GRADE 11 NOVEMBER 2015 MATHEMATICS P1 - …...6.2 Determine the effective interest rate if an investment earns interest at a nominal interest rate of 11,5% per annum, compounded quarterly

NATIONAL SENIOR CERTIFICATE

GRADE 11

NOVEMBER 2015

MATHEMATICS P1

MARKS: 150 TIME: 3 hours

This question paper consists of 9 pages.

*Imat1*

2 MATHEMATICS P1 (EC/NOVEMBER 2015)

Copyright reserved Please turn over

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 non-

graphical), 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.

(EC/NOVEMBER 2015) MATHEMATICS P1 3


QUESTION 1 1.1 Solve for . Round off to TWO decimal places, if necessary. 1.1.1 (2) 1.1.2

(4) 1.2 Given the following inequalities: and 1.2.1 Solve for if (4) 1.2.2 Solve for if (2) 1.2.3 If it is given that is a natural number, solve for if and

(2) 1.3 Given:

1.3.1 Determine the value of:

(3) 1.3.2 Hence determine the value of:

(2) [19] QUESTION 2 2.1 Simplify the following expressions without the use of a calculator.

2.1.1 √ √ √

(3)

2.1.2 [

√ ]

(4)

2.2 Solve for : (3)

2.3 If , determine the value of

(5) [15]



QUESTION 3 3.1 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:

3.2.1 and . (2) 3.2.2 and (2) 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. (2) 4.3 Determine the value of the eighteenth term. (2)

4.4 If , determine the value of (2) [7] QUESTION 5 5.1 The following number pattern has a constant second difference. 5.1.1 Write down the value of the constant second difference. (1) 5.1.2 Determine an expression for the -th term of the number pattern. (4) 5.1.3 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. (2) 5.1.4 Determine the units digit of the ( ) term. (2) 5.2 The -th term of a number pattern is as follows:

if is an even number and if is an uneven number.

5.2.1 Determine the value of (3) 5.2.2 Determine the value of if (5) [17]



QUESTION 6 6.1 The price of a new school bus is R540 000. 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. (2)

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 R15 000 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. 6.3.1 Calculate the value of Vishnu’s investment after 8 years, including the cash

bonus. (3) 6.3.2 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 R23 564. 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 . (2) 7.6 Write down the equation if is translated so that is the new point of

intersection of the asymptotes. (2) [18]



QUESTION 8 The sketch below shows the graph of The point is a point on the graph.

8.1 Show that (2) 8.2 Determine the -intercept of (2) 8.3 Write down the range of (1) 8.4 Determine , rounded off to three decimal places. (2) 8.5 Write down the equation if is reflected about the -axis followed by a translation of

2 units to the left. (2) [9]

𝑦 𝑓

𝑂 𝑥



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 9.1 Determine the coordinates of (2) 9.2 Determine the equation of (4) 9.3 Determine the coordinates of the turning point of . (2) 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 (2) [14]

𝑓

𝑦 𝑥

𝑂

𝐷

𝐴 𝐵 𝑥

𝑦



𝑇 𝑃

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:

x 41 read Teen Vogue. x 34 read People’s Magazine. x 40 read Drum. x 18 read Teen Vogue and Drum. x 8 read all three magazines. x 75 read at least one magazine. x and

The Venn-diagram below shows the above information. 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. (2) [9]

𝐷

𝑒

𝑓

𝑏

𝑎

𝑑

𝑐

𝑆



QUESTION 11 11.1 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 30 47 77 Fail 7 16 23 Total 37 63 100 A driver is randomly selected from the 100 drivers. 11.2.1 Determine the probability that a female that failed is selected. (2) 11.2.2 Determine the probability that the driver passed, given it is a male. (2) 11.3 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

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.

2 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2015)


NOTE/LET OP: x 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. x Consistent accuracy applies in ALL aspects of the memorandum.

Volgehoue akkuraatheid geld deurgaans in ALLE aspekte van die memorandum. x 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.

x 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 1 1.1.1

or/of

9 factors/faktore 9 -values/waardes

(2) 1.1.2

√

√

or/of

9 standard form/standaardvorm 9 substitution/substitusie 99 -values/waardes

(4)

1.2.1

OR/OF [ ] OR/OF

9 standard form/standaardvorm 9 factors/faktore 9 critical values/kritieke waardes 9 solution/oplossing

(4)

Answer ONLY: 1 mark. SLEGS antwoord: 1 punt.

