = showing foci, directrices 4 and intercepts with the aes. (c) Let a + y b = be an ellipse with foci S and S and eccentricity e. Carefully prove for any point P (,y ) on the ellipse that PS+ PS =a. (d) Let P (a cos θ, b sin θ) and Q ( a cos(θ + α),bsin(θ + α) ) be two distinct points on the ellipse a + y =. Let O be the origin. b (i) Show that line OP has equation (b sin θ) (a cos θ)y =0. (ii) Show that the perpendicular distance from Q to OP is ab sin α a cos θ + b sin θ. (iii) Hence show that the area of the triangle OPQ is independent of θ. Eam continues net page...

5 SGS Trial Form VI Mathematics Etension... Page 5 QUESTION FOUR (5 marks) Use a separate writing booklet. Marks (a) Factorise 5 as the product of real linear and quadratic factors. You may leave your answer in terms of trigonometric ratios. (b) y The graph of a certain function y = f() is sketched above. Draw neat half-page sketches of the following graphs. (i) y = f() (ii) y = e f() (iii) y = f() (c) The line y = + 4 is tangent to the cubic y = at A, and intersects the cubic again at B. (i) Show that the -coordinates of the points A and B are the roots of the cubic = 0. (ii) Eplain briefly why the cubic equation in part (i) must have a double root, and use this fact to find the -coordinates of the points A and B. Eam continues overleaf...

6 SGS Trial Form VI Mathematics Etension... Page 6 QUESTION FIVE (5 marks) Use a separate writing booklet. Marks (a) y h a a The diagram above shows a parabola with verte (0,h) and zeroes = a and = a. Show that the shaded area is 4 ha square units. (b) y z A certain dome tent is designed with an elliptical base 4 + y =, as in the diagram above. Cross-sections perpendicular to the base are segments of a parabola. The height of each parabolic segment is equal to the width of its base. (i) Use your result in part (a) to show that a typical cross-section has area 8. (ii) Show that the volume of the tent is 64 cubic units. 9 Eam continues net page...

7 SGS Trial Form VI Mathematics Etension... Page 7 QUESTION FIVE (Continued) (c) (i) Epand (cos θ + i sin θ) 7 using the binomial theorem. (ii) Use de Moivre s theorem to establish the identity tan 7θ = 7 tan θ 5 tan θ + tan 5 θ tan 7 θ tan θ + 5 tan 4 θ 7 tan 6. θ (iii) Use your result from part (ii) to solve the polynomial equation =0. (iv) Use product-of-roots to determine the value of tan π 7 tan π 7 tan π 7. (v) Use a careful argument involving roots to determine the value of cot π 7 + cot π 7 + cot π 7. Eam continues overleaf...

8 SGS Trial Form VI Mathematics Etension... Page 8 QUESTION SIX (5 marks) Use a separate writing booklet. Marks (a) y Q Z W P R Y X S ( Let P ct, c ) ( and Q ct, c ) be any two points on the rectangular hyperbola y = c t t ( with 0 <t <t. Let O be the origin. Let R ct, c ) ( and S ct, c ) be the t t points diametrically opposed to P and Q respectively. (that is, POR and QOS are straight). (i) Prove that P, Q, R and S are the vertices of a parallelogram. (ii) Show that the tangent at P has equation + t y =ct, and then write down the equation of the tangent at Q. (iii) Show that the point of intersection W of the tangents at P and at Q has coordinates,. ( ) ct t c t + t t + t (iv) Let the tangents at Q and R meet at X, the tangents at R and S meet at Y, and the tangents at S and P meet at Z. Write down, without working, the coordinates of X, Y and Z. (v) Show that WXYZ is also a parallelogram. Eam continues net page...

9 SGS Trial Form VI Mathematics Etension... Page 9 QUESTION SIX (Continued) (b) Let I n = ( ) n d, where n is any integer. (i) Show that I = π. (ii) Show that I n = n I n, for n 0. (Hint: Integration by parts is not required.) (iii) Hence find I 5. QUESTION SEVEN (5 marks) Use a separate writing booklet. Marks (a) A particle of mass m kilograms is launched vertically upwards in a highly resistive medium at a velocity 5 m/s. It is subject to the force of gravity and to a resistance due to motion of magnitude mv 00. Take g = 0 m/s and upwards as positive. The equation of motion is ẍ = g v 00. (i) Show the height above the point of launch and the velocity v are related by dv d = v v (ii) Find constants A, B and C such that 00v v = A v +0 + Bv + C v 0v (iii) Hence find the maimum height reached by the particle, giving your answer correct to the nearest centimetre. Eam continues overleaf...

10 SGS Trial Form VI Mathematics Etension... Page 0 QUESTION SEVEN (Continued) (b) A D M O C B L K In the diagram above, the points A, B, C and D lie on a circle with centre O. The intervals AB and DC produced meet at K, and the intervals AD and BC produced meet at M. The circles BCK and DCM meet again at L. Assume additionally that A, O, C and L are collinear. Copy or trace the diagram into your answer book. (i) Prove that the centre F of the circle DCM lies on CM. (ii) Prove that K, L and M are collinear. (iii) Prove that a circle may be drawn through D, B, K and M. Label the centre of this circle X and describe its location. (iv) Prove that there is a circle passing through B, X, F and D. Hint: Let DMB = α. (The circle also passes through O, L and the centre of circle BCK, but you need not prove these facts and should not assume them.) Eam continues net page...

11 SGS Trial Form VI Mathematics Etension... Page QUESTION EIGHT (5 marks) Use a separate writing booklet. Marks (a) (i) Show that the polynomial P (z) =7z +(i 48)z +99z 64 i. may be written as a difference of cubes: P (z) =8(z ) (z i). (ii) Hence solve the polynomial equation P (z) = 0. 4 You may wish to make use of the cube roots of unity, ω and ω in your solution, where ω = + i. Epress your solutions in the form a + ib, where a and b are real. (b) Let be any positive real number. Define recursively two linked sequences a 0, a, a,... and b 0, b, b,... by a 0 =(+ ), b0 = a i+ = (a i + b i ), for all i 0 b i+ = a i+ b i, for all i 0. (i) Let θ = tan, so that = tan θ. Prove by induction that for all n 0, 4 a n b n = cos θ n and sin θ b n = n sin θ n. (ii) Prove that sin θ θ as n, and write down lim b n n b. n + (iii) Let =, and evaluate three iterations to find a and b. You should record each answer correct to four decimal places. (iv) Hence estimate π 4. (v) With =, find algebraically how many iterations need to be taken before the ratio b n+ differs from by less than 0 0. b n END OF EXAMINATION

12 SGS Trial Form VI Mathematics Etension... Page The following list of standard integrals may be used: n d = n + n+, n ; 0, if n<0 d =ln, > 0 e a d = a ea, a 0 cos a d = sin a, a 0 a sin a d = cos a, a 0 a sec a d = tan a, a 0 a sec a tan a d = sec a, a 0 a a + d = a tan a, a 0 a d = sin, a > 0, a <<a a ( a d =ln + ) a, >a>0 ( + a d =ln + ) + a NOTE : ln = log e, > 0