NYGH 2022-S4EOY-IM2 with answer

Class Register Number Name End-of-Year Examination 2022 Secondary 4 INTEGRATED MATHEMATICS 2 2 hour 30 minutes Monday 3 October 0845 – 1115 Additional Materials: Nil READ THESE INSTRUCTIONS FIRST 1. Write your name, register number and class on all the work y ou hand in. 2. Write in dark blue or black ink. 3. You may use an HB pencil for any diagrams or graphs. 4. Do not use staples, paper clips, glue or correction tape/flu id. 5. Write your answers and working on the separate writing paper provided, unless otherwise stated. 6. Answer all questions. 7. Omission of essential working will result in loss of marks. 8. The use of an approved scientific calculator is expected, wh ere appropriate. 9. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degree to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. 10. At the end of the examination, fasten all your work securel y together. 11. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 6 printed pages. Setter(s): NYGH/HCI NANYANG GIRLS' HIGH SCHOOL [Turn over

2 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x = a acbb 2 42  Binomial expansion (a + b) n = a n +       1 n a n  1b +       2 n a n  2b2 + . . . +       r n a n  r br + . . . + b n , where n is a positive integer and ! ))......(( )!(! ! r rnnn rnr n r n 11 =        . 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  )(2 1sin)(2 1sin2coscos )(2 1cos)(2 1cos2coscos )(2 1sin)(2 1cos2sinsin )(2 1cos)(2 1sin2sinsin BABABA BABABA BABABA BABABA     Formulae for △ABC ௔ ୱ୧୬ ஺ൌ ௕ ୱ୧୬ ஻ൌ ௖ ୱ୧୬ ஼ 𝑎ଶ ൌ𝑏 ଶ ൅𝑐 ଶ െ2 𝑏 𝑐c o s𝐴 △ൌ 1 2 𝑏𝑐 sin 𝐴

3 [Turn Over 1 The equation of a curve is 225 7yx x  . (i) Express 225 7xx in the form 2()ax b c  , where a, b and c are constants. Hence sketch the curve, indicating clearly the turning point and the intercepts. [5] (ii) Find the range of values of m for which 225 7yx x  intersects the line 9ym x at two distinct points. [3] 2 (a) Solve 145 ( 2 ) 1 6 0xx   . [4] (b) The electric power, P watts, is given by 2P IR where I amperes is the current flowing through the circuit and R ohms is the resistance. Calculate the resistance of a 72 watts light bulb that draws  31 amperes of current. Leave your answer in the form 3ab , where a and b are integers. [3] 3 The expression 6(1 )p is equal to 23 4 5 61 6 15 20 15 6p pp p p p   . By using this result or otherwise, find the expression of 26(1 )x x in ascending powers of x up to the term in 3x . [4] 4 It is given that 75 (1 ) ( 21 ) 1 21 x AB x xxx     , where A and B are constants. (i) Determine the value of A and of B. [4] (ii) Hence, evaluate 2 0 75 d(1 ) ( 21 ) x xxx   . [3] 5 (a) Solve 3231 19 2 0xx x   , showing your working clearly. [4] (b) When a polynomial ()P x is divided by (1 )x and (2 )x , the remainders are 2 and 3 respectively. Find the remainder when ()P x is divided by (1 ) (2 )xx . [ 4 ]

4 6 A fact sheet on caffeine dependence from Johns Hopkins Medical Center states that the half-life of caffeine in the body is between 4 and 6 hours. Half-life is the time taken for the amount of caffeine in the body to decrease by half. The total amount of caffeine y mg in the body t hours after drinking the coffee can be modelled by the function tya b where a and b are constants. Rachel drinks a cup of coffee which has 120 mg of caffeine. Assuming that the half-life of caffeine in her body is 5 hours, (i) find the value of a and of b, [3] (ii) calculate the time it will take when there is only 20 mg of caffeine left in her body. [2] 7 It is given that 3 sin 3cos sin( )x xR xA  where R is a positive real number, the angles x and A are in radians and A is acute. (i) Find the value of R and of A. [2] (ii) Hence or otherwise, solve 3s i n 3c o s 3xx , where 02 x  . [3] 8 (a) Solve 24 1log 9 log 2 1 2xx   . [4] (b) Solve 3cos(180 4 ) 2sin 2 5yy   , where 0 360y   . [5] 9 (a) Sketch the graph of 1 sin 32yx  for 02 x   , indicating clearly the turning points and the intercept(s). [3] (b) The graph of 1 sin 32yx is obtained through two transformations from sinyx . Describe these two successive transformations. [2] 10 (i) Express 1 sin 4 sin 32 x x in the form sin cosab x c x , where a, b and c are constants. [1] (ii) Hence, show that 79 1sin cos cos sin sin 3 sin 522 2 2 2 xx x x x x  . [3]

5 [Turn Over 11 (i) Find an expression for   2 7d e3 l nd x x xx   . [3] (ii) Hence find the value of 22 7 1 2 el n d3 xx xx  , leaving your answer to 2 decimal places. [4] 12 The equation of a curve is 11 1cos 4 sin 242 2yx x . Point A(x, y) lies on the curve. (i) Find an expression for d d y x . [2] (ii) The variables x and y are such that y is decreasing at a rate of 0.05 units per second. Find the rate of change of x when x = 2.4. [3] (iii) Find the equation of the tangent at A when 2x  . Leave your answer i n t h e f o r m y ax b c , where a, b and c are constants. [ 4 ] 13 Cylinders make good drinking cans. The shape can stand upright on tables and fits comfortably into hands. The base radius of a closed cylindrical drinking can is x cm and the height of the cylinder is h cm. V cm3 and S cm2 represent the fixed volume and the total surface area of the cylinder respectively. (i) S h o w t h a t 2 22 VSx x . [2] (ii) The volume of a typical cylindrical drinking can is 320 cm3. Given that x and h can vary, find the value of x and of h for which S has a stationary value and determine whether this value of S is a maximum or a minimum. [5] (iii) Sam claimed that regardless of the volume, the height of a cylindrical drinking can is twice its radius for the total surface area to have a stationary value. Do you agree? Justify your answer. [3] [You do not need to determine the nature of the stationary val ue.]

6 14 A particle travels in a straight line. The velocity, v m/s, of the particle, t seconds after passing A, is given by 2 54vt t . The particle comes to rest momentarily first at point B and then at point C. The diagram shows part of the velocity-time graph of the particle. This diagram is not drawn to scale. (i) Find the values of t when the particle is at rest. [2] (ii) Find the total shaded area. [4] (iii) Explain the significance of your answer obtained in (ii). [1] (iv) The particle has zero acceleration at the point D. Determine with full working, whether D is nearer to A or to C. [5] End of Paper v O t

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2022 Sec 4 IM2 EOY Exam Answers 1(i) 2 58 12 48x  1(ii) 9o r 1mm  2(a) 1 or 3xx 2(b) 72 36 3 3 231 6 9 10 ...xx x   4(i) A = 2 B = 3 4(ii) 4.61 5(a) 2 or 1.85 or 0.180xx  5(b) x + 1 6(i) a = 120 1 15 51 or 22b  6(ii) 12.9 hours 7(i) 23R 3A  (ii) 11 or 26x   8(a) 10 or 1 (rejected)x 8(b) 135 or 315y 

9(a) (b) Scale along y-axis by a factor of 1 2 . Translate in positive y-direction by 3 units. 10(i) 7sin cos22 x x 11(i) 2 723 3 l nxxex  11(ii) 18964.78 (to 2 dp) 12(i) sin 4 cos 2