GRADE 11 NOVEMBER 2015 MATHEMATICS P1

Transcription

1 NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *Imat1* This question paper consists of 9 pages.

2 2 MATHEMATICS P1 (EC/NOVEMBER 2015) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of ELEVEN questions. Answer ALL the questions. 2. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in determining your answer. 3. You may use an approved scientific calculator (non-programmable and nongraphical), unless stated otherwise. 4. Answers only will not necessarily be awarded full marks. 5. If necessary, round off answers to TWO decimal places, unless stated otherwise. 6. Diagrams are NOT necessarily drawn to scale. 7. Number the answers correctly according to the numbering system used in this question paper. 8. Write neatly and legibly.

3 (EC/NOVEMBER 2015) MATHEMATICS P1 3 QUESTION Solve for. Round off to TWO decimal places, if necessary (4) 1.2 Given the following inequalities: and Solve for if (4) Solve for if If it is given that is a natural number, solve for if and 1.3 Given: Determine the value of: (3) Hence determine the value of: [19] QUESTION Simplify the following expressions without the use of a calculator (3) [ ] (4) 2.2 Solve for : (3) 2.3 If, determine the value of (5) [15]

4 4 MATHEMATICS P1 (EC/NOVEMBER 2015) QUESTION Solve for and in the following simultaneous equations. and (6) 3.2 Determine the nature of the roots of the quadratic equation if the following conditions are given: and and 3.3 Determine for which value(s) of will have no real solution. (4) [14] QUESTION 4 The first term of a linear number pattern is and the constant difference is 4.1 Write down the values of the second and third terms of the number pattern. (1) 4.2 Determine an expression for the -th term of the number pattern. 4.3 Determine the value of the eighteenth term. 4.4 If, determine the value of [7] QUESTION The following number pattern has a constant second difference Write down the value of the constant second difference. (1) Determine an expression for the -th term of the number pattern. (4) The first forty terms of the number pattern are all prime numbers. Determine the forty-first term and show that it is not a prime number Determine the units digit of the ( ) term. 5.2 The -th term of a number pattern is as follows: if is an even number and if is an uneven number Determine the value of (3) Determine the value of if (5) [17]

5 (EC/NOVEMBER 2015) MATHEMATICS P1 5 QUESTION The price of a new school bus is R The value of the bus decreases at 11% per annum according to the diminishing-balance method. Calculate the value of the bus after 8 years. 6.2 Determine the effective interest rate if an investment earns interest at a nominal interest rate of 11,5% per annum, compounded quarterly. (3) 6.3 Vishnu and Landi receive R each. They decide to invest the money for a period of 8 years as follows: Vishnu: Simple interest at per annum. At the end of the 8 years Vishnu receives a cash bonus of 3% on the principal amount. Landi: Interest at per annum, compounded monthly Calculate the value of Vishnu s investment after 8 years, including the cash bonus. (3) Calculate the value of Landi s investment after 8 years. (3) 6.4 James invests a certain amount for 5 years. The investment earns interest at per annum, compounded monthly, for the full term. James withdraws R2 000 from the account after 18 months. After 5 years the value of the investment is R What amount did James initially invest? (5) [16] QUESTION 7 Given the following two functions: and 7.1 Determine the -intercept of (3) 7.2 Sketch neat graphs of and on the same set of axes. Clearly show all intercepts with the axes as well as asymptotes. (5) 7.3 Write down the equation of the vertical asymptote of. (1) 7.4 Determine the coordinates of the points of intersection of and. Show all calculations. (5) 7.5 Write down the equation of if is the reflection of about the line. 7.6 Write down the equation if is translated so that is the new point of intersection of the asymptotes. [18]

6 6 MATHEMATICS P1 (EC/NOVEMBER 2015) QUESTION 8 The sketch below shows the graph of The point is a point on the graph. y f O x 8.1 Show that 8.2 Determine the -intercept of 8.3 Write down the range of (1) 8.4 Determine, rounded off to three decimal places. 8.5 Write down the equation if is reflected about the -axis followed by a translation of 2 units to the left. [9]

7 (EC/NOVEMBER 2015) MATHEMATICS P1 7 QUESTION 9 The sketch below shows the graph of the function The straight line with equation intersects at. The points and are the -intercepts of Point is a point on y D y x A O B x f 9.1 Determine the coordinates of 9.2 Determine the equation of (4) 9.3 Determine the coordinates of the turning point of. 9.4 Point is a point on the straight line so that is parallel to the -axis. Determine the length of (3) 9.5 Calculate the average gradient between and (1) 9.6 For which value(s) of is [14]

