1 霹雳怡保培南独立中学 SEKOLAH MENENGAH POI LAM (SUWA) SECOND MONTHLY TEST 2022 ADVANCED MATHEMATICS (II) ___________________________________________ DATE : 17.08.2022 (WEDNESDAY) TIME : 0830 – 1000 (1 ½ HOURS) ____________________________________________ NAME : _____________________ REGISTRATION NUMBER: _____________ CLASS : S3LI INSTRUCTIONS TO CANDIDATES 1. Begin each question on a fresh page. 2. Use only blue or black ink to write your answers and use pencil for drawing only. 3. Do not copy the questions, but the answer to each question should be clearly numbered. 4. Show all mathematical working clearly. Geometric figures should be drawn where necessary. 5. Unless otherwise specified, the prescribed electronic calculators may be used. 6. Arrange the answer scripts in numerical order and tie them together. 7. The total number of marks for this paper is 60. Do Not Turn Over This Page Until You Are Told To Do So ________________________________________________________________________________ This document consists of 3 printed pages (including this page) Prepared by : Ms Chai Siew Yin Checked by : Mr Yeow Ghee Ruey Signature : _________________ Signature : __________________
2 1. (a) Solve the equation cos−1 + cos−1 3 = 2 , where x 0 . [4] (b) Given the inverse trigonometric identity sin −1 + cos−1 = 2 , where – 1 x 1. If sin −1 + sin −1 = 2 3 and cos−1 − cos−1 = 3 , find the values of x and y. [4] 2. Prove that sin 4 + cos 4 = 1 − 1 2 sin 22. Hence, find the general solution of the trigonometric equation 2sin 4 + 2cos 4 = sin 2. [4] 3. (a) Given that p, q and r are three propositions. By using a truth table, prove that → (q r) ((p q) r). [5] (b) Evaluate − 3+ 12 1+ 8 . [3] 4. (a) Given that = cos + sin , prove that + 1 = 2 cos . [2] Using the expansion of + 1 4 , show that 4 = 1 8 cos 4 + 1 2 cos 2 + 3 8 . [4] (b) Given that = 1 2 + 3 2 , prove that (i) 1 = , where is the conjugate of z , (ii) 2 = − , (iii) 3 =− 1. [6] 5. (i) The tangent at the point P , on the rectangular hyperbola = 2 meets the x - axis at A and the y - axis at B. The normal at P meets the line y = x at C and the line y = – x at D. Find the coordinates of the points A, B, C and D. [4] (ii) The normal at P meets the hyperbola again at another point Q and the mid-point of PQ is M. Prove that the equation of the locus of the point M is 2 2 − 2 2 + 4 3 3 = 0. [6]
3 6. An ellipse is symmetrical about the x and y axes and passes through the point (2, 4). Suppose the major axis of the ellipse is three times the minor axis, where the major axis is parallel to the x - axis, find the equation of the ellipse. [3] 7. (i) Find the expansion of 1 + 3 3 2 in ascending powers of x up to the term 3 , and state the range of values of x for which the expansion is valid. [3] (ii) Hence, find the value of 1.15 3 2 correct to 4 decimal places. [2] 8. Use mathematical induction to show that cos + cos 3 + . . . . . + cos 2 − 1 = sin 22 sin , ∈ ℕ. [5] 9. (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] END OF PAPER