NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 009() MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheet. Please tur over

2 Mathematics/P DoE/November 009() INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios 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 your aswers. A approved scietific calculator (o-programmable ad o-graphical) may be used, uless stated otherwise. If ecessary, aswers should be rouded off to TWO decimal places, uless stated otherwise. Diagrams are NOT ecessarily draw to scale. ONE diagram sheet for aswerig QUESTION 3.3 is attached at the ed of this questio paper. Write your cetre umber ad eamiatio umber o these sheets i the spaces provided ad place them i the back of your ANSWER BOOK. Number the aswers correctly accordig to the umberig system used i this questio paper. It is i your ow iterest to write legibly ad to preset the work eatly. Please tur over

3 Mathematics/P 3 DoE/November 009() QUESTION. Solve for :.. ( ) = 30 (3) = 0 (Correct to ONE decimal place) (4)..3 5 < 4 9 (4). Solve simultaeously for ad y i the followig set of equatios: y = 3 y y 7 = 0 (5).3 Calculate the eact value of: (Show ALL calculatios.) (3).4 Simplify completely without the use of a calculator: QUESTION ( ) 8 + (3) []. Tebogo ad Matthew's teacher has asked that they use their ow rule to costruct a sequece of umbers, startig with 5. The sequeces that they have costructed are give below. Matthew's sequece: Tebogo's sequece: 5 ; 9 ; 3 ; 7 ; ; 5 ; 5 ; 3 5 ; 78 5 ; ; Write dow the th term (or the rule i terms of ) of:.. Matthew's sequece (3).. Tebogo's sequece (). Nomsa geerates a sequece which is both arithmetic ad geometric. The first term is. She claims that there is oly oe such sequece. Is that correct? Show ALL your workigs to justify your aswer. (5) [0] Please tur over

4 Mathematics/P 4 DoE/November 009() QUESTION 3 99 Give: (3t ) t = 0 3. Write dow the first THREE terms of the series. () 3. Calculate the sum of the series. (4) [5] QUESTION 4 The followig sequece of umbers forms a quadratic sequece: 3 ; ; 3 ; 6 ; ; 4. The first differeces of the above sequece also form a sequece. Determie a epressio for the geeral term of the first differeces. (3) 4. Calculate the first differece betwee the 35 th ad 36 th terms of the quadratic sequece. () 4.3 Determie a epressio for the th term of the quadratic sequece. (4) 4.4 Eplai why the sequece of umbers will ever cotai a positive term. () [] QUESTION 5 Data regardig the growth of a certai tree has show that the tree grows to a height of 50 cm after oe year. The data further reveals that durig the et year, the height icreases by 8 cm. I each successive year, the height icreases by 9 8 of the previous year's icrease i height. The table below is a summary of the growth of the tree up to the ed of the fourth year. Tree height (cm) Growth (cm) First year Secod year Third year Fourth year Determie the icrease i the height of the tree durig the seveteeth year. () 5. Calculate the height of the tree after 0 years. (3) 5.3 Show that the tree will ever reach a height of more tha 3 cm. (3) [8] Please tur over

5 Mathematics/P 5 DoE/November 009() QUESTION 6 Sketched below are the graphs of P ad Q are the poits of itersectio of f ad g. g =. + f ( ) = ad ( ) + y g f P Q g 0 6. Show that the coordiates of P ad Q are P( ; ) ad Q( ; ) respectively. (6) 6. A ais of symmetry of the graph of g is a straight lie defied as y = m + c, where m > 0. Write dow the equatio of this straight lie i the form y = h() = K () 6.3 Determie the equatio of h i the form y = K () 6.4 Show algebraically that g ( ) + g = g( ). g( ). ( 0 or ) (3) [3] Please tur over

6 Mathematics/P 6 DoE/November 009() QUESTION 7 The graphs of f ( ) = 3cos ad g ( ) = si( 60 ) are sketched below for [ 80 ; 90 ]. y 3 f B A g Write dow the rage of f. () 7. If A( 97,37 ; 0,38), write dow the coordiates of B. (3) 7.3 Write dow the period of g ( 3). () 7.4 Write dow a value of for which g( ) f ( ) is a maimum. () [8] Please tur over

7 Mathematics/P 7 DoE/November 009() QUESTION 8 Sketched below is the graph of ( ) = log. y f O f Write dow the domai of f. () 8. Write dow the equatio of f i the form y = () 8.3 Write dow the equatio of the asymptote of f. () 8.4 Eplai how, usig the graph of f, you would sketch the graphs of: 8.4. ( ) = log () g 8.4. h ( ) = 5 (3) 8.5 Use the graph of f to solve for where log < 3. (3) [0] Please tur over

