005-CE A MATH HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 005 ADDITIONAL MATHEMATICS :00 pm 5:0 pm (½ hours) This paper must be answered in English 1. Answer ALL questions in Section A and any FOUR questions in Section B.. Write your name in the answer book provided. For Sections A, there is no need to start each question on a fresh page.. All workings must be clearly shown. 4. Unless otherwise specified, numerical answers must be exact. 5. In this paper, vectors may be represented by bold-type letters such as u, but candidates are expected to use appropriate symbols such as u in their working. 6. The diagram in the paper are not necessarily drawn to scale.
FORMULAS FOR REFERENCE A + B A B sin( A ± B) = sin Acos B ± cos Asin B sin A + sin B = sin cos A + B A B cos( A ± B) = cos Acos B sin Asin B sin A sin B = cos sin tan A ± tan B A + B A B tan( A ± B) = cos A + cos B = cos cos 1 tan Atan B A + B A B sin Acos B = sin( A + B) + sin( A B) cos A cos B = sin sin cos Acos B = cos( A + B) + cos( A B) sin A sin B = cos( A B) + cos( A + B) ************************************* Section A (6 marks) Answer ALL questions in this section. 1. Solve 4 x = x +. ( marks). Find sin x cos x dx. (4 marks) 1
. Let A be a variable point on the curve xy =. B is the point (1, ) and P is the mid-point of AB. Find the equation of the locus of P. (4 marks) 4. In the expansion of n (1 x + x ), the coefficient of x is 15. (a) Find the value(s) of n. (b) Find the coefficient of x. 5. y y = x A y = 8x x O x In the figure, the shaded region is bounded by the curve y = 8x x, y = x and the x-axis. (a) Write down the coordinates of A. (b) Find the area of the shaded region.
d 6. (a) Find cos x. dx (b) Hence find sin x dx. 7. (a) Show that cos x + cos 4x + cos6x = cos 4x(1 + cos x). (b) sin x cos 4x Hence show that = sin x. cos x + cos 4x + cos 6x 8. In the figure, let OA = i + 4j, OB = 4i j. P is a point on AB such that AP : PB = t : 1 t. y A O P x B (a) Express OP in terms of t, i and j. (b) If OP bisects AOB, find the value of t.
9. (a) Show that the equation of the tangent to the curve x = t 1, y = t + 1 at (1, ) is L : x y + 1 = 0. (b) Find the equation(s) of the line(s) parallel to L whose distance from L is 5 units. (6 marks) 10. Given that α and β are the roots of the equation x tx ( t + 4) = 0, where t is a real constant. (a) Find α + β and αβ in terms of t. (b) If α = β, find t and α. (6 marks) 11. (a) Prove, by mathematical induction, that n( n + 1)( n + ) 1 + + 4 +... + n ( n + 1) = for all positive integers n. (b) Using the result of (a) and the identity n( n + 1) 1+ + +... + n =, find 1 100 + 99 + 98 +... + 100 1. (7 marks) 4
1. (a) A(5, ), B(, ) are two vertices of the square ABCD. Find two possible equations of AC. (b) A student says that the two possible equations of AC in (a) are tangent to a circle C centred at the origin. Determine whether the student is correct or not. (7 marks) 5
Section B (48 marks) Answer any FOUR questions in this section. Each question carries 1 marks. 1. (a) O, A, B and C are 4 points on a plane such that OA = a, OB = b and OC = m a + nb. (i) Express AB and AC in terms of m, n, a and b. (ii) Show that if A, B, C are collinear then m + n = 1. B Q C P O A (b) In the figure, POA is a straight line and OP = OA. Q is a point on OB such that OQ = kob, where 0 < k < 1 and PQ is produced to C such that PQ = (1 k) PC. Let OA = a and OB = b. (i) k k Show that OC = a + b. 1 k 1 k (ii) Given that C lies on the straight line AB. (1) Find the value of k. () If OC is perpendicular to OA and AOB = 10, OB find. OA 6 (7 marks)
14. The slope of the tangent T at any point (x, y) to the curve C is given 1 by. The x-intercept of C is 1. x 1 (a) Find the equation of C. (4 marks) (b) Find the equation of T parallel to the line x y + 5 = 0. Show that T passes through the origin. (4 marks) y T C O x (c) Find the volume generated by revolving the area of the shaded portion bounded by the tangent T, the curve C and the x-axis about the x-axis. (4 marks) 7
15. The figure shows a tetrahedron such that VA = VB = AB = a, VC = 5a and CA = CB = a. O is the foot of the perpendicular from V to the base ABC. M is the mid-point of AB. P is a point on BC such that BP = ra where 0 r. VMC = θ and VPO = φ. V a 5a C A θ O P M B φ (a) Express CM and VM in terms of a. Hence find the value of cos θ in surd form. (b) Using the results of (a), find sin θ and VO. (Give the answers in surd form.) (c) Express VP in terms of a and r. ( marks) Hence show that 45 sin φ = 8(r 8r + 1). 8
16. α, β are the roots of the equation x mx + ( m 1) = 0, where m is a real constant. (a) Show that α, β are real and distinct. (b) Find (α β ) in terms of m. ( marks) ( marks) (c) The graph y = x mx + ( m 1) cuts the x-axis at A and B. A circle C is drawn with AB as diameter. (i) Using the result of (b), show that the equation the circle C is x + y mx + m 1 = 0. (ii) When m = 1 and m = 4, two circles C and C are 1 formed. A circle C passes through the point (, 1) and the points of intersection of C of. C1 and C. Find the equation (7 marks) 9
17. In the figure, P(a, b) is a point moving on the parabola y = 4 x in the first quadrant. The tangent at P cuts the axes at A and B. y B y = 4 x P(a, b) O A x (a) Express the coordinates of A and B in terms of a. a 4 Hence show that the area of AOB is S = + a +. 4 a (b) 8 When b =, B is moving down at a rate of 4 unit per second. (i) Find the rate at which point A is moving out. (ii) A student says that the area of AOB is decreasing at this moment. Determine whether the student is correct or not. (7 marks) END OF PAPER 10