- X 1 00/303 NA TIONAL QUALIFICA 2003 TIONS WEDNESDAY, 21 MAY 10.30 AM - 12.00 NOON MATHEMATICS HIGHER Units 1,2 and 3 Paper 2 L Read Carefully 1 Calculators may be used in this paper. 2 Fullcreditwillbe givenonlywherethe solutioncontainsappropriateworking. 3 Answersobtainedby readingsfromscaledrawingswill not receiveanycredit. '-" \v: SCOTTISH7\. QUALIFICATIONS AUTHORITY LIB X100/303 613/26570 11111111111111111 1111111111 11 I11 1111111111 11111111 @ ALL questions = 6x3 - 1. lex) should be attempted. Marks 5x2 - 17x + 6. (a) Show that (x - 2) is a factor off(x). (b) Expressf(x) 4 in its fully factorised form. 2. The diagram shows a sketch of part y of the graph of a trigonometric function whose equation is of the form y = a sin(bx) + c. Determine ~-- - ----------- 5 3 the values of a, band c. ~ x -3 3. The incomplete graphs off(x) = x2 + 2x y and g(x) = x3 - x2 - 6x are shown in the diagram. The graphs intersect at A( 4, 24) and the origin. Find the shaded area enclosed between the curves. 5 x y=g(x) "-' - - 4. (a) Find y = x3 the + 2x2- equation of the 3x + 2 at the point tangent where x to the curve with equation 5 = 1. (b) Show that this line is also a tangent to the circle with equation x2 + y2 - 12x - 10y + 44 = 0 and state the coordinates of the point of contact. 6 [Turn over [XI00/303] Page three - Marks s. The diagram a function j. y shows the graph of y = f(x) f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2). (a) Sketch the graph of y =f(-x). x (b) On the same diagram, sketch the graph of y = 2f( -x). 2 2 -3 6. 4 Hf(x) = cos(2x) - 3 sin(4x), find the exact value of f'( ~). ...JJ 7. Part of the graph of y = 2sin(x O) + Scos(x O) is shown in the diagram. (a) Express y = 2sin(x y y = 2sin(x O) + Scos(x O) O) + Scos(x O) in the form ksin(x °+ a O)where k > 0 and 0 ~ a < 360. 4 0 (b) Find the coordinates of the minimum turning point P. 8. An open water tank, in the shape of a triangular prism, has a capacity of 108 litres. The tank is to be lined on the inside in order to make .it watertight. 3 ~ ....... .:-:<::<..:>:>:::-:.:.. . ~ . ) The triangular cross-section of the tank is right-angled and isosceles, with equal sides of length x cm. The tank has a length of l cm. (a) Show that the surface area to be lined, A cm2, is given by A(x) = x2 + 43,2000. x (b) Find the value of x which minimises this surface area. [XlOOj303] Page four ~ 3 5 Marks 9. The diagram shows vectors a and b. If Ia I = 5, Ib I = 4 and a.(a + b) = 36, find the size of the acute angle e between a and b. 10. 11. Solve the equation places. 3cos(2x) + 10cos(x) -1 4 = 0 for 0:::; x:::;It, correct to 2 decimal 5 (i) Sketch the graph of y = ax + 1, a> 2. (a) 2 (ii) On the same diagram, sketch the graph of y = ax+l, a> 2. (b) Prove tL. loga that the graphs intersect at a point ~ -- -- the x-coordinate is 3 ( a ~ 1). [END OF QUESTION [X100j303] where Page five PAPER]

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