8 Mathematics/P 8 DoE/November 009() QUESTION 9 9. A photocopier valued at R4 000 depreciates at a rate of 8% p.a. o the reducigbalace method. After how may years will its value be R5 000? (4) 9. A car that costs R is advertised i the followig way: 'No deposit ecessary ad first paymet due three moths after date of purchase.' The iterest rate quoted is 8% p.a. compouded mothly. 9.. Calculate the amout owig two moths after the purchase date, which is oe moth before the first mothly paymet is due. (3) 9.. Herschel bought this car o March 009 ad made his first paymet o Jue 009. Thereafter he made aother 53 equal paymets o the first day of each moth. (a) Calculate his mothly repaymets. (3) (b) Calculate the total of all Herschel's repaymets. () 9..3 Hashim also bought a car for R He also took out a loa for R30 000, at a iterest rate of 8% p.a. compouded mothly. He also made 54 equal paymets. However, he started paymets oe moth after the purchase of the car. Calculate the total of all Hashim's repaymets. (4) 9..4 Calculate the differece betwee Herschel's ad Hashim's total repaymets. () [6] QUESTION 0 0. Differetiate f () from first priciples if f ( ) = + 3. (5) dy 0. Evaluate: if y = d 3 () [7] Please tur over

9 Mathematics/P 9 DoE/November 009() QUESTION 3 Give: f ( ) = Calculate the -itercepts of the graph of f. (5). Calculate the coordiates of the turig poits of the graph of f. (5).3 Sketch the graph of f, showig clearly all the itercepts with the aes ad turig poits. (3).4 Write dow the -coordiate of the poit of iflectio of f. ().5 Write dow the coordiates of the turig poits of h ( ) = f ( ) 3. () [7] QUESTION A tourist travels i a car over a moutaious pass durig his trip. The height above sea level of 3 the car, after t miutes, is give as s( t) = 5t 65t + 00t + 00 metres. The jourey lasts 8 miutes.. How high is the car above sea level whe it starts its jourey o the moutaious pass? (). Calculate the car's rate of chage of height above sea level with respect to time, 4 miutes after startig the jourey o the moutaious pass. (3).3 Iterpret your aswer to QUESTION.. ().4 How may miutes after the jourey has started will the rate of chage of height with respect to time be a miimum? (3) [0] Please tur over

10 Mathematics/P 0 DoE/November 009() QUESTION 3 A steel maufacturer makes two kids of products, product A ad B, havig parts that must be cut, assembled ad fiished. The maufacturer is aware that it ca sell as may products as it ca produce. Let ad y be the umber of uits of product A ad product B that are maufactured every day respectively. The costraits that gover the maufacture of the products are represeted below ad the feasible regio is shaded. 4 y 3 Number of uits of product B Feasible Regio Number of uits of product A 3. Write dow the costraits i terms of ad y that represet the above iformatio. (7) 3. If product A yields a profit of R30 per item ad product B yields R40 per item, write dow the equatio idicatig the daily profit i terms of ad y. () 3.3 Determie the umber of uits of product A ad product B that the maufacturer eeds to produce i order to maimise his daily profit. A diagram is provided o DIAGRAM SHEET. () 3.4 The maufacturer would like the maimum profit to be at (6 ; ) for the profit equatio P = m + c. Determie the values of m which will satisfy this coditio () [3] TOTAL: 50

11 Mathematics/P DoE/November 009() INFORMATION SHEET: MATHEMATICS b ± b 4 ac = a A = P( + i) A = P( i) A = P( i) A = P( + i) i= i= = ar F = f i ( r ) a = r [( + i) ] i f ( + h) f ( ) '( ) = lim h 0 h i= ; r i= ( + ) i = [ ( + i) ] P = i a = r ( a + ( i ) d ) = ( a + ( ) d ) i= i ar ; < r < d = ( ) ( ) + y y M + y + y ; y = m + c y y = m ) ( a) + ( y b) = r ( y y m = m = taθ I ΔABC: si a A b c = = a b c = + bc. cos A area Δ ABC = ab. si C si B si C ( 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 ( i ) = σ = i= f ( A) P( A) = P(A or B) = P(A) + P(B) P(A ad B) ( S) y ˆ = a + b b ( ) ( y y) = ( )

12 Mathematics/P DoE/November 009() CENTRE NUMBER: EXAMINATION NUMBER: DIAGRAM SHEET QUESTION y 3 Number of uits of product B Number of uits of product A

MATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12

Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please