NYGH 2016-S3EOY-IM2 with ans

Class Register Number Name 南洋女子中学校 NANYANG GIRLS' HIGH SCHOOL End-of-Year Examination 2016 Secondary Three INTEGRATED MATHEMATICS 2 2 hours Tuesday 11 Oct 2016 0845-1045 READ THESE INSTRUCTIONS FIRST INSTRUCTIONS TO CANDIDATES 1. Write your name, register number and class in the spaces at the top of this page. 2. Answer questions 1 - 11 before attempting question 12 (Bonus Question). 3. Write your answers and working on the separate writing paper provided. 4. Omission of essential working will result in loss of marks. 5. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION FOR CANDIDATES 1. The number of marks is given in brackets [ ] at the end of each question or part question. 2. The total number of marks for this paper is 80. 3. You are reminded of the need for clear presentation in your answers. Setter: A. Low This document consists of 6 printed pages. NANYANG GIRLS' HIGH SCHOOL

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0 , x = a acbb 2 42−±− . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ∓ sin A sin B tan (A ± B) = BA BA tantan1 tantan ∓ ± sin 2A = 2 sin A cos A cos 2A = cos 2A – sin 2A = 2 cos 2A – 1 = 1 – 2 sin 2A tan 2A = A A 2tan1 tan2 − Formulae for ΔABC C c B b A a sinsinsin == a2 = b2 + c2 − 2bc cos A Δ = 21ab sin C 3. MENSURATION Arc length = rθ, where θ is in radians Sector area = θ2 2 1r, where θ is in radians

[Turn over 3 1 A cup of hot water is left on a table to cool. After t minutes, the temperature of the water, T °C is given by T = 27 + 30e–0.02t. (i) State the initial temperature of the cup of water. [1] (ii) Find the time it takes for the cup of water to cool to 41°C. [2] (iii) The temperature of the cup of water will not go below x °C. Write down the largest value of x given that x is an integer. [1] 2 The graph of y = f(x) cuts the coordinate axes at the points (4, 0) and (0, –6). Find the coordinates of the points where the following graphs cut the coordinate axes: (a) y = f –1 (x), [2] (b) y = f (–2x) , [2] (c) y=32f(x). [2] 3 (a) It is given that y=3x2−4x+5. (i) Express y in the form a(x−h)2+k. [2] (ii) Hence, (a) write down the equation of the line of symmetry of y = 3x2 – 4x +5, [1] (b) state the value of c, given that the line y = c is tangent to the curve y=3x2−4x+5. [1] (b) Find the range of values of b for which 6x2 – 7bx + 2b2 + 6 ≥ 0 for all real values of x. [3] 4 The roots of the quadratic equation 3x2 + 6x + 5 = 0 are α and β. Find a quadratic equation whose roots are and . [5] 5 Solve the equation 3 + 8cot2 x = 18cosec x, for 0° ≤ x ≤ 360°. [5] α22ββ22α

[Turn over 4 6 In the diagram, A, B, C and D are points on a circle with centre O and diameter AC. It is given that AC = 7 cm and ∠CAB=0.8radians. (i) Explain briefly why ∠BOC=1.6radians. [1] (ii) Find the perimeter of the region bounded by the arc BC and the lines AB and AC. [3] (iii) Find the area of the major segment ABCD. [3] 7 It is given that cosA=−13, where 180° ≤ A ≤ 360°. Find, without the use of calculators, the value of (i) tan A, [2] (ii) cosec A, [1] (iii) sin (A + 60°), giving your answer in the form a+b6. [3] 8 (a) Find the remainder when 4x3 + 3x2 – 3x + 14 is divided by 2x – 1 . [2] (b) (i) Given that 4x3 + 3x2 – 3x + 14 = (x + 2)(4x2 + Bx + C) find the value of B and of C.has only one real root. Hence, show that 4x3 + 3x2 – 3x + 14 has only one real root. [3] (ii) Express x2−5x+194x3+3x2−3x+14as partial fractions. [5] A C B 7 cm 0.8 D O

