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
NOVEMBER 2012
MATHEMATICS: PAPER II
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 14 pages, an Answer/Diagram Sheet of 4 pages (i to iv) and
an Information Sheet of 2 pages (i – ii). Please check that your paper is complete.
2. Please detach the Answer/Diagram Sheet from the middle of your question paper. Write
your examination number in the space provided in your Answer Book and the
Answer/Diagram Sheet.
3. Any changes made to a diagram must be shown on the Answer/Diagram Sheet, and not on
the question paper. Please hand in the Answer/Diagram Sheet with your Answer Book.
4. Answer ALL the questions.
5. Please note that diagrams are not necessarily drawn to scale.
6. All necessary working details must be shown.
7. Approved non-programmable and non-graphical calculators may be used, unless otherwise
stated.
8. Ensure that your calculator is in DEGREE mode.
9. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2012 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 14
SECTION A
QUESTION 1
PLEASE ENSURE THAT YOUR CALCULATOR IS IN DEGREE MODE
(a) Given: E 4;3 , F 0; 1 and G t;1 .
Determine the value of t for which
(1) E, F and G all lie on the same straight line. (4)
(2) FEG is right angled at F. (2)
(b) In the diagram below, ABCD is a parallelogram.
A is the point 2;5 , D is the point 4;6 and B is on the x-axis.
The equation of line CD is given by 2 y x 16.
y
D
C
O B x
(1) ˆ correct to one decimal digit.
Determine ABO (3)
(2) Determine the equation of AB in the form y mx c. (3)
(3) Determine the co-ordinates of B. (2)
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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
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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
NOVEMBER 2012
MATHEMATICS: PAPER II
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 14 pages, an Answer/Diagram Sheet of 4 pages (i to iv) and
an Information Sheet of 2 pages (i – ii). Please check that your paper is complete.
2. Please detach the Answer/Diagram Sheet from the middle of your question paper. Write
your examination number in the space provided in your Answer Book and the
Answer/Diagram Sheet.
3. Any changes made to a diagram must be shown on the Answer/Diagram Sheet, and not on
the question paper. Please hand in the Answer/Diagram Sheet with your Answer Book.
4. Answer ALL the questions.
5. Please note that diagrams are not necessarily drawn to scale.
6. All necessary working details must be shown.
7. Approved non-programmable and non-graphical calculators may be used, unless otherwise
stated.
8. Ensure that your calculator is in DEGREE mode.
9. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2012 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 14
SECTION A
QUESTION 1
PLEASE ENSURE THAT YOUR CALCULATOR IS IN DEGREE MODE
(a) Given: E 4;3 , F 0; 1 and G t;1 .
Determine the value of t for which
(1) E, F and G all lie on the same straight line. (4)
(2) FEG is right angled at F. (2)
(b) In the diagram below, ABCD is a parallelogram.
A is the point 2;5 , D is the point 4;6 and B is on the x-axis.
The equation of line CD is given by 2 y x 16.
y
D
C
O B x
(1) ˆ correct to one decimal digit.
Determine ABO (3)
(2) Determine the equation of AB in the form y mx c. (3)
(3) Determine the co-ordinates of B. (2)
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