, Gr 12 Further Studies Mathematics - Paper 1 Content
2 hours, 200 marks, write on paper
Algebra (Key: Done in Gr 11; Done this year with Ms Colly; Mrs White to do/done)
• Complex numbers
• Solving equations
• Dividing polynomials
• Solving inequalities
• Partial fractions
• Absolute values
o Interpret and apply to different situations
o Graphs
• Logs and exponential functions
o 𝑒 and ln 𝑥
o Use log laws
o Draw graphs
• Mathematical induction
Calculus: (Key: Done in Gr 11; Done this year with Ms Colly; Mrs White to do/done)
• Trig
o Convert between angles in degrees and radians
o Manipulate trig statements, Solve problems, Find general solution of equations
o Measurement: Find length of arc of circle, Area of sectors and segments of circle
• Limits (repeated this year)
• Continuity: identify, apply (repeated this year)
• Apply theorem “A function that is differentiable at a point is continuous at that point”
• Derivatives
1 1
o First principles (including √𝑥; √𝑎𝑥 + 𝑏; 𝑎𝑥+𝑏; ) *see note on classroom*
√𝑎𝑥+𝑏
o Rules of differentiation: Power rule, chain rule, product rule, quotient rule
o Including 𝑒 𝑥 , ln 𝑥, trig functions
o Higher order derivatives (in regular maths)
o Implicit differentiation
• Newton-Raphson method
• Sketch and interpret graphs
o Split domain and composite functions
o Polynomial functions
o Rational functions
• Interpretation (same as regular maths, just different derivatives)
o Max and min values, Increasing/decreasing function, Concavity, etc
o Optimisation, rates of change, etc (same ideas as core Maths)
• Fundamental theorem of calculus
• Integration
o Including: 𝑒 𝑥 , ln 𝑥, trig functions
o Methods: Direct anti derivatives, simplification of trig functions, u-method, integration with given
substitution, integration by parts
• Area under a curve/between curve
o Definite integral
o Riemann sums (interpret formula, not needed to do long way) (Needs more work)
• Volume generated by rotating function about 𝑥-axis
,
,
,
,12/4 Numbers
Imaginary and complex
·
↓
2 = -
1
S
=
(i) i
e.g.it3 ;72.;3
=
(i)
=
-1
(i4)18.;
=
(i) =i2 = -
i
(I)'8.-i
=
=
i?
(i) -1. 1 1
=
=
-
-- i
e.g. N -
16 1
= -
1x16 N 1x11b
=
-
ix
=
4 4i =
/
This is a COMPLEX NUMBER:
real a +
bi where a, b ER
unreal
part
part
Adding e.g.:(2 Si) (8-3i) 10 2i
+ +
=
+
Subtracting e.g.:(2 +Si) -(8-3i) = -
6 8i +
Multiplying e.g.:(2 +Si) (8-3i) 16-6: 40i-15:
=
+
=
16 34: 15
+
+
31
=
34;
+
(2 +5i)2 (2 Si)(2 + 5i)
squaring e.g.:
=
+
=
4 10i +
10i
+ 25;2
+
=-
21 20 i
+
,131
Division with ComplexNumbers
(2 5i)(8 -3i)
+ =
31 34:
+
(31 34:(8 3i) 102:2
i
+
+
248 93i 272i
+ +
+
=
(8 3i) 8 3i)
- +
64 24i + -
24i -
9;2
146 365:
+
-
73
2
=
5i
+
3
Remember: The CONJUGATE of
(x 2)(x 2)
-
bi
+
a bi
+ is a
=> x2 -
4
,>
Math 127
Complex Numbers Worksheet Name: Memo
Perform the indicated operations and then completely simplify.
1. (3 + 5i) + (4 − 3i) 9. (5 + 7i)(7 − 5i)
7 zi
=
+
35
=
-
25i 49i
+ -
35:2
=
70 24i +
2. (11 − 6i) + 7i 12 − i
10.
=11 i
+
5
= -
jior 2,4-0,2i
3. (13 − 5i) − (7 − 11i)
6 6i
L2 − 3i (i
= +
11.
2i x i
= 3 2i +
-
2
=-- 1 or -1,5-i
4. 17 − (5 − 6i)
12
=
6i
+
(((2
5 + 6i i) +
12.
