GRADE 12 SEPTEMBER 2015 MATHEMATICS P1

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1 NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 10 pages, icludig a iformatio sheet.

2 MATHEMATICS P1 (EC/SEPTEMBER 015) INSTRUCTIONS AND INFORMATION Read the followig istructios carefully before aswerig the questios. 1. This questio paper cosists of ELEVEN questios. Aswer ALL the questios.. Clearly show ALL calculatios, diagrams, graphs, et cetera that you have used i determiig your aswer. 3. You may use a approved scietific calculator (o-programmable ad ographical), uless stated otherwise. 4. Aswers oly will ot ecessarily be awarded full marks. 5. If ecessary, roud off aswers to TWO decimal places, uless stated otherwise. 6. Diagrams are NOT ecessarily draw to scale. 7. Number the aswers correctly accordig to the umberig system used i this questio paper. 8. Write eatly ad legibly. 9. A iformatio sheet with formulae is icluded at the ed of the questio paper. Please tur over

3 (EC/SEPTEMBER 015) MATHEMATICS P1 3 QUESTION Give: ( )( ) Solve for if () 1.1. Solve for rouded to two decimal places, if (5) The turig poit of ( ) = (x + 3) (3x 1) is ( 1 ; 8 ). (a) How must the graph of be traslated for it to have equal roots? () (b) Hece, write dow the values of for which ( ) + = 0 will have o real solutios. (1) 1. Solve for ad simultaeously i the followig equatios: 1.3 Solve for if ( )( ). (4) 1.4 Solve for if (4) [4] QUESTION.1 Give the followig arithmetic sequece: (6).1.1 Determie the value of the 50 th term. (3).1. Calculate the sum of the first fifty terms. (). The followig represets the first three terms of a arithmetic sequece: Calculate the value(s) of (4).3 Cosider the geometric series:.3.1 For which value(s) of will the series coverge? (3).3. It is give that: Calculate the value of (3).4 The sum of the first three terms of a geometric sequece is The sum of the fourth, fifth ad sixth term of the same sequece is Determie the value of the commo ratio ( ). (5) [0] Please tur over

4 4 MATHEMATICS P1 (EC/SEPTEMBER 015) QUESTION 3 Cosider the followig umber structure: Row 1 3 Row 6 9 Row Row Row The secod term i each row produces the followig umber patter: 3.1 Determie a expressio for the -th term of the umber patter give above. (4) 3. Determie the value of the fifth term i Row 0. (3) [7] QUESTION Patrick opes a savigs accout o 1 Jauary 01. He makes a immediate paymet of R 000 ito the accout ad thereafter a mothly paymet of R1 00 at the ed of each moth. The last paymet is made o 31 December 013. Iterest is calculated at 8% per year, compouded mothly Calculate the value of Patrick s ivestmet o 31 December 013. (5) 4.1. Patrick decides ot to withdraw the moey o 31 December 013. He makes o further paymets ad the ivestmet ears the same iterest rate. Calculate the value of the ivestmet o 31 May 014. (3) 4. Lilly takes out a loa to the value of R She repays the loa by meas of equal mothly istalmets which she makes at the ed of each moth. The first istalmet is made three moths after the gratig of the loa ad the last istalmet is eight years after the gratig of the loa. The iterest rate is 15% per year, compouded mothly Calculate the value of the equal mothly istalmets. (5) 4.. Covert the iterest rate to a effective iterest rate, rouded to two decimal places. () [15] Please tur over

5 (EC/SEPTEMBER 015) MATHEMATICS P1 5 QUESTION 5 The sketch shows the graphs of ( ) ( ) ad ( ) is the poit of itersectio of the asymptotes of. is the -itercept of. The graph of passes through the origi. is parallel to the -axis. f y g B A O x g 5.1 Write dow the equatio of i the form y. () 5. Write dow the domai of. (1) 5.3 Calculate the value(s) of if ( ) (3) 5.4 Determie the rage of () 5.5 If ( ) is the equatio of oe of the axes of symmetry of, determie the coordiates of () 5.6 Hece determie the equatio of (4) 5.7 For which value(s) of is ( ) (1) [15] Please tur over

