NATIONAL SENIOR CERTIFICATE GRADE 11


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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over
2 Mathematics/P DoE/November 007 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios: This questio paper cosists of 0 questios. Aswer ALL the questios. Show clearly ALL calculatios, diagrams, graphs, etc. which you have used i determiig the aswers. A approved scietific calculator (oprogrammable ad ographical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Number the aswers correctly accordig to the umberig system used i this questio paper. Diagrams are NOT ecessarily draw to scale. It is i your ow iterest to write legibly ad to preset the work eatly. A iformatio sheet with formulae is attached. Please tur over
3 Mathematics/P 3 DoE/November 007 QUESTION. Solve for x (correct to TWO decimal places where ecessary):.. (a) ( x + 3 )( x ) = x + (b) Hece or otherwise, solve for x if x + 3x 4 < 0 (3).. x + 3x = (5). Solve simultaeously for x ad y i the followig system of equatios: x + y = 3 ad x + y = 5xy (9).3 If f ( x) = x x, show by completig the square that ( x ) = ( x ) f. [5] QUESTION. Simplify: x 8x + 36x. Give: M = + x + 5 x.... Show that M is a ratioal umber if x =,5 Determie the values of x for which M is a real umber. (3) (3).3 Eri had to fid the product of 007 ad ad the calculate the sum of the digits of the aswer. Eri arrived at a aswer of. Is she correct? Show ALL the calculatios to motivate your aswer. (5) [5] Please tur over
4 Mathematics/P 4 DoE/November 007 QUESTION 3 The umber patter, 5,, 9, is such that the sequece of 'secod differeces' is a costat. 3. Determie the 5 th umber i the patter. () Derive a formula for the th umber i the patter. What is the 00 th umber i the patter? (7) () [0] QUESTION 4 A rubber ball is bouced from a height of 4 metres ad bouces cotiuously as show i the diagram below. Each successive bouce reaches a height that is half the previous height. 4 bouce bouce bouce If the patter of the maximum height reached durig each bouce cotiues, what maximum height will the ball reach durig the 6 th bouce? Determie a algebraic expressio for the maximum height reached i the th bouce. () 4.3 After how may bouces will the ball reach a maximum height of metres? 5 [0] Please tur over
5 Mathematics/P 5 DoE/November 007 QUESTION 5 5. After 4 years of reducig balace depreciatio, a asset has a 4 of its origial value. The origial value was R Calculate the depreciatio iterest rate, as a percetage. (Correct your aswer to decimal place.) (5) 5. Jabu ivests a certai sum of moey for 5 years. She receives iterest of % per aum compouded mothly for the first two years. The iterest rate chages to 4% per aum compouded semiaually for the remaiig term. The moey grows to R at the ed of the 5year period Calculate the effective iterest rate per aum durig the first year. Calculate how much moey Jabu ivested iitially. (6) 5.3 The expediture of the Departmet of Health (i billios of rads) is idicated i the followig table. (We take 003 as t = 0, 004 as t = ad so o.) Year Time (t), i years 0 3 Expediture (E), i billios of rads,5 3 3, Plot the four data poits i your aswer book, as accurately as you ca. () 5.3. Make a cojecture about the relatioship betwee the expediture ad time. () Use your cojecture to write dow the equatio of E as a fuctio of t. () Use your equatio to predict the expediture of the Departmet of Health i 00 (i billios of rads) () [] Please tur over
6 Mathematics/P 6 DoE/November 007 QUESTION 6 Below is a sketch graph of parabola, f, ad straight lie, g. P(; 8) is the turig poit of f. f cuts the yaxis at (0; 6) ad g cuts the yaxis at (0; ). f ad g itersect at B ad C. B is a poit o the xaxis y P(; 8) 6 f A O B g x > C 6. Show that f ( x) = x + 4x + 6. (6) 6. Calculate the average gradiet of ( x) f betwee x = ad x = 3. (3) 6.3 Show that the equatio of g is g ( x) = x. (3) Calculate the coordiates of C. (6) 6.5 If h(x) = f( x), explai how the graph of h may be obtaied from the graph of f. () 6.6 Write dow the equatio of h. () [] Please tur over
7 Mathematics/P 7 DoE/November 007 QUESTION 7 8 = x 8 Give: f ( x) Write dow the domai of f. For what value of x is f (x) = 0? Determie the value of p, if A (0; p) lies o the graph of f. Write dow the equatios of the asymptotes of f. Draw a eat sketch graph of f, idicatig the asymptotes ad itercepts with the axes, o the diagram sheet provided. () () () () [] QUESTION 8 The graph of ( ) x f x = + a. (a is a costat) passes through the origi as show below. y f O x Show that a =. Determie the value of f ( 5) correct to FIVE decimal places. Determie the value of x, if P (x; 0,5) lies o the graph of f. If the graph of f is shifted uits to the right to give the fuctio h, write dow the equatio of h. () () (3) () [9] Please tur over
8 Mathematics/P 8 DoE/November 007 QUESTION 9 Give the fuctio f (x) = cos (x 30º) for x [360º; 360º]. O 0 Determie: 9. The period of the fuctio g, if g (x) = f (x) () 9. The rage of the fuctio h, if h (x) = f (x) () 9.3 The amplitude of the fuctio q, if q (x) = f (x) + () [6] Please tur over
9 Mathematics/P 9 DoE/November 007 QUESTION 0 Two fragraces A ad B are used to make the perfumes Laughter ad Joy. Laughter You require 3 g of fragrace A ad 4 g of fragrace B to produce litre of Laughter. Oe litre of Joy requires 9 g of fragrace A ad 6 g of fragrace B. At least 3 litres of Laughter eeds to be produced per week. At the begiig of a particular week the compay has 7 g of fragrace A ad 30 g of fragrace B. Let x ad y be the umber of litres of Laughter ad Joy respectively that are produced per week State algebraically, i terms of x ad y, the costraits that apply to this problem for this week. Represet the costraits graphically o the graph paper provided ad shade the feasible regio. If the profit o l of Laughter is R30 ad the profit o l of Joy is R50, express the profit, P, i terms of x ad y. Determie how may litres of each perfume must be produced i this week to esure a maximum profit. Calculate the maximum possible profit. TOTAL: (5) (8) () () [] 50
10 Mathematics/P DoE/November 007 NAME/EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 0 0.
11 b ± x = b 4 ac a INFORMATION SHEET: MATHEMATICS INLIGTINGSBLAD: WISKUNDE A = P( + i) A = P( i) A = P( i) i= = A = P( + i) i= ( + ) i = ( a + ( i ) d ) = ( a + ( ) d ) i= i= ar x F = f i ( r ) a = r [( + i) ] i f ( x + h) f ( x) '( x) = lim h 0 h ; r = i a r i ar = ; < r < x[ ( + i) ] P = i d = ( x ) ( ) x + y y M x + x y + y ; y = mx + c y y = m x ) y y m = m = taθ x x ( x a) + ( y b) = r ( x I ABC: si a A b c = = a b c = + bc. cos A area ABC = ab. si C si B si C ( α + β ) = siα.cos β cosα. siα si( α β ) = siα.cos β cosα. siα si + cos ( α + β ) = cosα.cos β siα. si β cos ( α β ) = cosα.cos β + siα. si β cos α si α cos α = si α si α = siα. cosα cos α ( xi x) = i= fx x ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) ( S ) =
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