1 霹雳怡保培南独立中学 SEKOLAH MENENGAH POI LAM (SUWA) FINAL EXAMINATION 2022 ADVANCED MATHEMATICS (II) PAPER 2 SUBJECTIVE QUESTIONS ___________________________________________ DATE : 12.10.2022 (WEDNESDAY) TIME : 1335 – 1535 (2 HOURS) ____________________________________________ NAME : _____________________ REGISTRATION NUMBER: _____________ CLASS : S3LI INSTRUCTIONS TO CANDIDATES 1. This subject comprises two papers: Paper 1: Multiple-choice questions (40%), Paper 2: Subjective questions (60%). 2. Paper 2 Subjective questions Section A: Trigonometry (2 questions); Section B: Algebra (4 questions); Section C: Analytic Geometry (2 questions); Section D: Calculus (4 questions). Choose six questions from a total of twelve questions including at least one but at most two questions from each section. 3. Begin each question on a fresh page. 4. Use only blue or black ink to write your answers and use pencil for drawing only. 5. Do not copy the questions, but the answer to each question should be clearly numbered. 6. Show all mathematical working clearly. Geometric figures should be drawn where necessary. 7. Unless otherwise specified, the prescribed electronic calculators may be used. 8. Arrange the answer scripts in numerical order and tie them together. 9. Mathematics Formula Sheet is on pages 2 – 4. Do Not Turn Over This Page Until You Are Told To Do So __________________________________________________________________________________ This document consists of 8 printed pages (including this page) Prepared by : Ms Chai Siew Yin Checked by : Mr Yeow Ghee Ruey Signature : _________________ Signature : __________________
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5 Section A : Trigonometry (Choose at least one but at most two questions in this section.) 1. (a) As shown in the diagram, VABC is a pyramid. O is the circumcentre of ∆ABC. VO is perpendicular to the plane ABC. Given that AB = 8 cm, AC = 6 cm, BC = 5 cm and VO = 12 cm. V A O B C (i) Find the length of AO ; [3] (ii) Find the angle between the line VA and the plane ABC. [2] (Give all your answers correct to two decimal places) (b) Solve the equation tan−1�√2( + 1)� − tan−1 � −1 √2 � = tan−1 √2 3 . [5] 2. (a) In ∆OAB, OA = a , OB = b and ∠AOB = 2θ , C is a point on AB such that OC bisects ∠AOB. If OC = c , prove that 1 + 1 = 2 cos θ . A a C θ c θ O b B [2] (b) Given that sin α + sin β = , cos α + cos β = , prove that cos(α + β) =2 − 2 2+2 . [5] (c) Find the general solution of the equation cos2 x + 3cos2 2x = cos 2 3x. [3]
6 Section B : Algebra (Choose at least one but at most two questions in this section.) 3. (a) The probability of winning a grand prize each time playing a game is 0.1. Find the probability of winning at most 1 grand prize when playing the game for 20 times. Give your answer correct to 4 decimal places. [2] (b) Solve the inequality 84.3− 243 ≥ 32 . [3] (c) (i) Find the expansion of (1 + 3) 3 2 in ascending powers of x up to the term x 3, and state the range of values of x for which the expansion is valid. [4] (ii) Hence, find the value of (1.15) 3 2 , correct to four decimal places. [2] 4. (a) Construct a truth table for (∼ p ∨ q) ∧ p. [2] (b) Given that =+2 22+3+6 , where x ∈ R . Find the maximum value and the minimum value of y. [4] (c) Solve the equation 2 − √22 + 6 + 1 = 1 – 3x . [4] 5. (a) (i) Decompose 3+1 (+1)(+2) into partial fractions. [3] (ii) Hence, find the value of 4 1×2×3 + 7 2×3×4 + 10 3×4×5 + ⋯ + 31 10×11×12 . [2] (b) In the diagram , OABC is a parallelogram. Given that � OA����⃗ = a , � OC����⃗ = c , OD : DA = 3 : 1 , OE : EC = 1 : 2 , lines CD and BE intersect at point F. If � CF���⃗ = CD�����⃗ , EF����⃗ = EB�����⃗ , express OF�����⃗ in two different ways, and hence find the values of k and l. C B F c E O a D A [5]
7 6. (a) Use mathematical induction to show that cos + cos 3 + …… + cos(2 − 1) = sin 2 2 sin , n ∈ N. [6] (b) Given the polynomial () = 4 − 123 + 622 − 140 + 125, (i) show that 2 − 4 + 5 divides (), [2] (ii) hence, find all the complex number z that satisfies the equation f(z) = 0. [2] Section C : Analytic Geometry (Choose at least one but at most two questions in this section.) 7. Consider a hyperbola x 2 – y 2 = a 2 with parametric equation = sec θ and = tan θ . (a) Prove that the equation of the tangent to the hyperbola at point ( sec θ, tan θ) is x – ysin θ = acos θ; [3] (b) Let P and Q be two points on the hyperbola corresponding to θ = φ and θ = π 2 + φ respectively. Let N be the intersection point of the tangents to the hyperbola at P and Q. Prove that the coordinates of the point N is � cos∅ − sin∅ ,(cos ∅ + sin∅) cos ∅ − sin∅ � ; [3] (c) Find the Cartesian equation of the locus of N , and state which type of conic section this locus is. [4] 8. (a) Given that the line L is parallel to the line 2 + + 6 = 0 and is a tangent to the parabola y 2 = 8x , find the equation of L. Hence, find the shortest distance between the line 2 + + 6 = 0 and the parabola y 2 = 8x . [6] (b) Given that the line 5x – y + a = 0 is a tangent to the circle 3x² + 3y² – 2x + 4y + b = 0 at the point (c, –1), find the values of a, b and c . [4]
8 Section D : Calculus (Choose at least one but at most two questions in this section.) 9. (a) Without using a calculator, evaluate ∫ sin 2 cos 3 d. π 0 [4] (b) Find lim→∞ � −1 � . [3] (c) Find ∫ sin3 cos4 d . [3] 10. (a) Given that () = 22 − 1 , use the definition of derivative ′ () = limℎ→0 (+ℎ)− () ℎ to prove that ′ () = 4. [3] (b) Given that (+2)2 3(3−1) = A − 1 + B+C 2++1 , (i) Find the values of A, B and C. [4] (ii) Find ∫ (+2)2 3( 3−1) dx. [3] 11. (a) Given that any point (x , y) on a curve y = f(x) satisfies the equation ( − 2) d d + ( − 1) = 0. If the curve passes through the point (–2, 4), find the equation of the curve. [4] (b) A rectangle has two vertices on the x–axis , the other two on the part of the parabola = 25 − 2 that is above the x–axis. Find the maximum area of this rectangle. [6] 12. (a) Given that the area of an equilateral triangle is increasing at 8√3 cm 2 per second. When the side length is 16 cm, find the rate of change of the side length. [3] (b) Without using a calculator, evaluate ∫ sin 2 d . π 0 [3] (c) Use integration to find the area of the region in the ellipse 2 + 2 4 = 1 which lies in the first quadrant. [4]
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