8 8 MATHEMATICS P1 (EC/NOVEMBER 2015) QUESTION 10 A survey regarding their favourite magazine(s) was conducted among 84 high school girls. Three magazines, namely Teen Vogue, Drum and People s Magazine were used in the survey. The results are as follows: 41 read Teen Vogue. 34 read People s Magazine. 40 read Drum. 18 read Teen Vogue and Drum. 8 read all three magazines. 75 read at least one magazine. and The Venn-diagram below shows the above information. S T P f d e a D b c 10.1 Determine the values of and (7) 10.2 Determine the probability that a randomly selected girl reads at least two of the three magazines. [9]

9 (EC/NOVEMBER 2015) MATHEMATICS P1 9 QUESTION The Venn-diagram below shows two independent events, and Determine the values of probabilities and Show all calculations. (5) 11.2 The following contingency table shows information on the drivers tests of 100 drivers tested at a test centre in Port Elizabeth. Male Female Total Pass Fail Total A driver is randomly selected from the 100 drivers Determine the probability that a female that failed is selected Determine the probability that the driver passed, given it is a male William writes a Mathematics examination and an Accounting examination. He estimates that he has a 40% chance of passing the Mathematics examination. He estimates that he has a 60% chance of passing the Accounting examination. He estimates that he has a 30% chance of passing both. Determine the probability that William will fail Mathematics and Accounting. (3) [12] TOTAL: 150

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17 NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 11 NOVEMBER 2015 MATHEMATICS P1/WISKUNDE V1 MEMORANDUM MARKS/PUNTE: 150 Hierdie memorandum bestaan uit 14 bladsye./ This memorandum consists of 14 pages.

18 2 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) NOTE/LET OP: If a candidate answered a question TWICE, mark the FIRST attempt ONLY. Indien ʼn kandidaat ʼn vraag TWEE keer beantwoord het, merk SLEGS die EERSTE poging. Consistent accuracy applies in ALL aspects of the memorandum. Volgehoue akkuraatheid geld deurgaans in ALLE aspekte van die memorandum. If a candidate crossed out an attempt of a question and did not redo the question, mark the crossed-out attempt. Indien ʼn kandidaat ʼn poging vir ʼn vraag deurgetrek het en nie die vraag weer beantwoord het nie, merk die poging wat deurgetrek is. The mark for substitution is awarded for substitution into the correct formula. Die punt vir substitusie word toegeken vir substitusie in die korrekte formule. QUESTION 1/VRAAG or/of Answer ONLY: 1 mark. SLEGS antwoord: 1 punt. factors/faktore -values/waardes Penalise 1 mark for incorrect rounding off. Penaliseer 1 punt vir verkeerde afronding. standard form/standaardvorm substitution/substitusie or/of If answer left in surd form: 3 marks. Indien antwoord in wortelvorm gelaat: 3 punte. -values/waardes (4) OR/OF OR/OF [ ] standard form/standaardvorm factors/faktore critical values/kritieke waardes solution/oplossing (4) x Note/Let op: If/As x or/of x : max./maks. 2 marks/punte. If correct graphical solution but concludes incorrectly: max. 3 marks. As korrekte grafiese oplossing, maar maak verkeerde gevolgtrekking: maks. 3 punte.

19 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V solution/oplossing { } 4 values/waardes 7 values/waardes If/As x, 1 mark/punt ( ) Answer ONLY: 1 mark. SLEGS antwoord: 1 punt. squaring/kwadrering simplification/vereenvoudiging answer/antwoord (3) If answer = 8: 2 marks./as antwoord = 8: 2 punte. Used/gebruik m m If answer = 10: 1 mark./as antwoord = 10: 1 punt ( ) ( ) factors/faktore answer/antwoord If no factors shown but correct answer: 0 marks. As geen faktore maar regte antwoord: 0 punte. [19]

20 4 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) QUESTION 2/VRAAG Answer ONLY: 0 marks. SLEGS antwoord: 0 punte. answer/antwoord OR/OF ( ) ( ) [ ] answer/antwoord (3) [ ] [ ] and/en [ ] OR/OF rule/reël rule/reël answer/antwoord (4) 2.2 answer/antwoord (3)

21 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V OR/OF If/As then/dan Answer ONLY: 0 marks. SLEGS antwoord: 0 punte. (5) [15]

22 6 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) QUESTION 3/VRAAG / or/of or/of or/of or/of substitution/substitusie standard form/standaardvorm factors/faktore -values /-waardes -values/-waardes (6) method/metode Roots are rational and unequal Wortels is rasionaal en ongelyk rational AND unequal/ rasionaal EN ongelyk No mark for answer if no method is shown. Geen punt vir antwoord as geen metode aangedui word nie. If candidate includes non-real : max 1 mark. As kandidaat nie-reëel insluit: maks 1 punt No mark for answer if no method is shown. Geen punt vir antwoord as geen metode aangedui word nie. No marks if contradictions are given, e.g. unequal. Geen punt as teenstrydighede genoem word nie, bv. ongelyk. Roots are nonreal/wortels is nie-reëel. answer/antwoord 3.3 substitution/substitusie For non-real solution/ Vir nie-reële wortels: ( )( ) statement/stelling critical values/kritieke waardes answer/antwoord (4) [14]