Penalise 1 mark for incorrect rounding off. Penaliseer 1 punt vir verkeerde afronding.

If answer left in surd form: 3 marks. Indien antwoord in wortelvorm gelaat: 3 punte.

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.

(EC/NOVEMBER 2015) MATHEMATICS P1/WISKUNDE V1 3


1.2.2

9 9 solution/oplossing

(2) 1.2.3 { }

9 4 values/waardes 9 7 values/waardes

(2) 1.3.1

( )

9 squaring/kwadrering 9 simplification/vereenvoudiging 9 answer/antwoord

(3)

1.3.2

( ) (

)

9 factors/faktore 9 answer/antwoord

(2)

[19]

If/As 𝑥 , 1 mark/punt.


If answer = 8: 2 marks./As antwoord = 8: 2 punte. Used/gebruik 𝑚

𝑚 If answer = 10: 1 mark./As antwoord = 10: 1 punt

If no factors shown but correct answer: 0 marks. As geen faktore maar regte antwoord: 0 punte.



QUESTION 2/VRAAG 2 2.1.1 √ √

√

√ √ √

√ √

OR/OF √ √

√

( ) ( )

[ ]

9 √ √ 9 √ 9 answer/antwoord 9

9

9 answer/antwoord

(3) 2.1.2

[

√ ]

[

]

[

] OR/OF

9 and/en

9 rule/reël 9 rule/reël 9 answer/antwoord

(4) 2.2

9 9 9 answer/antwoord

(3)

Answer ONLY: 0 marks. SLEGS antwoord: 0 punte.



2.3

OR/OF If/As then/dan

9 9 9 9 9 9 9 9

9

9

(5) [15]




QUESTION 3/VRAAG 3 3.1

or/of or/of

or/of or/of

9 / 9 substitution/substitusie 9 standard form/standaardvorm 9 factors/faktore 9 -values /-waardes 9 -values/-waardes

(6) 3.2.1

Roots are rational and unequal Wortels is rasionaal en ongelyk

9 method/metode 9 rational AND unequal/ rasionaal EN ongelyk

(2)

3.2.2

Roots are nonreal/Wortels is nie-reëel.

9 9 answer/antwoord

(2) 3.3

For non-real solution/ Vir nie-reële wortels: ( √ )( √ ) √ √

9 substitution/substitusie 9 statement/stelling 9 critical values/kritieke waardes 9 answer/antwoord

(4) [14]

No mark for answer if no method is shown. Geen punt vir antwoord as geen metode aangedui word nie.

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 marks if contradictions are given, e.g. unequal. Geen punt as teenstrydighede genoem word nie, bv. ongelyk.



QUESTION 4/VRAAG 4 4.1 9 ;

(1) 4.2

OR/OF

9 9 9 9 substitution/substitusie

(2) 4.3 OR/OF

9 substitution/substitusie 9 answer/antwoord

(2) 4.4

OR/OF


9 expansion/uitbreiding 9 answer/antwoord

(2) [7]




QUESTION 5/VRAAG 5 5.1.1 2 9 answer/antwoord

(1) 5.1.2

OR/OF

9 -value/waarde 9 -value/waarde 9 -value/waarde 9 answer/antwoord 9 -value/waarde 9 -value/waarde 9 -value/waarde 9 answer/antwoord

(4) 5.1.3

is also a factor/ is ook ŉ faktor. Factors/faktore: and/en is not prime/nie priem nie.

9 9 argument

(2) 5.1.4 Units digits/enesyfers:

Decimal/desimaal Units digit/enesyfer

9 identify pattern/identifiseer patroon 9 answer/antwoord

(2) 5.2.1

9 9 9 answer/antwoord

(3)





5.2.2

9 9 answer/antwoord 9 9 answer/antwoord 9 choice/keuse

(5) [17] 1 3 5 7 9 11 13 15

5 -3 -19 -43 -75 -115 -163 -219

2 4 6 8 10 12 14 16 18 20 22 24 26 28

-14 -24 -34 -44 -54 -64 -74 -84 -94 -104 -114 -124 -134 -144

30 32 34 36 38 40 42 44

-154 -164 -174 -184 -194 -204 -214 -224



(2)

6.2 (

)

9 substitution/substitusie 9 9 answer/antwoord

(3) 6.3.1

R25 890

9 9

9 answer/antwoord

(3)

6.3.2 (

)

R26 009,69

9


(3)

If ONLY/Indien SLEGS: 𝑘 𝑘 𝑘 4 marks/punte. If continues and mentions that 𝑘 is uneven: 5 marks. As voortgaan en meld dat 𝑘 is onewe: 5 punte

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.

