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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
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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]
Mathematics/P1 4 WCED/May 2015Competency Test
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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]
Mathematics/P1 6 WCED/May 2015Competency Test
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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
Mathematics/P1 8 WCED/May 2015Competency Test
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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]
Mathematics/P1 10 WCED/May 2015Competency Test
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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
Mathematics/P1 11 WCED/May 2015Competency Test
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ANSWER
QUESTION 5
Mathematics/P1 12 WCED/May 2015Competency Test
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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
Mathematics/P1 13 WCED/May 2015Competency Test
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ANSWER
QUESTION 6
Mathematics/P1 14 WCED/May 2015Competency Test
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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]
Mathematics/P1 15 WCED/May 2015Competency Test
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ANSWER
QUESTION 7
Mathematics/P1 17 WCED/May 2015Competency Test
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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
Mathematics/P1 18 WCED/May 2015Competency Test
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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]
Mathematics/P1 19 WCED/May 2015Competency Test
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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
Mathematics/P1 20 WCED/May 2015Competency Test
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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]
Mathematics/P1 21 WCED/May 2015Competency Test
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CANDIDATE’S ANSWER
2.1 6561
; 01000
2.2 18
19 9
1010T
= 66,62
2.3
9
101
10
S
= 90
Mathematics/P1 22 WCED/May 2015Competency Test
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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
Mathematics/P1 23 WCED/May 2015Competency Test
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CANDIDATE’S ANSWER
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
Mathematics/P1 24 WCED/May 2015Competency Test
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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]