MATHEMATICS P1 COMPETENCY TEST ENGLISH 2015 · COMPETENCY TEST MARKERS MATHEMATICS P1 May 2015 MARKS: 100 TIME: 2 hours This paper consists of 25 pages. This competency test consists

COMPETENCY TEST

MARKERS

MATHEMATICS P1 May 2015

MARKS: 100

TIME: 2 hours

This paper consists of 25 pages.

This competency test consists of Section A (80 MARKS) and Section B (20 MARKS) Answer all questions.

Kindly print

NAME:

(first name) (surname)

INSTITUTION / SCHOOL:

Section A

Section B

Total

Mathematics/P1 2 WCED/May 2015Competency Test

Copyright reserved Please turn over

SECTION A 80 MARKS

INSTRUCTIONS

Provide full solutions to Questions 1 to 7 in the spaces provided.

Clearly show ALL calculations, diagrams, graphs, et cetera you have used indetermining the solutions.

QUESTION 1

1.1 Solve for x in

xx

5576 , if x (6)

1.2 If ,1 i calculate the value of 201 i (3)

[9]



ANSWER

QUESTION 1



QUESTION 2

2.1 The first term of an arithmetic sequence is 3 more than the first term of a

geometric sequence. The constant difference of the arithmetic sequence

and the common ratio of the geometric sequence are both 2. If the 50th term

of the arithmetic sequence equals 103, determine the sum of the first eighteen

terms of the geometric sequence. (6)

2.2 A quadratic sequence is defined with the following properties:

17161615 11 ,5 TTTT and 7525 T

Determine the first term of the sequence. (7)

[13]



ANSWER

QUESTION 2



QUESTION 3

In the diagram below the straight line, g, represents one of the axes of symmetry of the

hyperbola, f.

A( 1; 2) is a reflection of B(3 ; 6) in the straight line, g.

3.1 Determine the equation of g, if g has slope 1. (4)

3.2 If the equation of the other axis of symmetry of g is given by 1 xy ,

determine the value(s) of x for which 1)( xxf (7)

[11]

f

f

A( 1; 2)

B(3; 6)

y

x

g



ANSWER

QUESTION 3



QUESTION 4

Joan took out a home loan for R850 000 at an interest rate of 7% interest per annum,

compounded monthly. She plans to repay this loan over 25 years and her first payment is made

one month after her loan is granted.

4.1 Calculate the value of her monthly instalment. (4)

4.2 At the end of the third year an investment enables Joan to make an extra payment of

R100 000 in addition to the monthly instalment. How soon after the third year will

she pay off the loan? (7)

[11]



ANSWER

QUESTION 4



QUESTION 5

Given below is the sketch graph of g , the derivative of the function .g

32)(' 2 xxxg

5.1 Determine the coordinates of A and B, the x-intercepts g (3)

5.2 What does the x-intercepts of g tell you about the graph of g? (1)

5.3 If the graph of g is shifted vertically 5 units upwards, draw a possible graph of the

corresponding cubic function, g. Indicate the x-coordinate of the point of inflection. (3)

5.4 If qxxf 5)( is a tangent to the graph of ,g determine the value of q. (6)

[13]

A B

y

x



ANSWER

QUESTION 5



QUESTION 6

In the diagram below B is the point of inflection of the graph of a cubic function,

.2)( 23 dcxbxxxf The x-coordinate of B is 2

1.

A and

0;2

5C are the x-intercepts of f.

6.1 Determine the coordinates of A. (7)

6.2 Determine the values of m for which 0)( mxf will have three different

roots. (6)

[13]

f

y

x A



ANSWER

QUESTION 6



QUESTION 7

Learners at ABC High School are allowed to create their own individual passwords of seven

characters to sign in on the school’s computers. However, the first four characters must consist of 4

letters from the alphabet (excluding all the vowels). The last three characters must be three digits

ranging from 0 to 9. Letters and digits may be repeated.

7.1 How many different passwords can be formed? (3)

7.2 What is the probability that a student’s password will contain

7.2.1 at least one 9? (4)

7.2.2 exactly one 9? (3)

[10]



ANSWER

QUESTION 7



BLANK PAGE



SECTION B 20 MARKS

Instructions:

The following questions and solutions are taken from a mock mathematics examination.

Below each question a memorandum with mark allocation is provided.

Answers from a candidate are provided in each case.

Mark each question WITH A RED PEN and indicate each mark with a tick.

Indicate the tick at the appropriate step where the mark is given and circle any mistakesthat you deduct marks for.

