NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II – INFORMATION SHEET Page i of ii
INFORMATION SHEET
–b ± b 2 – 4ac
x =
2a
n n
n (n + 1)
∑1 = n
i =1
∑i
i =1
=
2
n
Tn =a + (n − 1)d S=
n [2a + (n − 1)d ]
2
−1 a(r n − 1) a
Tn = ar n= Sn ;r ≠1 =
S∞ ; −1 < r < 1
r −1 1− r
f ( x + h) − f ( x )
f ′ ( x ) = lim
h →0 h
=
A P (1 + n i ) =
A P (1 − n i )
=
A P (1 + i ) n =
A P (1 − i ) n
(1 + i )n − 1 1 − (1 + i )− n
F =
x P =
x
i i
x + x2 y1 + y 2
= ( x2 – x1 ) 2 + ( y 2 – y1 ) 2 M 1 ;
2
d
2
=
y mx + c y – y1 = m ( x – x1 )
y 2 – y1
m = m = tanθ
x2 – x1
( x – a )2 + ( y – b )2 =
r2
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,NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II – INFORMATION SHEET Page ii of ii
a b c
In ∆ABC: = =
sin A sin B sin C
a=
2
b 2 + c 2 – 2 b c.cos A
1
area ∆ ABC = a b.sinC
2
sin(α=
+ β) sin α .cos β + cos α .sin β sin(α – β ) = sin α .cos β – cos α .sin β
cos(α + β ) =
cos α .cos β – sin α .sin β =
cos(α – β ) cos α .cos β + sin α .sin β
cos2 α − sin2 α
cos 2 α
= 1 − 2 sin α
2
sin2 α = 2sin α .cos α
2cos2 α − 1
n
∑ ( xi − x )
2
∑ fx
x= σ2 =i = 1
n n
n ( A)
P ( A) = =
P ( A or B) P ( A) + P (B ) – P ( A and B )
n (S )
∑ ( x − x )( y − y )
ŷ= a + bx b=
∑ (x − x )
2
IEB Copyright © 2021
, NATIONAL SENIOR CERTIFICATE EXAMINATION
SUPPLEMENTARY EXAMINATION – MARCH 2019
MATHEMATICS: PAPER I
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 11 pages and an Information Sheet of 2 pages (i–ii).
Please check that your question paper is complete.
2. Read the questions carefully.
3. Answer all the questions.
4. Number your answers exactly as the questions are numbered.
5. You may use an approved non-programmable and non-graphical calculator unless
otherwise stated.
6. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
Answers only will NOT necessarily be awarded full marks.
7. Diagrams are not necessarily drawn to scale.
8. If necessary, round off answers to ONE decimal place, unless stated otherwise.
9. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2019 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I – SUPPLEMENTARY Page 2 of 11
SECTION A
QUESTION 1
(a) Consider the following arithmetic sequence:
( x + 5) ; (37 − x ) ; ( x + 13) …
(1) Determine the value of x. (3)
(2) Determine the general term of the sequence in the form: Tn = ... (2)
(b) The sum of the first three terms of a geometric sequence is 91 and its
common ratio is 3, determine the first term of the sequence. (3)
375
(c) In a convergent geometric series, S2 = 90 and S∞ = . Determine its first
4
term and its common ratio. (6)
(d) The share price of a certain company formed a quadratic pattern over a
specific time interval.
The share price at the end of each day for the first 5 days was:
Day 1: R 32 699
Day 2: R 32 896
Day 3: R 33 091
Day 4: R 33 284
Day 5: R 33 475
(1) Show that the pattern is quadratic. (2)
(2) Determine a formula for the n th term of the pattern. (6)
(3) At the end of which day, will the share price be at its maximum? (3)
[25]
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INFORMATION SHEET
–b ± b 2 – 4ac
x =
2a
n n
n (n + 1)
∑1 = n
i =1
∑i
i =1
=
2
n
Tn =a + (n − 1)d S=
n [2a + (n − 1)d ]
2
−1 a(r n − 1) a
Tn = ar n= Sn ;r ≠1 =
S∞ ; −1 < r < 1
r −1 1− r
f ( x + h) − f ( x )
f ′ ( x ) = lim
h →0 h
=
A P (1 + n i ) =
A P (1 − n i )
=
A P (1 + i ) n =
A P (1 − i ) n
(1 + i )n − 1 1 − (1 + i )− n
F =
x P =
x
i i
x + x2 y1 + y 2
= ( x2 – x1 ) 2 + ( y 2 – y1 ) 2 M 1 ;
2
d
2
=
y mx + c y – y1 = m ( x – x1 )
y 2 – y1
m = m = tanθ
x2 – x1
( x – a )2 + ( y – b )2 =
r2
IEB Copyright © 2021 PLEASE TURN OVER
,NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II – INFORMATION SHEET Page ii of ii
a b c
In ∆ABC: = =
sin A sin B sin C
a=
2
b 2 + c 2 – 2 b c.cos A
1
area ∆ ABC = a b.sinC
2
sin(α=
+ β) sin α .cos β + cos α .sin β sin(α – β ) = sin α .cos β – cos α .sin β
cos(α + β ) =
cos α .cos β – sin α .sin β =
cos(α – β ) cos α .cos β + sin α .sin β
cos2 α − sin2 α
cos 2 α
= 1 − 2 sin α
2
sin2 α = 2sin α .cos α
2cos2 α − 1
n
∑ ( xi − x )
2
∑ fx
x= σ2 =i = 1
n n
n ( A)
P ( A) = =
P ( A or B) P ( A) + P (B ) – P ( A and B )
n (S )
∑ ( x − x )( y − y )
ŷ= a + bx b=
∑ (x − x )
2
IEB Copyright © 2021
, NATIONAL SENIOR CERTIFICATE EXAMINATION
SUPPLEMENTARY EXAMINATION – MARCH 2019
MATHEMATICS: PAPER I
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 11 pages and an Information Sheet of 2 pages (i–ii).
Please check that your question paper is complete.
2. Read the questions carefully.
3. Answer all the questions.
4. Number your answers exactly as the questions are numbered.
5. You may use an approved non-programmable and non-graphical calculator unless
otherwise stated.
6. Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
determining your answers.
Answers only will NOT necessarily be awarded full marks.
7. Diagrams are not necessarily drawn to scale.
8. If necessary, round off answers to ONE decimal place, unless stated otherwise.
9. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2019 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I – SUPPLEMENTARY Page 2 of 11
SECTION A
QUESTION 1
(a) Consider the following arithmetic sequence:
( x + 5) ; (37 − x ) ; ( x + 13) …
(1) Determine the value of x. (3)
(2) Determine the general term of the sequence in the form: Tn = ... (2)
(b) The sum of the first three terms of a geometric sequence is 91 and its
common ratio is 3, determine the first term of the sequence. (3)
375
(c) In a convergent geometric series, S2 = 90 and S∞ = . Determine its first
4
term and its common ratio. (6)
(d) The share price of a certain company formed a quadratic pattern over a
specific time interval.
The share price at the end of each day for the first 5 days was:
Day 1: R 32 699
Day 2: R 32 896
Day 3: R 33 091
Day 4: R 33 284
Day 5: R 33 475
(1) Show that the pattern is quadratic. (2)
(2) Determine a formula for the n th term of the pattern. (6)
(3) At the end of which day, will the share price be at its maximum? (3)
[25]
IEB Copyright © 2019
