VIEWS: 1,200 PAGES: 8 CATEGORY: Algebra POSTED ON: 10/21/2012
Program Latih Tubi SPM 2012 Matematik Tambahan Kertas 2
1 3472/2 TOPIC: SIMULTANEOUS EQUATIONS 1 1. Solve the simultaneous equations x + y = 1 and y2 – 10 = 2x. [5 marks] 2 2. Solve the simultaneous equations x + 2y = 1 and x2 + 2y2 + xy = 5. Give your answers correct to three decimal places. [5 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 2 3472/2 TOPIC: TRIGONOMETRIC FUNCTIONS 1. (a) Prove that cosec2 x – 2 sin2 x cot2 x = cos 2x. [2 marks] (b) (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2. (ii) Hence, using the same axes, draw a suitable straight line x to find the number of solutions to the equation 3(cosec2 x 2 sin2 x – cot2 x) = 1 for 0 ≤ x ≤ 2.. State the number of solutions. [6 marks] 2. (a) Sketch the graph of y = 2 sin 2x for 0 ≤ x ≤ 2. [4 marks] (b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the equation + 2 sin 2x = 0 for 0 ≤ x ≤ 2.. State the number of solutions. [3 marks] x 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 3 3472/2 3. (a) Sketch the graph of y = | 3cos 2x | for 0 ≤ x ≤ 2. [4 marks] (b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the x equation 2 - |3cos 2x | = for 0 ≤ x ≤ 2. State the number of solutions. [3 marks] 2 4. (a) Sketch the graph of y = 1 + 3 sin x for 0 ≤ x ≤ 2. [4 marks] (b) Hence, using the same axis, sketch a suitable straight line to find the number of solutions to the equation 6π sin x = 4π – 3x for 0 ≤ x ≤ 2. State the number of solutions. [3 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 4 3472/2 TOPICS: COORDINATE GEOMETRIC AND DIFFERENTIATION 1. A curve has a gradient function px2 – 4x, where p is a constant. The tangent to the curve at the point (1, 3) is parallel to the straight line y + x – 5 = 0. Find (a) the value of p. [3 marks] (b) the equation of the curve. [3 marks] 2 2. A curve with a gradient function 2x – has a turning point at (k, 8). x2 (a) Find the value of k. [3 marks] (b) Determine whether the turning point is a maximum or a minimum point. [2 marks] (c) Find the equation of the curve. [3 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 5 3472/2 TOPICS: QUADRATIC FUNCTIONS 1. Diagram 2 shows the curve of a quadratic function f(x) = −x2 + kx – 5. The curve has a maximum point at B(2, p) and intersects the f(x)-axis at point A. (a) State the coordinates of A. [1 mark] (b) By using the method of completing the square, find the value of k and of p. [4 marks] (c) Determine the range of values of x, If f(x) ≥ − 5. [2 marks] f(x) x 0 B(2, p) A Diagram 2 2. The gradient function of a curve is hx2 – kx, where h and k are constants. The curve has a turning point at (3, −4). The gradient of the tangent to the curve at the point x = −1 is 8. (a) the value of h and of k.. [5 marks] (b) the equation of the curve. [3 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 6 3472/2 TOPIC: COORDINATE GEOMETRIC 3 1. Solution by scale drawing is not accepted. y P( , k) 2 Diagram 10 shows a triangle OPQ. Point S lies on the line PQ. (a) A point W moves such that its distance S(3, 1) from point S is always 2½ units. Find the equation of the locus of W. [3 marks] O x Q Diagram 10 (b) It is given that point P and point Q lie on the locus of W. Calculate (i) the value of k. (ii) the coordinate of Q. [5 marks] (c) Hence, find the area, in unit2, of triangle OPQ. [2 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 7 3472/2 2. Solution by scale drawing is not accepted. Diagram 3 shows the triangle AOB where O is the origin. Point C lies on the straight line AB . (a) Calculate the area, in unit2 , of the triangle AOB. [2 marks] (b) Given AC : CB = 3 : 2, find the coordinate of C. [2 marks] . y A(−3, 4) C x O B(6,−2) Diagram 3 (c) A point P moves such that its distance from point A is always twice its distance from point B. (i) Find the equation of the locus of P. (ii) Hence, determine whether this locus intersects the y-axis. [6 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012 8 3472/2 TOPIC: VECTORS 5 2 k 1. Given that AB = , OB = and CD = , find 7 3 5 (a) the coordinates of A. [2 marks] (b) the unit vector in the direction of OA . [2 marks] (c) the value of k, if CD is parallel to AB . [2 marks] 2. In diagram 6, ABCD is a quadrilateral. The diagonals BD and AC intersect at point R. Point P lies on AD. D 1 1 It is given that AP = AD, BR = BD , 3 3 AB = x and AP = y. C P R (a) Express in terms of x and y : (i) BD A B (ii) AR [3 marks] Diagram 6 (b) Given that DC k x y and AR = h AC , where h and k are constant, find the value of h, and of k. [4 marks] 3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012