Cambridge IGCSE Mathematics - Additional 0606 — 2017 Feb/March Paper 2 · Variant 2

2 The value, V dollars, of a car aged t years is given by V = 12000 e -0.2t . (i) Write down the value of the car when it was new. [1] 2 (ii) Find the time it takes for the value to decrease to of the value when it was new. [2] 3

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3 The polynomial p ( x) is x 4 - 2x 3 - 3x 2 + 8x - 4 . (i) Show that p ( x) can be written as ( x - 1 )( x 3 - x 2 - 4 x + 4 ) . [1] (ii) Hence write p ( x) as a product of its linear factors, showing all your working. [4]

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4 Find the set of values of k for which the line y = 3x + k and the curve y = 2x 2 - 3 x + 4 do not intersect. [4]

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5 x 3 + 5 20 The diagram shows a trapezium made from a rectangle and a right-angled triangle. The dimensions, in centimetres, of the rectangle and triangle are shown. The area, in square centimetres, of the trapezium is 13 + 5 5 . Without using a calculator, find the value of x in the form p + q 5 , where p and q are integers. [5]

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6 (a) (i) Express - 8x 9 x -3 in the form axb, where a and b are constants to be found. [2] 3 9 6 -3 (ii) Hence solve the equation - 8x x =- 6250 . [2] ` `j j (b) It is given that y = log a ( ax) + 2 log a ( 4x - 3) - 1, where a is a positive integer. (i) Explain why x must be greater than 0.75. [1] - 24x + 9x) . [3] (ii) Show that y can be written as log a ( 16x 3 2 (iii) Find the value of x for which y = log a ( 9x) . [2]

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7 (a) Calculate the magnitude and bearing of the resultant velocity of 10 ms -1 on a bearing of 240° and 5 ms -1 due south. [5] (b) A car travelling east along a road at a velocity of 38 kmh -1 passes a lorry travelling west on the same road at a velocity of 56 kmh -1 . Write down the velocity of the lorry relative to the car. [2]

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8 The points A(3, 7) and B(8, 4) lie on the line L. The line through the point C(6, −4) with gradient 6 meets the line L at the point D. Calculate (i) the coordinates of D, [6] (ii) the equation of the line through D perpendicular to the line 3y - 2x = 10 . [2]

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9 (a) Find e 2x + 1 dx . [2] y x d (b) (i) Given that y = , find y. [3] ln x d x J 1 1 1 N - + (ii) Hence find 2 2 OO d x . [3] y KK ln x ( ln x) x L P

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10 Solve, for 0° G x G 360° , the equation 3 (i) cot ( 2x - 10 °) = , [4] 4 (ii) sin 2 x - cos 2 x = cos x . [5]

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11 The functions f and g are defined by x 2 - 2 f ( x) = for x H 2 , x x 2 - 1 g ( x) = for x H 0 . 2 (i) State the range of g. [1] (ii) Explain why fg(1) does not exist. [2] 2 c (iii) Show that gf ( x) = ax + b + 2 , where a, b and c are constants to be found. [3] x (iv) State the domain of gf. [1] 2 -1 x + x + 8 (v) Show that f ( x) = . [4] 2

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12 The diagram shows a shape made by cutting an equilateral triangle out of a rectangle of width x cm. x cm The perimeter of the shape is 20 cm. (i) Show that the area, A cm2, of the shape is given by J N 6 + 3 A = 10x - KK OO x 2 . [3] 4 L P (ii) Given that x can vary, find the value of x which produces the maximum area and calculate this maximum area. Give your answers to 2 significant figures. [4]

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