NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad iformatio sheet. Please tur over
Mathematics/P DBE/Feb. Mar. 04 INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios... 3. 4. 5. 6. 7. 8. 9. 0. This questio paper cosists of 3 questios. Aswer ALL the questios. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig the aswers. Aswers oly will ot ecessarily be awarded full marks. You may use a approved scietific calculator (o-programmable ad o-graphical), uless stated otherwise. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. ONE diagram sheet for QUESTION 3 is attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o this diagram sheet i the spaces provided ad isert the diagram sheet iside the back cover of your ANSWER BOOK. A iformatio sheet with formulae is icluded at the ed of this questio paper. Number the aswers correctly accordig to the umberig system used i this questio paper. Write eatly ad legibly. Please tur over
Mathematics/P 3 DBE/Feb. Mar. 04 QUESTION. Solve for i each of the followig:.. 35 = 0 (3).. 6 0 (4)..3 9. =.3 (3). Give: f ( ) = 5 + c Determie the value of c if it is give that the solutios of ( ) = 0 f are 5 ± 4. (3).3 Solve for ad y if: 3 0 = 3 3 ad y + = 0. (5) [8] QUESTION. A geometric sequece has T = 3 0 ad T = 40. 4 Determie:.. The commo ratio ().. A formula for T (3). The followig sequece has the property that the sequece of umerators is arithmetic ad the sequece of deomiators is geometric: 4 ; ; 5 5 ;..... Write dow the FOURTH term of the sequece. ().. Determie a formula for the th term. (3)..3 Determie the 500 th term of the sequece. ()..4 Which will be the first term of the sequece to have a NUMERATOR which is less tha 59? (3) [3] Please tur over
Mathematics/P 4 DBE/Feb. Mar. 04 QUESTION 3 3. Give the arithmetic sequece: w 3 ; w 4 ; 3 w 3.. Determie the value of w. () 3.. Write dow the commo differece of this sequece. () 3. The arithmetic sequece 4 ; 0 ; 6 ;... is the sequece of first differeces of a quadratic sequece with a first term equal to 3. Determie the 50 th term of the quadratic sequece. (5) [8] QUESTION 4 I a geometric series, the sum of the first terms is give by ifiity of this series is 0. S = p ad the sum to 4. Calculate the value of p. (4) 4. Calculate the secod term of the series. (4) [8] QUESTION 5 5. Draw the graphs of + y = 6 ad + y = 4 o the same set of aes i your ANSWER BOOK. (4) 5. Write dow the coordiates of the poits of itersectio of the two graphs. () [6] Please tur over
Mathematics/P 5 DBE/Feb. Mar. 04 QUESTION 6 6 Cosider: f ( ) = + 3 6. Write dow the equatios of the asymptotes of the graph of f. () 6. Write dow the domai of f. () 6.3 Draw a sketch graph of f i your ANSWER BOOK, idicatig the itercept(s) with the aes ad the asymptotes. (4) 6.4 The graph of f is traslated to g. Describe the trasformatio i the form ( ; y)... if the aes of symmetry of g are y = + 3 ad y = +. (4) [] QUESTION 7 The graph of f ( ) = a( p) + q where a, p ad q are costats, is give below. Poits E, F( ; 0) ad C are its itercepts with the coordiates aes. A( 4 ; 5) is the reflectio of C across the ais of symmetry of f. D is a poit o the graph such that the straight lie through A ad D has equatio g ( ) = 3. y B A( 4 ; 5) C E g O f F( ; 0) D 7. Write dow the coordiates of C. () 7. Write dow the equatio of the ais of symmetry of f. () 7.3 Calculate the values of a, p ad q. (6) 7.4 If ( ) = 4 + 5 f, calculate the -coordiate of D. (4) 7.5 The graph of f is reflected about the -ais. Write dow the coordiates of the turig poit of the ew parabola. () [4] Please tur over
Mathematics/P 6 DBE/Feb. Mar. 04 QUESTION 8 Give the graph of g( ) A is the -itercept of g. = log. 3 P ; 9 is a poit o g. y P( / 9 ; ) O A g 8. Write dow the coordiates of A. () 8. Sketch the graph of the graph. g idicatig a itercept with the aes ad ONE other poit o (3) 8.3 Write dow the domai of g. () [5] Please tur over
