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
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NATIONAL SENIOR CERTIFICATE EXAMINATION
MAY 2021
MATHEMATICS: PAPER I
EXAMINATION NUMBER
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 25 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 on the question paper and hand it in at the
end of the examination. Remember to write your examination number
in the space provided.
4. You may use an approved non-programmable and non-graphical calculator unless
otherwise stated.
5. 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.
6. Diagrams are not necessarily drawn to scale.
7. If necessary, round off answers to ONE decimal place, unless stated otherwise.
8. There are two blank pages at the end of the paper that you may use if you run out
of space when answering a question. If you use this space, clearly indicate this at
the question and write the question number of your answer in the extra space.
9. It is in your own interest to write legibly and to present your work neatly.
FOR OFFICE USE ONLY: MARKER TO ENTER MARKS
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTAL
19 6 17 14 15 23 13 16 21 6 /150
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, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I – MAY Page 2 of 25
SECTION A
QUESTION 1
(a) Given: f ( x ) = −3 x 2 + 1
(1) Determine the average gradient of f ( x ) between x = 1 and x = 2.
(3)
(2) Determine f '( x ) from first principles.
(5)
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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
, TOTAL
MARKS
Please paste the barcoded
label here
NATIONAL SENIOR CERTIFICATE EXAMINATION
MAY 2021
MATHEMATICS: PAPER I
EXAMINATION NUMBER
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 25 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 on the question paper and hand it in at the
end of the examination. Remember to write your examination number
in the space provided.
4. You may use an approved non-programmable and non-graphical calculator unless
otherwise stated.
5. 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.
6. Diagrams are not necessarily drawn to scale.
7. If necessary, round off answers to ONE decimal place, unless stated otherwise.
8. There are two blank pages at the end of the paper that you may use if you run out
of space when answering a question. If you use this space, clearly indicate this at
the question and write the question number of your answer in the extra space.
9. It is in your own interest to write legibly and to present your work neatly.
FOR OFFICE USE ONLY: MARKER TO ENTER MARKS
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTAL
19 6 17 14 15 23 13 16 21 6 /150
IEB Copyright © 2021 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I – MAY Page 2 of 25
SECTION A
QUESTION 1
(a) Given: f ( x ) = −3 x 2 + 1
(1) Determine the average gradient of f ( x ) between x = 1 and x = 2.
(3)
(2) Determine f '( x ) from first principles.
(5)
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