L2 − i ((z i) +
10 5i+12i
+
+
6:2
-
4 -
i2
5. 4(3 − i) 4 17i
=
+
S
12-4i
I 5ior
=
= +
0,8 3,4i
+
L3 − 2i3(1 7i) +
13.
L1 − 7i (1 7i) +
2
3 21i
+
-
2i -
14:
=
6. 3i(7 − 2i) 1 -
49:2
17 192
21i-bi2
+
=
=
50
6 21i
-+6i 0,34 0,382
=
+
or
+
14. i3
-- i
7. (5 − i)(7 + 4i)
35 20i-7i
= +
-
4;2
39 13i
=
+
15. i1183
. .
=L ·L
8. (7 + 5i)(7 − 5i) = (it)2.i
=
49 -
35i 35i
+ -
25:2 =- i
=
74
16. i137
=;136.;
= i
,GRADE 11 TERM 1 2024
FS TUTORIAL 1
EXERCISE 1
Question 1
1. Determine the value of the following:
a. ! !" b. ! !#
c. ! !$ d. ! !%%&
2. Expand the following, leaving answers in terms of " ± $!
a. (3 + 2!)" b. (1 − 2!)"
c. (2 + 4!)(3 − 2!) d. !(2 − 3!) − (5 − !)"
"
e. .√2008! "%%' − !2
Question 2
Given two complex numbers 3 and 4 such that 3 = 4 − 3! and 4 = 2 + 2!, find the value of
the following:
a. 3 + 24 b. 23 − 34
c. 3" !
d.
"
e. 4∗ #
f.
!∗
g. Re(3 × 4∗ ) !"
h. Im # $
#$
Question 3
Determine the values of = and > !f 1 + 3! = (2 + !)(= + >!).
, EXERCISE 2
Question 1
Factorise the following fully in ℂ.
a. A " + 2 b. 2A " + 8
c. 3A " − 6 d. A " + 2A + 2
e. A " − 2A − 2 f. A # − 81
g. A ! + 27 h. A ! − 27
i. A! − A" + A − 1 j. A ! − 5A " + 8A − 6
2 hours, 200 marks, write on paper
Algebra (Key: Done in Gr 11; Done this year with Ms Colly; Mrs White to do/done)
• Complex numbers
• Solving equations
• Dividing polynomials
• Solving inequalities
• Partial fractions
• Absolute values
o Interpret and apply to different situations
o Graphs
• Logs and exponential functions
o 𝑒 and ln 𝑥
o Use log laws
o Draw graphs
• Mathematical induction
Calculus: (Key: Done in Gr 11; Done this year with Ms Colly; Mrs White to do/done)
• Trig
o Convert between angles in degrees and radians
o Manipulate trig statements, Solve problems, Find general solution of equations
o Measurement: Find length of arc of circle, Area of sectors and segments of circle
• Limits (repeated this year)
• Continuity: identify, apply (repeated this year)
• Apply theorem “A function that is differentiable at a point is continuous at that point”
• Derivatives
1 1
o First principles (including √𝑥; √𝑎𝑥 + 𝑏; 𝑎𝑥+𝑏; ) *see note on classroom*
√𝑎𝑥+𝑏
o Rules of differentiation: Power rule, chain rule, product rule, quotient rule
o Including 𝑒 𝑥 , ln 𝑥, trig functions
o Higher order derivatives (in regular maths)
o Implicit differentiation
• Newton-Raphson method
• Sketch and interpret graphs
o Split domain and composite functions
o Polynomial functions
o Rational functions
• Interpretation (same as regular maths, just different derivatives)
o Max and min values, Increasing/decreasing function, Concavity, etc
o Optimisation, rates of change, etc (same ideas as core Maths)
• Fundamental theorem of calculus
• Integration
o Including: 𝑒 𝑥 , ln 𝑥, trig functions
o Methods: Direct anti derivatives, simplification of trig functions, u-method, integration with given
substitution, integration by parts
• Area under a curve/between curve
o Definite integral
o Riemann sums (interpret formula, not needed to do long way) (Needs more work)
• Volume generated by rotating function about 𝑥-axis
,
,
,
,12/4 Numbers
Imaginary and complex
·
↓
2 = -
1
S
=
(i) i
e.g.it3 ;72.;3
=
(i)
=
-1
(i4)18.;
=
(i) =i2 = -
i
(I)'8.-i
=
=
i?