6 6 MATHEMATICS P1 (EC/SEPTEMBER 015) QUESTION 6 Give ( ) ad ( ) 6.1 Write dow the -itercept of (1) 6. Determie the -itercepts of (3) 6.3 Determie the coordiates of the turig poit of () 6.4 Sketch the graph of Clearly show the itercepts with both axes as well as the coordiates of the turig poit. () 6.5 Determie the coordiates of poit a poit o where the gradiet of the taget to at is equal to 6. (4) 6.6 Determie the equatio of the straight lie passig through the poits ( ) ad ( ) 6.7 Write dow the equatio of i the form ( ) ( ) if ( ) ( ) (3) (3) [18] QUESTION Give: ( ) Determie ( ) from first priciples. (5) 7. Give the followig: ad Determie the followig: 7..1 (1) 7.. () The straight lie ( ) is a taget to the curve of fuctio at Calculate ( ) ( ) (3) (3) [14] Please tur over

7 (EC/SEPTEMBER 015) MATHEMATICS P1 7 QUESTION 8 The sketch below shows the graph of ( ) The -itercepts of are ( ) ( ) ad ( ) ad are the turig poits of f ad is the -itercept of The sketch is ot draw to scale. y B f O x D A 8.1 Write dow the value of. (1) 8. Determie the coordiates of ad. (5) 8.3 Determie the value of where the cocavity of chages. () 8.4 Determie the coordiates of the poit o with a maximum gradiet. () 8.5 Determie for which value(s) of is ( ) ( ) (3) [13] Please tur over

8 8 MATHEMATICS P1 (EC/SEPTEMBER 015) QUESTION 9 The graph below shows the sketch of ( ) is the poit ( ) ad is the poit ( ) ad are poits o is parallel to the -axis ad is parallel to the -axis. is a rectagle. y Q(q ) R( ) x P S f T 9.1 Write dow the coordiates of i terms of. (1) 9. Show that the area ( ) of rectagle ca be expressed as follows: 9.3 Determie the maximum area ( ) of rectagle. (4) [7] () Please tur over

9 (EC/SEPTEMBER 015) MATHEMATICS P1 9 QUESTION ad are two evets i a sample space. ( ) ad ( ) Determie ( ) (1) Determie ( ) if ad are mutually exclusive evets. () Determie ( ) if ad are idepedet evets. () 10. A blue ( ) ad gree ( ) bucket are filled with balls. The blue bucket cotais 5 white ( ) ad 3 red ( ) balls. The gree bucket cotais white ad 7 red balls. A bucket is radomly selected ad oe ball is thereafter radomly draw from the bucket. QUESTION Draw a tree diagram to represet the above iformatio. Clearly idicate the probability of each brach of the tree. Show all possible outcomes. (4) 10.. Determie the probability that a red ball is draw. (3) [1] The Easter Cape requires ew codes for umber plates. The ew codes cosist of four letters followed by four digits, as show below. All codes ed with EC. BCDF 3856 EC The vowels (A, E, I, O, U) ad Q may ot be used ad digits 1 to 9 are used. Letters ad digits may be repeated Determie how may umber plates with differet codes ca be made. (3) 11. Determie the probability that a code that is radomly selected will cosist of eve digits which are ot the same. () [5] TOTAL: 150 Please tur over

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11 10 MATHEMATICS P1 (EC/SEPTEMBER 015) INFORMATION SHEET: MATHEMATICS b b 4 ac x a A P( 1 i) A P( 1 i) A P( 1 i) A P( 1 i) T a ( 1) d S a ( 1 d ) 1 ar 1 T ar S F f '( x 1 i 1 i x) lim h 0 f ( x h) f ( x) h r 1 ; r 1 x[1 (1 i) ] P i ( ) ( ) x1 x y1 y d x x1 y y1 M ; y mx c y y m x ) x a y b r I ABC: si cos si a A 1 ( x1 1 S a ; 1 r 1 1 r y y1 m m ta x x b c a b c 1 bc. cos A area ABC ab. si C si B si C si.cos cos. si si si.cos cos. si cos.cos si. si cos cos.cos si. si cos si cos 1 si si si. cos cos 1 xi x i1 fx x ( A) P( A) P(A or B) = P(A) + P(B) P(A ad B) yˆ a bx S b x x ( y y) ( x x) Please tur over

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