Wrong formule/verkeerde formule: 0 marks/punte.

Wrong formule/verkeerde formule: 0 marks/punte.

If ONLY/Indien SLEGS: 1 mark/punt.

Wrong formule/ Verkeerde formule: 1 mark/punt for/vir 𝑖

If answer given as : 2 marks. Indien antwoord gegee as : 2 punte.



6.4

[ ( )

] (

)

R14 642,83 OR/OF

( )

(

)

( )

(

)

OR/OF

[ ( )

] (

)

( )

( )

9

9 ( )

9

9 ( )

9 answer/antwoord 9

9 ( )

9

9 ( )

9 answer/antwoord 9

9 ( )

9 ( )

9 (

)

9 answer/antwoord (5)

[16]

𝑇 𝑇 𝑇

𝑃 𝑃

Wrong formule/ verkeerde formule: 1 mark/punt for/vir 𝑖




Let/stel

9 9 simplify/vereenvoudig 9 answer/antwoord

(3) 7.2

9 -intercept/afsnit 9 asymptote/asimptoot 9 shape/vorm 9 -intercept/afsnit 9 -intercept/afsnit

(5) 7.3 9 answer/antwoord (1) 7.4

or/of or/of

9 equation/vergelyking 9 simplify/vereenvoudig 9 -values/waardes 9 9

(5) 7.5 9 9

(2) 7.6

9 9

(2) [18]

𝑔

𝑦

𝑂

𝑦

𝑥

Answer ONLY 𝑥 : 2 marks.

SLEGS antwoord 𝑥 : 2 punte.




9 substitution/substitusie 9 simplify/vereenvoudig

(2) 8.2

9 9

(2) 8.3 9 answer/antwoord

(1) 8.4


(2) 8.5 9

9 (2)

[9] QUESTION 9/VRAAG 9 9.1

Let/stel

9 9 answer/antwoord

(2) 9.2

(

9 setting up equation/ opstel van vergelyking 9 substitution/substitusie 9 -value/waarde 9 simplification/vereenvoudiging

(4)

NOTE/LET WEL: No reference can be made to 9.3/Geen verwysing kan na 9.3 gemaak word nie.

Do not penalise if not in coordinate form. Moenie penaliseer indien nie in koördinaatvorm nie.



9.3

OR/OF

( )

( )

( )

( )

or/of or/of 6,125

(

)

9

9

(2) 9.4 :

Let/stel units/eenhede

9 substitute/vervang 9 9 answer/antwoord

(3) 9.5 ;

9 answer/antwoord

(1) 9.6 or/of 9

9 (2)

[14] QUESTION 10/VRAAG 10 10.1

OR/OF

9 9 9 9 equation/vergelyking 9 9 9

(7) 10.2 at least 2 out of 3/ten minste 2 uit 3)

or/of

or/of 9 9 answer/antwoord (2)

[9]

Do not penalise if not in coordinate form. Moenie penaliseer indien nie in koördinaatvorm nie.

𝐶𝐴:𝑏 𝑎

𝐶𝐴: 𝑒 𝑎 𝑑

𝐶𝐴:𝑓 𝑑



QUESTION 11/VRAAG 11 11.1 For independent events/vir onafhanklike

gebeurtenisse: and/en

9 rule/reëel 9 substitution/substitusie 9 9 -value/waarde 9 -value/waarde

(5) 11.2.1 female failing/vroulik druip)

9 9

(2) 11.2.2 pass, given male/slag, gegee manlik)

9 9

(2) 11.3 or and

of en fails both/druip albei) OR/OF fails both/druip albei)

9 rule/reël 9 0,7 9 answer/antwoord 9 9 9 answer/antwoord

(3) [12] TOTAL/TOTAAL: 150

𝐴/𝑅 𝑀/𝑊


9

Documents

GRADE 11 NOVEMBER 2015 MATHEMATICS P1 - …...6.2 Determine the effective interest rate if an investment earns interest at a nominal interest rate of 11,5% per annum, compounded quarterly