Indicate with CA (consistent accuracy) next to the tick if you are marking with thecandidate’s mistake.

Indicate the total mark allocated for the question.

Each question will be assessed as follows:

2 marks for ticks correctly allocated

1 mark for mistake(s) identified

1 mark for CA marking

1 mark for reasonable total marks allocated

Total: 5 marks



QUESTION 1

Given

xxxxf 627)( 23

1.1 Determine the values of x for which .0)( xf (5)

1.2 Determine the values of x for which .0)( xf (3)

[8]

PROPOSED MEMORANDUM

1.1

79,0or 08,1 14

1722

)7(2

)6)(7(422

0627 or 0

0)627(

2

2

2

xx

x

xxx

xxx √ factors

√ correct subst in formula

√ x = 0√ 79,0x√ 08,1x

(5)

1.2 79,00or 08,1 xx √ 08,1x

√√ 79,00 x

(3) TOTAL [8]



CANDIDATE’S ANSWER

1.1 0627 23 xxx

08,1 of 79,0 14

1722

)7(2

)6)(7(4222

06227 )

xx

x

xx-x

1.2 062237 xxx

08,1 of 79,0

: valuesCritical

06227 )

xx

xx-x

79,0 1,08

08,1 79,0 x



QUESTION 2

Given

... 0 100

729 0

10

81 0 9 0 10

Assume that this number pattern continues consistently.

2.1 Write down the next two terms of the sequence. (2)

2.2 Calculate the value of the 19th term of the series (correct to two decimal places) (3)

2.3 Calculate the sum to infinity of the series. (2)

[7]

PROPOSED MEMORANDUM

2.1 6561 ; 0

1000

√ √ one mark for each term in thecorrect order

(2)

2.2 ...

100

729

10

81 9 10 is a GS with r = 0,9

110)9,0(10GS theof 10T19T

= 3,87

√ the value of r√ the value of n in geometric

series√ answer

(3)2.3

9,01

10

S

= 100

√ Substitution in correctformula

√ answer (2)

TOTAL [7]




2.1 6561

; 01000

2.2 18

19 9

1010T

= 66,62

2.3

9

101

10

S

= 90



QUESTION 3

In the sketch below the parabola, f, cuts the y-axis at 2, and passes through the points P( 1; 1) and Q(2; 2).

3.1 Determine the equation of the parabola. (5)

3.2 If ,31)( xfxh determine the equation of .1h (4)

PROPOSED MEMORANDUM

3.1 22 bxaxy

Subst ( 1;1) yields 2)1()1(1 2 ba 3 ba …………..(1)

Subst (2; 2) yields 2)2()2(2 2 ba 024 ba …………..(2)

Solving (1) & (2) simultaneously, yields a = 1 & b = 2

22)( 2 xxxf

√ value of c

√ Substitute point ( 1;1)

√ Substitute point (2; 2)

√ value of a√ value of b

(5)3.2 31)( 2 xxf

2)( xxh

21 : yxh

xy

√ f in form qpxaxf 2)(√ equation of h√ swopping x and y√ equation of h-1

(4)

TOTAL [9]

x

y

O

Q(2; 2) 2

P( 1;1)

f




3.1 2 xaxy

21)1(3 a

a33 1a

2 xaxy

22)( xxxf

22)( 2 xxxf

3.2 41)( 2 xxf

1)( 2 xxh

12 yx

1 xy



QUESTION 4

Two independent relay teams want to qualify for the next Olympic Games. The probability

that the two teams run under the qualifying time is 9

4 and

7

3 respectively. Calculate the

probability that one of the relay teams will run under the qualifying time in their next race.

[4]

PROPOSED MEMORANDUM

P(A and B) = P(A) P(B)

= 7

3

9

4

= 21

4

P(A or B) = P(A) + P(B) – P(A and B)

63

4363

12272821

4

7

3

9

4

= 68,25%

√ P(A) P(B) =21

4

√ P(A or B) = P(A) + P(B) – P(A and B)

√ substitution

√ answer

[4]




P(A and B) = P(A) + P(B)

= 4 3

9 7

= 55

63

P(A or B) = P(A) + P(B) – P(A and B)

4 3 55

9 7 6328 27 55

630

Documents

MATHEMATICS P1 COMPETENCY TEST ENGLISH 2015 · COMPETENCY TEST MARKERS MATHEMATICS P1 May 2015 MARKS: 100 TIME: 2 hours This paper consists of 25 pages. This competency test consists