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'" "" X100/901 ~ .- iii NA TIONAL QUALIijICATIONS 2003 MATHEMATICS HIGHER WEDNEI"SDAY, 21 MAY 9.00 AM - 10.10 AM ~ ; 1jj ~ Units 1,2 and 3 Paper 1 (Non~calculator) ~ Read Carefully 1 Calculators may NOTbe u§ed in this paper. 2 Fullcreditwill'be givenonlywherethesolutioncontainsappropriateworkin£!. 3 Answersobtainedby readingsfromscaledrawingswill not receiveanycredit IOi 'ii i!I '" ~ Eo seams ') ;; ~ QUALIFICATION~/ AUTHORITY LIB X100/301 6/3/26570 1I11I1~1111111111111111111~ IIII~IIII III11 @ ~ ALL questions should be attempted. Marks 1. Find the equation of the line which passes through the point (-1, 3) and is perpendicular to the line with equation 4x + y - 1 = O. 3 = x2 + 6x + 11 in the 2 2. (a) Writef(x) form (x + a)2 + b. (b) Hence or otherwise sketch the graph of y 3. Vectors u and v are defined by u Determine =f(x). 2 = 3i + 2j and v = 2i - whether or not u and v are perpendicular 3j + 4k. 2 to each other. 4. A recurrence relation is defined by Un+t=pUn + q, where -1 < P < 1 and Uo=12. 2 (a) If Ut = 15 and U2 = 16, find the values of p and q. (b) Find the limit of this recurrence '" 5. Given that f(x) relation as n ~ 5 = -JX + x;, findf'(4). 6. A and B are the points (-1, -3, 2) and A (2, -1,1) respectively. Band C are the points of trisection of AD, that is AB = BC = CD. Find the coordinates of D. 7. 2 00. Show that the line with equation with equation y = x2 + 3x + 4. y = 2x 3 + 1 does not intersect the parabola 5 t 8. Find 4 Io(3xdx+ l)t . 9. Functions f(x) =-.L x-4 and g(x) = 2x + 3 are defined on suitable domains. =f(g(x)). 2 (b) Write down any restriction on the domain of h. 1 (a) Find an expression for hex) where hex) [Turn over for Questions [Xl 00/301] Page three 10 to 12 on Page four '" 10. A is the point (8, 4). the line OA is inclined Marks y at an al'lgle p radians to the x-axis. (a) Find the exact values of: A(8,4) (i) sin(2p); (ii) cos(2p). x 0 5 ~ The line OB is inclined at an angle 2p radians to the x-axis. (b) Write down the gradient of OB. exact val'ue y B of' the 1 "' x .~ 11. ~ . 0, A and B are the centres of the three circles shown in the diagram below. . The . two outer circles are congruent and each touches the smallest circle. Circle centre A has equatiopJx -12)2 -t",(y+",5)2=,25. . The three centres lie on a parabola whose axis of symmetry is shown by tbe broken line through A. y " x (a) (i) Shte the coordinates 2 3 of A and find the length of the line OA. (ii) Hence find the equation of the circle with centre B. (b) The equation of the parabola can be written in the form y Find the values of p and q. =px(x + q). 2 \ 12. Simplify 3loge(2e) - 2loge(3e) expressing your answer in the form A + loge B - loge C where",A, B andC are whole numbers. [END OF QUESTION iii rX1'OO/3011 Pafle four PAPER] 4 '" I X1'OO/3Q3 - ~-~->~= - C'~c.., NATIONAL QUALIFICA 2{)03 r ~~> -_.~-~-WEDNESDAY, 10.30 AM TIONS - 21 MAY 12.00 NOON ~-~ ,,- _0 ~~- -~-~ MATHEMATICS HIGHER Units 1,2 and 3 Paper L 11' Read,Car~fu'ily 0 1 Calculators may be usediii;l,tlilispaper. 2 Fullcreditwill be giVenonlywherethe solutioncontainsappropriateworking. 3 Answersobtainedby readingsfromscaledra)/Vings will not receiveanycredit. III IiI r, !> \v[ SCOTTISH./"'QUALIFICATIONS AUTHORITY LI B X 100/303 6/3/26570 l'IIIII.IIIIII:IIIIIIIIII.IIII.I!II;III!IIII~lllljllllIIII @ ALL {'}.uestions should be attempted'. Marks 1. f(x)::::6x3 - 5x2 - 17x + 6. (a) Show that (x - 2) is a factor off(x). 4 (b) Expressf(x) in its fully factorised form. 2. The diagram s9-°ws a sketch of part of the graph of a trigonometric function whose equatioFl is of' the fprm y = a sin(bx) + c. Determine - - ~- J - - - - - - - -~ - - - - - - - - - - - - - 5 - - - - - - - - - - - - - - - - - 3 the values of a, b aQd c. x - - - - ~- - - - - - - - - - -- - - - -- "-- -- -- -3 3. The incomF>letegraphs off(x) GC x2'f- 2x y and g(x) = x3 - x2"",;6xare shown in the diagram. The grawhs iEl}[email protected]>rseCF at A( 4, 24) and the origip. FiEld the shaded area enclosed between the curves. 5 x y 4. (a) Find y the equation = x3 + 2X2 - (b) Show that of the tangent to the curVe = g(x) with equation 5 3x + 2 at the point where x::::1. this line is also x2 + y2 - 12x - 10y + 44 = a tangent 0 and state to the circle the coordinates contact. with equation of the point of 6 - ~'fu'rfl over [X100/303] Page three Marks 5. The diagram a functionj. 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). 6. 7. 2 2 -3 4 Hf(x) = cos(2x) - 3 sin(4x), find the exact value of 1'( ~). '" Part of the graph of y = 2sin(x O) + Scos(x O)is shown in the diagram. y y = 2sin(x O)+ Scos(x O) (a) Express y = 2sin(xO) + Scos(XO) in the form ksin(x o + a O)where k > 0 and 0:::;a < 360. 4 0 (b) Find the coordinates of the minimum turning point P. 8. 3 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. ~ ....... . ..:-::<:»::::::::::-::-:-..,. 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 I cm. (a) Show that the surface area to be lined, A cm2, is given by A(x) = x2 + 432000. x (b) Find the value of x which minimises this surface area. ----- [XIOO/303] Page four 3 5 Marks 9. The.diagram shows v~ctors a and b. If Ia I =!5, Ib I == 4 anda.(a + b) ==36, find tBe size of the acute angle e between a and b. Ut Solve the equation 3cos(2x) + l'Ocos(x) ~ 4" 1 = 0 for 0 ::;;x ::;;1t, correct to 2 decimal 5 places. [ 11. (a) (i) Sketch the graph of y ;::.ax+ l, a> 2. 2 (ii) On the same diagram, sketch'Phe graph of y = a~+l, a> 2. (b) Prove that the graphs intersect at a point where is 3 ). loga ( a ~ 1 [END OF QUESTION PAPER] " f / [XlOO/303] the x-cooEdinate Page five