(i) -1. 1 1
=
=
-
-- i
e.g. N -
16 1
= -
1x16 N 1x11b
=
-
ix
=
4 4i =
/
This is a COMPLEX NUMBER:
real a +
bi where a, b ER
unreal
part
part
Adding e.g.:(2 Si) (8-3i) 10 2i
+ +
=
+
Subtracting e.g.:(2 +Si) -(8-3i) = -
6 8i +
Multiplying e.g.:(2 +Si) (8-3i) 16-6: 40i-15:
=
+
=
16 34: 15
+
+
31
=
34;
+
(2 +5i)2 (2 Si)(2 + 5i)
squaring e.g.:
=
+
=
4 10i +
10i
+ 25;2
+
=-
21 20 i
+
,131
Division with ComplexNumbers
(2 5i)(8 -3i)
+ =
31 34:
+
(31 34:(8 3i) 102:2
i
+
+
248 93i 272i
+ +
+
=
(8 3i) 8 3i)
- +
64 24i + -
24i -
9;2
146 365:
+
-
73
2
=
5i
+
3
Remember: The CONJUGATE of
(x 2)(x 2)
-
bi
+
a bi
+ is a
=> x2 -
4
,>
Math 127
Complex Numbers Worksheet Name: Memo
Perform the indicated operations and then completely simplify.
1. (3 + 5i) + (4 − 3i) 9. (5 + 7i)(7 − 5i)
7 zi
=
+
35
=
-
25i 49i
+ -
35:2
=
70 24i +
2. (11 − 6i) + 7i 12 − i
10.
=11 i
+
5
= -
jior 2,4-0,2i
3. (13 − 5i) − (7 − 11i)
6 6i
L2 − 3i (i
= +
11.
2i x i
= 3 2i +
-
2
=-- 1 or -1,5-i
4. 17 − (5 − 6i)
12
=
6i
+
(((2
5 + 6i i) +
12.
L2 − i ((z i) +
10 5i+12i
+
+
6:2
-
4 -
i2
5. 4(3 − i) 4 17i
=
+
S
12-4i
I 5ior
=
= +
0,8 3,4i
+
L3 − 2i3(1 7i) +
13.
L1 − 7i (1 7i) +
2
3 21i
+
-
2i -
14:
=
6. 3i(7 − 2i) 1 -
49:2
17 192
21i-bi2
+
=
=
50
6 21i
-+6i 0,34 0,382
=
+
or
+
14. i3
-- i
7. (5 − i)(7 + 4i)
35 20i-7i
= +
-
4;2
39 13i
=
+
15. i1183
. .
=L ·L
8. (7 + 5i)(7 − 5i) = (it)2.i
=
49 -
35i 35i
+ -
25:2 =- i
=
74
16. i137
=;136.;
= i
,GRADE 11 TERM 1 2024
FS TUTORIAL 1
EXERCISE 1
Question 1
1. Determine the value of the following:
a. ! !" b. ! !#
c. ! !$ d. ! !%%&
2. Expand the following, leaving answers in terms of " ± $!
a. (3 + 2!)" b. (1 − 2!)"
c. (2 + 4!)(3 − 2!) d. !(2 − 3!) − (5 − !)"
"
e. .√2008! "%%' − !2
Question 2
Given two complex numbers 3 and 4 such that 3 = 4 − 3! and 4 = 2 + 2!, find the value of
the following:
a. 3 + 24 b. 23 − 34
c. 3" !
d.
"
e. 4∗ #
f.
!∗
g. Re(3 × 4∗ ) !"
h. Im # $
#$
Question 3
Determine the values of = and > !f 1 + 3! = (2 + !)(= + >!).
, EXERCISE 2
Question 1
Factorise the following fully in ℂ.
a. A " + 2 b. 2A " + 8
c. 3A " − 6 d. A " + 2A + 2
e. A " − 2A − 2 f. A # − 81
g. A ! + 27 h. A ! − 27
i. A! − A" + A − 1 j. A ! − 5A " + 8A − 6
