1 霹雳怡保培南独立中学 SEKOLAH MENENGAH POI LAM (SUWA) FINAL EXAMINATION 2021 ADVANCED MATHEMATICS (II) PAPER 2 SUBJECTIVE QUESTIONS ___________________________________________ DATE : 16.11.2021 (TUESDAY) TIME : 1345 – 1545 (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 : ______CHAI_____ Signature : __________________
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5 Section A : Trigonometry (Choose at least one but at most two questions in this section.) 1. (a) Prove that tan 3 − tan 2 − tan = tan 3 tan 2 tan . [3] (b) Solve the equation 3sin−1 − cos−1 = sin−1 1 2 . [3] (c) Find the general solution of the equation 5sin2 + sin 2 − 3cos2 = 2. [4] 2. (a) Express = sin − √3 cos in the form of sin ( − ), where > 0 and 0° < < 90°, find the values of R and . [3] If sin − √3 cos = 4−6 4− , find the range of values of a. [3] (b) The angle of elevation from point B to the top of a tower is measured to be . Walking horizontally towards the tower a distance of 30 m, at point C, the angle of elevation now is 2. Moving horizontally another 10√3 m towards the tower, at point D, the angle of elevation increases to 4. Find the value of and the height of the tower. [4] Section B : Algebra (Choose at least one but at most two questions in this section.) 3. (a) Given that three sides of a right-angled triangle are in an arithmetic sequence and that the area of this triangle is 24. Find the length of the hypotenuse. [3] (b) Find the value of n if the sum of the first n terms of a geometric sequence 4 1 2 , 3, 2, 1 1 3 , … … .. is just greater than 13.49. [3] (c) A game is played by three players A, B and C. Each of them has 5 electric bulbs, numbered 1, 2, 3, 4, 5 respectively in front of them. When the whistle is blown, each player will select a bulb and switch it on, and the number shown on the selected bulb is recorded as a, b, c corresponding to A, B, C respectively. Find (i) the probability that a, b, c are all different numbers, [2] (ii) the probability that < < . [2] 4. (a) Solve the inequality 2−4 2−3+2 ≥ + 1. [3] (b) Given that () is a function and () = () + (), prove by mathematical induction that ( ) = (), where ∈ . [3]
6 (c) (i) Given that +2 (+1)2 ≡ ∙2 + (+1)∙2 , find the value of A and of B. [2] (ii) Hence, find the value of ∑ +2 (+1)2 ∞ =1 . [2] 5. (a) Solve the equation √3 − 5 − √ + 2 = √ − 6. [4] (b) Given the complex numbers 1 = 1 + √3 and 2 = 1 − . (i) Find the trigonometric forms of 1 and 2. [2] (ii) Without using a calculator, find all the complex numbers z that satisfy the equation 3 = 1 3 2 12. Express your answers in the form + , where x and y are real numbers. [4] 6. (a) If , and are the roots of the equation 4 3 + 3 2 − 2 + 1 = 0, find the equation whose roots are (1 − 1 ) , (1 − 1 ) and (1 − 1 ). [4] (b) Resolve () = 2−4 (1− 2)(1−2) into partial fractions. [3] Hence, expand () in a series of ascending powers of x up to the term containing 2 , and state the range of values of x for which the expansion is valid. [3] Section C : Analytic Geometry (Choose at least one but at most two questions in this section.) 7. (a) By using the method of differentiation, show that the gradient at point ( 2 , 2) of a parabola 2 = 4 is 1 . Suppose the tangents to the parabola 2 = 4 at the points ( 2 , 2) and ( 2 , 2) meet at R where ≠ , find the coordinates of R. Prove that the area of Δ = 1 2 2 | − | 3 . [6] (b) = + is a tangent to the parabola 2 = 3. Without finding the value of m, determine the coordinates of its point of contact. [4] 8. (a) A straight line intersects the hyperbola 2 2 − 2 2 = at points 1 and 2. (i) Suppose M is the midpoint of 1 and 2 , find the coordinates of M. [4] (ii) If the coordinates of M is (h, k), find the equation of l. [3]
7 (b) The line = + intersects the rectangular hyperbola = 16 at the points P and Q. Suppose R is the midpoint of the line segment PQ, find the coordinates of R in terms of m and c. Hence, find the equation of the locus of R in terms of m. [3] Section D : Calculus (Choose at least one but at most two questions in this section.) 9. (a) Find the integral ∫ 7−4 2 2−3−2 . [3] (b) Using substitution = 4sin2, or otherwise, evaluate ∫ √ 4− 2 0 . [3] (c) A body moves along a line with an acceleration of 2 2 = 4 (4−)2 , where s meters is the distance travelled for the time t seconds. Its velocity is 2 m/s when = 0. Find the total distance travelled by the body from = 1 to = 2. [4] 10. (a) The volume of the water in a trough is (2 3 − 2 2 + 5) cm3 when the depth of the water is x cm. Water is poured into the trough at a constant rate of 180 cm3 per second. Find the rate of water level rise when the depth of the water in the trough is 5 cm. [3] (b) In the figure, at point A, y = 0, while at the point B, x = 0. Given that point A is a maximum point, while B is a minimum point and dy dx = −6x 2 + 6x. Find the area of the shaded region. [4] y A O x B (c) If 3 + = 2 cos , find the value of d 2 d 2 at the point (0, 1). [3]
8 11. (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] 12. (a) Given that the area of an equilateral triangle is increasing at 8√3 cm2 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 ∫ x sin 2x dx . π 0 [3] (c) Use integration to find the area of the region in the ellipse x 2 + y 2 4 = 1 which lies in the first quadrant. [4] END OF PAPER
