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 2015
MATHEMATICS: PAPER I
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 12 pages and an Information Sheet of 2 pages (i – ii). Please
check that your 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. Round off your answers to one decimal digit where necessary.
7. All the necessary working details must be clearly shown.
8. It is in your own interest to write legibly and to present your work neatly.
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, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I Page 2 of 12
SECTION A
QUESTION 1
(a) Solve for x :
(1) ( x − 3)( x + 1) =
5 (3)
3x
(2) 9 2 x−1
= (3)
3
(3) 2 2 − 7 x = −36 x (3)
(b) Determine, in terms of k, the co-ordinates of the points of intersection of the graphs
of =
y kx + k and y =x 2 + 2kx + k , where k ∈ � . (5)
(c) Given: 9 x 2 + nx + 49 =
0
(1) Express the roots of the equation in terms of n. (2)
(2) For what value(s) of n will the roots be equal? (2)
[18]
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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 2015
MATHEMATICS: PAPER I
Time: 3 hours 150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1. This question paper consists of 12 pages and an Information Sheet of 2 pages (i – ii). Please
check that your 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. Round off your answers to one decimal digit where necessary.
7. All the necessary working details must be clearly shown.
8. It is in your own interest to write legibly and to present your work neatly.
IEB Copyright © 2015 PLEASE TURN OVER
, NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I Page 2 of 12
SECTION A
QUESTION 1
(a) Solve for x :
(1) ( x − 3)( x + 1) =
5 (3)
3x
(2) 9 2 x−1
= (3)
3
(3) 2 2 − 7 x = −36 x (3)
(b) Determine, in terms of k, the co-ordinates of the points of intersection of the graphs
of =
y kx + k and y =x 2 + 2kx + k , where k ∈ � . (5)
(c) Given: 9 x 2 + nx + 49 =
0
(1) Express the roots of the equation in terms of n. (2)
(2) For what value(s) of n will the roots be equal? (2)
[18]
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