Mathematics/P 7 DBE/Feb. Mar. 04 QUESTION 9 Susa buys a car for R350 000. She secures a loa at a iterest rate of 7% p.a., compouded mothly. The mothly istalmet is R6 300. She pays the first istalmet oe moth after the loa was secured. 9. Calculate the effective aual iterest rate o the loa. Leave your aswer correct to TWO decimal places. (3) 9. How may moths will it take to repay the loa? (5) 9.3 Calculate the value of the fial istalmet. (5) 9.4 The value of the car depreciates at i % p.a. After 3 years its value is R5 000. Calculate i. (3) [6] QUESTION 0 0. Give: f ( ) = 0.. Determie f () from first priciples. (5) 0.. For which value(s) of will f ( ) > 0? Justify your aswer. () 0. Evaluate 3 0.3 Give: 4( ) dy if y =. () d 4 y = ad 3 = w dy Determie. (4) dw 3 0.4 Give: f ( ) = a + b + c + d Draw a possible sketch of y = f () if a, b ad c are all NEGATIVE real umbers. (4) [7] Please tur over
Mathematics/P 8 DBE/Feb. Mar. 04 QUESTION The graph of 3 f ( ) = + a + b + c is sketched below. The -itercepts are idicated. y B 0 4 A. Calculate the values of a, b ad c. (4). Calculate the -coordiates of A ad B, the turig poits of f. (5).3 For which values of will f ( ) < 0? (3) [] QUESTION A small busiess curretly sells 40 watches per year. Each of the watches is sold at R44. For each yearly price icrease of R4 per watch, there is a drop i sales of oe watch per year.. How may watches are sold years from ow? (). Determie the aual icome from the sale of watches i terms of. (3).3 I what year ad at what price should the watches be sold i order for the busiess to obtai a maimum icome from the sale of watches? (4) [8] Please tur over
Mathematics/P 9 DBE/Feb. Mar. 04 QUESTION 3 A sweet factory produces two types of toffees, Taffy ad Chewy, ad stores them i cotaiers. The quatities of butter ad sugar (i kilograms) used i each cotaier of toffies are as follows: o Taffy toffees cotai 40 kg of butter ad 64 kg of sugar for every cotaier of toffees. o Chewy toffees cotai 50 kg of butter ad 40 kg of sugar for every cotaier of toffees. Each week, the factory has a maimum of 000 kg of butter ad 560 kg of sugar available. The factory must produce at least 5 cotaiers of Taffy toffees per week. Let ad y be the umber of cotaiers of Taffy ad Chewy toffees produced each week, respectively. 3. Write dow all the costraits which describe the productio process above. (5) 3. Sketch the system of costraits (iequalities) o the graph paper o DIAGRAM SHEET, clearly idicatig the feasible regio. (4) 3.3 Write dow the maimum umber of cotaiers of Taffy toffees that ca be produced uder these coditios. () 3.4 If the profit eared per week by the factory is R 400 per cotaier of Taffy toffees ad R 000 per cotaier of Chewy toffees, what amout of each type of toffee eeds to be produced i order to make a maimum profit per week? (3) [4] TOTAL: 50
Mathematics/P DBE/Feb. Mar. 04 CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION 3. 70 y 65 60 55 50 Number of 00 cotaiers kgs of Chewy of Chewy Toffees toffees 45 40 35 30 5 0 5 0 5 5 0 5 0 5 0 5 30 35 40 45 50 55 Number 00 of kgs cotaiers of Taffy toffees Taffy toffees
Mathematics/P DBE/Feb. Mar. 04 b ± b 4 ac = a A = P( + i) A = P( i) INFORMATION SHEET A = P( i) A = P( + i) i= = i= ( + ) i = T = ar a( r ) S = F = f [( + i) ] i f ( + h) f ( ) '( ) = lim h 0 h r T a + ( ) d = S = ( a + ( d ) ; r [ ( + i) ] P = i ( ) ( ) + y + y d = + y y M ; y = m + c y y = m ) ( a) + ( y b) = r I ABC: si a A area ABC ( b c = = a = b + c bc. cos A si B si C = ab. si C S ) a = ; < r < r y y m = m = taθ ( α + β ) = siα.cos β cosα. si β si( α β ) = siα.cos β cosα. si β 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 ( ; y) ( cosθ y siθ ; y cosθ + siθ ) ( i ) = σ = i= f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) y ˆ = a + b ( S ) b ( ) ( ) ( y y) =