Add Math (4047/02) Year t years $P
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1 Add Math (4047/0) Requirement : Answer all questions Total marks : 100 Duration : hour 30 minutes 1. The price, $P, of a company share on 1 st January has been increasing each year from 1995 to 015. The company claims that this increase is exponential and so can be modelled by an equation of form P P 0 ekt where P0 and k are constants and t is the time in years since 1 st January The table below gives values of P and t for some of the years 1995 to 010. Year t years $P (i) Plot a suitable straight line graph to show that the model is valid for the years 1995 to 010. [] (ii) Estimate the value of P0 and k. [3] (iii) Assuming that the model is still appropriate, estimate the price of a share on 1 st January 015. [] (i) Given that P P0 ekt, ln P ln P kt To plot ln P against t. 0 Year t years $P ln P Prepared by Mr Ang, Nov 016 1
2 Add Math (4047/0) ln P O t (ii) When t = 0, ln P 0.7, 0.7 ln P0, 0.7 P0 e.01. the gradient k (3 s.f.) 0.05 (iii) When t = 0, ln P 1.475, P e 4.37 (3 s.f.) the price of a share on 1 st January 015 is $4.37 Prepared by Mr Ang, Nov 016
3 Add Math (4047/0) 6. (i) Given that the coefficient of x in the expansion of 1 x 1 px is 16, find the two possible values of the constant p. [5] (ii) For each value of p, find the coefficient of x 3 in the expansion of 1 px 6. [3] From 1 x 1 4x 4x 1 px C 1 px C 1 px C 1 px C 1 px (ii) When the coefficient of x, 15p 4 p1 0 5p 8p 4 0 p p 1 4x 4x 1 4x 6px 4px 15p x 5 0 p or p 5 15p x 4 4 p15 p 16 p, the coefficient of x 3, C p When 5 p, the coefficient of x 3, C p Prepared by Mr Ang, Nov 016 3
4 3. (i) Using cos 3x cos x x Add Math (4047/0), show that cos3x may be expressed as cos x1 4sin x. (ii) Find all the values of x between 0 and 360 for which cos3x 15sin x cos x. [5] (i) Let cos 3x cos x x, cos x cos x sin x sin x cos xcos x 3sin x cos x1 4sin x cos x sin x cos x sin x cos x sin x (ii) cos 3x 15sin x cos x cos 1 4sin 15sin cos x x x x cos x 15sin x 8sin x 0 cos x sin x 18sin x 0 cos x 0 or sin x or When cos x 0, x 90 or x 70 When sin x, no solution as sin x 1 When 1 sin x, sin x 8 x sin 8 Principal angle, x 7. (1 d.p.) x 7. or x (1 d.p.) Therefore, x 7. or x 90 or x 17.8 or x 70 [3] Prepared by Mr Ang, Nov 016 4
5 4. The roots of the quadratic equation (i) Add Math (4047/0) x x 5 0 are and. Find 3 3 the value of, [5] (ii) a quadratic equation with roots and. [3] (i) (ii) or x x x x 5 0 Prepared by Mr Ang, Nov 016 5
6 Add Math (4047/0) 5. A D P \ / C B The diagram shows a circle passing through the vertices of a triangle ABC. The tangents to the circles at A and B intersect at the point P. The point D lies on AC such that DC = DB. (i) Prove that angle APB + angle ADB = 180 [5] (ii) Given that A, P, B and D lie on a circle, prove that PD and BC are parallel. [] (i) Let O be the centre of the circle. A D P \ O / C B APB AOB 180 (tangent from external point P) 1 ACB AOB ( s at circumference = ½ of at the centre of the circle) CDB 180 ACB ( CDB is an isosceles) CDB 180 AOB CDB APB CDB APB Since CDB ADB 180 (straight angle) APB ADB 180 Prepared by Mr Ang, Nov 016 6
7 Add Math (4047/0) (ii) A D P \ O / C B PDB PAB ( s in the same segment) 1 ACB DCB AOB ( s at circumference = ½ of at the centre of the circle) 1 DCB DBC AOB ( CDB is an isosceles) Since PAB 180 APB, PAB AOB 1 AOB (tangent from external point P) 1 Therefore, PDB AOB DBC, (alternate s) Hence, PD and BC are parallel Prepared by Mr Ang, Nov 016 7
8 6. It is given that y x x 5 3 (i) Add Math (4047/0). Obtain an expression for d y in the form ax bx 5, where a and b are dx integers, [3] (ii) Determine the values of x for which y is a decreasing function. [1] The variables x and y are such that, when x = 3, y is increasing at a rate of 0.35 units per second. (iii) Find the rate of change of x when x = 3. [] It is given further that the variable z is such that z = y. (iv) Show that, when x =3, z is increasing at twice the rate of y. [] (i) Given that y x x 5 3, dy dx dy x 5 x 5 6x dx ( ) d y 8x 17x 5 dx x 5 3 3x x 5 (ii) for which y is a decreasing function, d y 0 dx x x ( ) 1 (+) 1 (+) 8 x 1 x 8 dy dy dx (iii) dt dx dt. d y 0.35 dt, when x 3. when x 3, d y dx dx dt dx 0.05 dt Prepared by Mr Ang, Nov 016 8
