INTERNATIONAL SECONDARY CERTIFICATE EXAMINATION
NOVEMBER 2023
FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I
MARKING GUIDELINES
Time: 2 hours 200 marks
These marking guidelines are prepared for use by examiners and sub-examiners, all
of whom are required to attend a standardisation meeting to ensure that the
guidelines are consistently interpreted and applied in the marking of candidates'
scripts.
The IEB will not enter into any discussions or correspondence about any marking
guidelines. It is acknowledged that there may be different views about some matters
of emphasis or detail in the guidelines. It is also recognised that, without the benefit
of attendance at a standardisation meeting, there may be different interpretations of
the application of the marking guidelines.
IEB Copyright © 2023 PLEASE TURN OVER
,INTERNATIONAL SECONDARY CERTIFICATE: FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I – Page 2 of 18
MARKING GUIDELINES
QUESTION 1
1.1 Solve:
(a) In(2 + e − x ) = 2 Leave your answer in the form x = In(....)
e2 = 2 + e − x
e− x = e2 − 2
− x = In(e2 − 2)
x = −In(e2 − 2)
1
x = In 2
e −2
(b) 2x + 3 = 3 x + 4
2x + 3 = 3 x + 4 2x + 3 = −3 x − 4
or
7
x = −1 or x=−
5
a check reveals x = −1 only
1.2 Give, in standard ax 4 + bx3 + cx 2 + dx + e = 0 form, a quartic equation which has
x = 2 + 3 and 2 − i as roots. The values of a, b, c, d and e must be rational.
one quadratic has roots 2 + 3 and 2 − 3
the sum of these roots is 4 and the product is 1 so x 2 − 4x + 1 = 0
the other has roots 2 − i and 2 + i
the sum of these roots is 4 and the product is 5 so x 2 − 4x + 5 = 0
( x 2 − 4 x + 1)( x 2 − 4 x + 5) = 0
x 4 − 8x3 + 22x 2 − 24x + 5 = 0
1.3 Determine positive real values of a and b if:
(a + bi )(b + i ) = (2b + a)i
LHS = ab − b + (b2 + a)i
so, ab − b = 0 and b2 + a = 2b + a
so, b(a − 1) = 0
since b 0, a = 1
b = 2
IEB Copyright © 2023 PLEASE TURN OVER
, INTERNATIONAL SECONDARY CERTIFICATE: FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I – Page 3 of 18
MARKING GUIDELINES
1.4. Sketch the following functions on the axes provided. You should draw and give the
equations of any asymptotes as well as showing any intercepts with the axes.
(a) y = e− x − 1
(b) y = In( x + 1)
IEB Copyright © 2023 PLEASE TURN OVER
NOVEMBER 2023
FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I
MARKING GUIDELINES
Time: 2 hours 200 marks
These marking guidelines are prepared for use by examiners and sub-examiners, all
of whom are required to attend a standardisation meeting to ensure that the
guidelines are consistently interpreted and applied in the marking of candidates'
scripts.
The IEB will not enter into any discussions or correspondence about any marking
guidelines. It is acknowledged that there may be different views about some matters
of emphasis or detail in the guidelines. It is also recognised that, without the benefit
of attendance at a standardisation meeting, there may be different interpretations of
the application of the marking guidelines.
IEB Copyright © 2023 PLEASE TURN OVER
,INTERNATIONAL SECONDARY CERTIFICATE: FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I – Page 2 of 18
MARKING GUIDELINES
QUESTION 1
1.1 Solve:
(a) In(2 + e − x ) = 2 Leave your answer in the form x = In(....)
e2 = 2 + e − x
e− x = e2 − 2
− x = In(e2 − 2)
x = −In(e2 − 2)
1
x = In 2
e −2
(b) 2x + 3 = 3 x + 4
2x + 3 = 3 x + 4 2x + 3 = −3 x − 4
or
7
x = −1 or x=−
5
a check reveals x = −1 only
1.2 Give, in standard ax 4 + bx3 + cx 2 + dx + e = 0 form, a quartic equation which has
x = 2 + 3 and 2 − i as roots. The values of a, b, c, d and e must be rational.
one quadratic has roots 2 + 3 and 2 − 3
the sum of these roots is 4 and the product is 1 so x 2 − 4x + 1 = 0
the other has roots 2 − i and 2 + i
the sum of these roots is 4 and the product is 5 so x 2 − 4x + 5 = 0
( x 2 − 4 x + 1)( x 2 − 4 x + 5) = 0
x 4 − 8x3 + 22x 2 − 24x + 5 = 0
1.3 Determine positive real values of a and b if:
(a + bi )(b + i ) = (2b + a)i
LHS = ab − b + (b2 + a)i
so, ab − b = 0 and b2 + a = 2b + a
so, b(a − 1) = 0
since b 0, a = 1
b = 2
IEB Copyright © 2023 PLEASE TURN OVER
, INTERNATIONAL SECONDARY CERTIFICATE: FURTHER STUDIES MATHEMATICS (STANDARD): PAPER I – Page 3 of 18
MARKING GUIDELINES
1.4. Sketch the following functions on the axes provided. You should draw and give the
equations of any asymptotes as well as showing any intercepts with the axes.
(a) y = e− x − 1
(b) y = In( x + 1)
IEB Copyright © 2023 PLEASE TURN OVER
