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 2011
MATHEMATICS: PAPER II
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 14 pages, a Diagram Sheet (pages i to iv), an Answer Sheet
for Questions 4 and 5 and an Information Sheet (pages i to ii). Please check that your paper
is complete.
2. Write your examination number in the space provided in your Answer Book, the Diagram
Sheet and the Answer Sheet.
3. Answer ALL the questions. Answer Questions 4 and 5 on the Answer Sheet provided, and
hand this in with your Answer Book.
4. A Diagram Sheet is provided. Any changes made to a diagram must be shown on this sheet,
and not on the question paper. Please hand in the Diagram Sheet with your Answer Book.
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. All answers must be given correct to one decimal digit, unless otherwise stated.
10. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2011 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 14
SECTION A
QUESTION 1
(a) In the diagram below, trapezium ABCD is drawn with vertices A(−4;5), B(7; 2),
C(0; −2) and D(t ;1).
y
O x
(1) Determine the equation of the line passing through B and C. (3)
(2) Determine, correct to one decimal digit, the acute angle that the line BC makes
with the x-axis. (2)
(3) Determine the value of t. (3)
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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
NOVEMBER 2011
MATHEMATICS: PAPER II
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 14 pages, a Diagram Sheet (pages i to iv), an Answer Sheet
for Questions 4 and 5 and an Information Sheet (pages i to ii). Please check that your paper
is complete.
2. Write your examination number in the space provided in your Answer Book, the Diagram
Sheet and the Answer Sheet.
3. Answer ALL the questions. Answer Questions 4 and 5 on the Answer Sheet provided, and
hand this in with your Answer Book.
4. A Diagram Sheet is provided. Any changes made to a diagram must be shown on this sheet,
and not on the question paper. Please hand in the Diagram Sheet with your Answer Book.
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. All answers must be given correct to one decimal digit, unless otherwise stated.
10. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2011 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER II Page 2 of 14
SECTION A
QUESTION 1
(a) In the diagram below, trapezium ABCD is drawn with vertices A(−4;5), B(7; 2),
C(0; −2) and D(t ;1).
y
O x
(1) Determine the equation of the line passing through B and C. (3)
(2) Determine, correct to one decimal digit, the acute angle that the line BC makes
with the x-axis. (2)
(3) Determine the value of t. (3)
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