9 7. (i) Given that u x, express Add Math (4047/0) x1 x 6 as an equation in u. [3] (ii) x1 x Hence, find the values of x for which 6, giving your answer, where appropriate, to 1 decimal place. [4] (iii) Explain why the equation x1 x k has no solution if k 8. [3] (i) x1 x 6 1 x 4 x 6 0 u 8u1 0 (ii) u 8u1 0 u u 6 0 u or u 6 When u, x, x 1 When u 6, x 6, lg 6 x.6 (1 d.p.) lg (iii) Given that x1 x k, u u k 8 0 For the equation to have real solutions, the discriminant 0. k k 0 k 8 For the equation x1 x k to have no solution, then k 8. Prepared by Mr Ang, Nov 016 9
10 8. It is given that 3 Add Math (4047/0) f x x 3x 4x 1. (i) (ii) By showing clearly your working factorise f x. [3] Explain why the equation f x 0 has only one real root and state its value. [] (iii) Find the value of the constant k for which the graph of d y point at which d y f x kx has a stationary 0 x. [5] 3 (i) f , x 3 is a factor of By long division, (ii) Since (iii) f x 0, x x x f x 3x 4x 1 x x x 4 4 0, x 3 0 x 3 y f x kx 3 y x 3x 4x 1 kx dy d x d y d x x x x k 6 6 d y For 0 dx, x 1 When x 1, dy k dx dy 1 k dx To be a stationary point at x 1, d y 0 dx 01 k k 1 f x. Prepared by Mr Ang, Nov
11 Add Math (4047/0) 9. y B O x 3 y x x 3x A The diagram above shows part of the graph of 3 y x x 3x. The x-coordinate of the point A is 3. (i) Find the gradient of the curve at A. [4] The tangents to the curve at the points A and B are parallel. (ii) Find the x-coordinate of point B. [3] (iii) Showing all your working, find the total area of the shaded region bounded by the curve, the x-axis and the lines from A and B perpendicular to the x-axis. [4] (i) When dy dx x, 3 3x 4x 3 dy dx 3 3 the gradient of the curve at A is 1. (ii) 1 3x 4x 3 3x 4x 4 0 3x x 0 x or x 3 the x-coordinate of point B is. Prepared by Mr Ang, Nov
12 Add Math (4047/0) (iii) total area of the shaded region x x 3x dx x x 3x dx x x 3 3 x x x x unit 81 Prepared by Mr Ang, Nov 016 1
13 Add Math (4047/0) 10. A car, driven along a straight road, passes a signpost, A, with a speed of p km/h. A little later the car passes a second signpost, B, with a speed of 80 km/h. Between signposts A and B, the speed, v km/h, of the car is given by where t, the time after passing A, is measured in hours. 5t v 30e 0, (i) State the value of p. [1] (ii) Calculate, to the nearest second, the time taken to travel from A to B. [3] (iii) Calculate the distance between A and B. [5] (iv) Obtain an expression, in terms of t, for the acceleration of the car between A and B. [] (i) When t 0, 0 p 30e 0 50 (ii) At B, let t T, 5T 80 30e 0 5 e T 5T ln ln T hours (3 s.f.) 5 the time taken to travel from A to B, 100 seconds (iii) the distance between A and B ln 5 0 ln e 5 5t 5 0 dt 30 5e 5t 0 d ln 0 6 e 5t 0 t 5 6 ln ln 6 0 e 0 e ln ln (3 s.f.) the distance between A and B is 1.75 km t Prepared by Mr Ang, Nov
14 Add Math (4047/0) (iv) the acceleration of the car between A and B, dv 305e 5 t dt dv 5t 750e dt Prepared by Mr Ang, Nov
15 Add Math (4047/0) 11. The equation of a circle, C1, with centre A, is x y x y (i) Find the coordinates of A and the radius of C1. [4] (ii) Show that the point P(10, 7) lies on C1. [1] (iii) Find the equation of the tangent to C1 at P. [3] A second circle, C, has diameter AP. (iv) Find the equation of C. [3] (v) Find the equation of the tangent to C at P. [1] (i) x y x y 4 95 x x y y x y,1 (ii) P(10, 7) 1 10 A, radius= point P(10, 7) lies on C1 (iii) A,1, P(10, 7) Gradient of AP, m AP Gradient of tangent to C1 at P, m AP the equation of the tangent to C1 at P, y7 x y x (iv) Midpoint of AP, 6,4 10 radius 5 the equation of C, x y Prepared by Mr Ang, Nov
16 Add Math (4047/0) (v) the equation of the tangent to C at P = the equation of the tangent to C1 at P 4 61 y x 3 3 Prepared by Mr Ang, Nov
, correct to 4 significant figures?
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