[Turn over 5 9 (a) Solve the following equations. (i) 45 – 3x = 62x, [3] (ii) 3lny+lg2y=4. [3] (b) Solve the equation2z+z=3, giving your answer in the form a+b2 where a and b are integers. [4] 10 It is given that f(x) = 6cos2 x – 2sin2 x for 0≤x≤32π. Show that f(x) can be written as f(x) = 2+4cos2x. [2] Hence, (i) write down the amplitude and period of f(x). [2] (ii) solve the equation f(x) = 0 for 0≤x≤32π. [3] (iii) sketch the curve y = f(x) for 0≤x≤32π, showing clearly all the intercepts. [2] (iv) given that g(x) = 12f(x – π) , state the range of g(x) for the domain0≤x≤32π. [1] 11 The function f is given by f: x!x−2+3, x ∈ ℝ. (i) Find f 2(–4). [2] (ii) Sketch the graph of y = f(x), showing clearly the coordinates of the highest/lowest point and the intercept(s). [2] (iii) State the range of f. [1] (iv) Write down the range of values of c for which the equation x−2+3=x+c has exactly one solution. [1] The function g is given by g: x!x−2+3, x≤k. (v) Write down the largest value of k for which the function g –1 exists. [1] (vi) Find the function g –1 in a similar form and state its domain. [3]

[Turn over 6 Bonus Question 12 Find an expression for f(x) for which 4f(2x) – 5f 2x⎛⎝⎜⎞⎠⎟=3x. [3] End of Paper

[Turn over 7 2016 Sec 3 IM2 EOY Answers 1(i) 57°C 1(ii) 38.1 minutes 1(iii) 27°C 2(i) (−6,0) and (0, 4) 2(ii) −2,0()and (0, –6) 2(iii) (4, 0) and (0, 9) 3(a)(i) y=3x−23⎛⎝⎜⎞⎠⎟2+323 3(a)(ii)(a) x=23 3(a)(ii)(b) c=323 3(b) −12≤b≤12 4 60x2−36x+25=0 5 x = 23.6° or 156.4° 6(i) ∠BOC=2(0.8) = 1.6 rad (angle at centre is twice angle at circumference) 6(ii) 17.5 cm 6(iii) 35.2 cm2 7(i) 7(ii) −62 7(iii) −12−166 8(a)(i) 8(b)(i) B = 5 C = 7 Since the discriminant is 87 which, 4x2 5x + 7 = 0 has no real roots. Thus f(x) = 0 has one real root. 8(b)(ii) x2−5x+194x3+3x2−3x+14=1x+2+6−3x4x2−5x+7 9(a)(i) 0.895 9(a)(ii) 2.94 9(b) 21334−−−z=27−182

[Turn over 8 10 f(x) = (6cos2 x – 2sin2 x) =61+cos2x2⎛⎝⎜⎞⎠⎟−21−cos2x2⎛⎝⎜⎞⎠⎟ = 3 + 3cos 2x – 1 + cos 2x = 2 + 4 cos 2x 10(i) Amplitude = 4 Period = π 10(ii) x = π3,2π3,4π3 10(iii) 10(iv) −1≤g(x)≤3 11(i) 10 11(ii) 11(iii) f(x) ≥3 11(iv) c > 1 11(v) The largest value of k is 2 . 11(vi) g –1: x!5−x, x≥3. x0.1π0.2π0.3π0.4π0.5π0.6π0.7π0.8π0.9ππ1.1π1.2π1.3π1.4π1.5π -2 -1 0 1 2 3 4 5 6 x y -4 -3 -2 -1 0 1 2 3 4 5 6 70 1 2 3 4 5 6 (2, 3)

[Turn over 9 Bonus 12 f(x)=−23x−103x