23 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V1 7 QUESTION 4/VRAAG ; (1) 4.2 OR/OF substitution/substitusie 4.3 OR/OF substitution/substitusie answer/antwoord 4.4 substitution/substitusie answer/antwoord OR/OF Answer ONLY: 1 mark. SLEGS antwoord: 1 punt. expansion/uitbreiding answer/antwoord [7]

24 8 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) QUESTION 5/VRAAG answer/antwoord (1) value/waarde -value/waarde -value/waarde answer/antwoord OR/OF Answer ONLY: 3 marks. SLEGS antwoord: 3 punte. -value/waarde -value/waarde -value/waarde answer/antwoord (4) is also a factor/ is ook ʼn faktor. Factors/faktore: and/en is not prime/nie priem nie. argument Units digits/enesyfers: Decimal/desimaal Units digit/enesyfer Answer ONLY: 2 marks. SLEGS antwoord: 2 punte. identify pattern/identifiseer patroon answer/antwoord answer/antwoord (3)

25 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V If ONLY/Indien SLEGS: k k k answer/antwoord 4 marks/punte. If continues and mentions that k is uneven: 5 marks. As voortgaan en meld dat k answer/antwoord is onewe: 5 punte choice/keuse (5) [17] If expansion that leads to correct answer: 5 marks. As uitbreiding wat tot korrekte antwoord lei: 5 punte. If ONLY expansion: 2 marks. Indien SLEGS uitbreiding: 2 punte QUESTION 6/VRAAG Wrong formule/verkeerde formule: 0 marks/punte. substitution/substitusie answer/antwoord 6.2 ( ) Wrong formule/verkeerde formule: 0 marks/punte. substitution/substitusie answer/antwoord If answer given as : 2 marks. Indien antwoord gegee as : 2 punte. (3) R If ONLY/Indien SLEGS: 1 mark/punt. answer/antwoord (3) R26 009,69 ( ) Wrong formule/ Verkeerde formule: 1 mark/punt for/vir i substitution/substitusie answer/antwoord (3)

26 10 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) 6.4 T T T P OR/OF [ ( ) ] ( ) R14 642,83 ( ) ( ) P Wrong formule/ verkeerde formule: 1 mark/punt for/vir i ( ) ( ) answer/antwoord ( ) ( ) ( ) ( ) answer/antwoord OR/OF [ ( ) ] ( ) ( ) ( ) ( ) ( ) ( ) answer/antwoord (5) [16]

27 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V1 11 QUESTION 7/VRAAG Let/stel Answer ONLY x SLEGS antwoord x : 2 marks. : 2 punte. simplify/vereenvoudig answer/antwoord (3) 7.2 y g -intercept/afsnit asymptote/asimptoot shape/vorm y -intercept/afsnit -intercept/afsnit O x 7.3 answer/antwoord (1) (5) 7.4 equation/vergelyking or/of or/of simplify/vereenvoudig -values/waardes (5) [18]

28 12 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) QUESTION 8/VRAAG substitution/substitusie simplify/vereenvoudig 8.3 answer/antwoord 8.4 substitution/substitusie answer/antwoord 8.5 (1) [9] QUESTION 9/VRAAG Let/stel Do not penalise if not in coordinate form. Moenie penaliseer indien nie in koördinaatvorm nie. answer/antwoord 9.2 setting up equation/ opstel van vergelyking substitution/substitusie NOTE/LET WEL: No reference can be made to 9.3/Geen verwysing kan na 9.3 gemaak word nie. -value/waarde ( simplification/vereenvoudiging (4)

29 (EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V OR/OF ( ) ( ) ( ) ( ) ( ) or/of or/of 6,125 Do not penalise if not in coordinate form. Moenie penaliseer indien nie in koördinaatvorm nie. 9.4 : Let/stel substitute/vervang 9.5 ; units/eenhede answer/antwoord answer/antwoord (3) (1) 9.6 or/of [14] QUESTION 10/VRAAG CA: b a equation/vergelyking OR/OF CA: e a d CA: f d (7) 10.2 at least 2 out of 3/ten minste 2 uit 3) or/of or/of answer/antwoord [9]

30 14 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015) QUESTION 11/VRAAG For independent events/vir onafhanklike gebeurtenisse: and/en rule/reëel substitution/substitusie female failing/vroulik druip) pass, given male/slag, gegee manlik) 11.3 or and of en fails both/druip albei) OR/OF -value/waarde -value/waarde rule/reël 0,7 answer/antwoord Answer ONLY: 1 mark. SLEGS antwoord: 1 punt. (5) M/W A/R fails both/druip albei) answer/antwoord (3) [12] TOTAL/TOTAAL: 150