kzn department of education mathematics just in …
TRANSCRIPT
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KZN DEPARTMENT OF EDUCATION
MATHEMATICS
JUST IN TIME MATERIAL
GRADE 12
TERM 2 – 2020
TABLE OF CONTENTS
TOPIC PAGE
NO.
FUNCTIONS AND INVERSES
2 - 21
ANALYTICAL GEOMETRY
22 - 37
APPLICATION OF CALCULUS
38 – 46
This document has been compiled by the FET Mathematics Subject Advisors together with Lead
Teachers. It seeks to unpack the contents and to give more guidance to teachers.
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FUNCTIONS AND INVERSES
Weighting in final
examinations 35 ±3 marks out of 150 marks in Paper 1
Methodology
HINTS TO TEACHERS: FOCUS ON THE FOLLOWING
CHARACTERISTICS:
Definition of a function, vertical line test
Domain and the Range
General concept of the inverse of a function
Restricting domain of MANY TO ONE function to make its INVERSE
a function
Intercepts with the axes
Turning points (Minima and Maxima)
Asymptotes (horizontal and vertical) and line of symmetry
Shape of the graph
Intervals on which the function INCREASES/DECREASES
How f(x) has been transformed to generate f(-x), -f(x), f(x+a), a.f(x) and
x = f(y) where a R
Errors/Misconceptions Learners confuse domain and range
Reflection about the x- axis, y-axis and the line y=x.
FROM ATP:
DATES CURRICULUM STATEMENT
09/4
(1 day)
1. Definition of a function.
2. General concept of the inverse of a function.
3. Determine and sketch graphs of the inverse of the function defined by
𝑦 = 𝑎𝑥 + 𝑞
4. Focus on the following characteristics:
domain and range, intercepts with the axes, shape and symmetry, gradient, whether the function
increases/decreases.
14/4 – 15/4
(2 days)
5. Determine and sketch graphs of the inverse of the function defined by
𝑦 = 𝑎𝑥2
6. Determine how the domain of the function may need to be restricted (in order to obtain a one-to-one
function) to ensure that the inverse is a function.
7. Focus on the following characteristics:
domain and range, intercepts with the axes, turning points, minima, maxima, shape and symmetry,
average gradient (average rate of change), intervals on which the function increases/decreases.
16/4 – 20/4
(3 days)
8. Determine and sketch graphs of the inverse of the function defined by
𝑦 = 𝑏𝑥 for 𝑏 > 0, 𝑏 ≠ 1.
9. Focus on the following characteristics:
domain and range, intercepts with the axes, asymptotes (horizontal and vertical), shape and symmetry,
average gradient (average rate of change), intervals on which the function increases/decreases.
10. Understand the definition of a logarithm:
𝑦 = 𝑙𝑜𝑔𝑏𝑥 ⟺ 𝑥 = 𝑏𝑦, where 𝑏 > 0 and 𝑏 ≠ 1.
The graph of the function defined by 𝑦 = 𝑙𝑜𝑔𝑏𝑥 for both the cases 0 < 𝑏 < 1 and 𝑏 > 1.
21/4 – 24/4
(4 days) Further sketching and interpretation of graphs of functions and their inverses.
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FURTHER HINTS (OBJECTIVES) TO TEACHERS:
Be able to recognise the graph as being:
Linear ( qaxy ),
Quadratic(y = a(𝑥 + 𝑝)2 + q),
Hyperbolic (y = 𝑎
𝑥+𝑝 + q),
Exponential (y = a𝑏𝑥+𝑝 + q)
Understand the effects of the parameters a, p and q
Sketch the graphs, indicating all intercepts with the axes and turning points or asymptotes
(it is important to be able use and interpret functional notation)
Understand the definition of a FUNCTION
Understand the INVERSE of a function(swop x and y)
Know the inverse of xayaxyqaxy ,, 2 where a > 0
Be able to write the inverse in the form y = …
Understand the definition of a logarithm: y = log𝑏 𝑥 ↔ x = 𝑏𝑦 where b > 1
and b ≠ 1
Understand reflection about the y-axis (f(x) = f(-x)) and about the x-axis (f(x) = -f(x))
Understand reflection about the point (use mid-point formula)
Understand vertical translation (f(x) = f(x) + q) and horizontal translation
(f(x) = f(x + p))
If given a sketch of a function [f(x)], be able to determine the values of x for which:
1. f(x) < 0 OR f(x) > 0
2. f’(x) ≥ 0 OR f’(x) ≤ 0
3. 𝑓−1(x) < 0 OR 𝑓−1(x) ≥ 0
If given two functions (f(x) and g(x)) on the same set of axes, be able to determine the values of x for
which:
1. )()( xgxf 2. )()( xgxf OR 0)()( xgxf
3. )()( xgxf OR 0)()( xgxf 4. F(x).g(x)>0 OR f(x).g(x)<0
5. 𝑓(𝑥)
𝑔(𝑥)>0 6. G(x).𝑓1(x)>0
7. 𝑓−1(x) = 𝑔−1(x) 8. 𝑔′(x)>𝑓′(𝑥)
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THE PARABOLA
In Grade 11 you were introduced to the parabola in the forms shown in the table below. The table below
summarises the properties of the parabola which you learnt in Grade 11.
FO
RM
S
y = a(x + p)2 + q y = ax2+bx+c y = a(x – x1)(x – x2)
In this form – p is the x-
value of the turning
point and q is the y-
value of the turning
point. The parabola is
symmetrical about its
turning point.
In this form c is the y-
intercept. The parabola is
symmetrical about its
turning point. The x-value
of the turning point and
axis of symmetry can be
found using equation
bx = -
2a.
In this form x1 and x2 are x-
intercepts of the parabola.
Remember the x-value of the
turning point lies halfway
between the x-intercepts
(i.e 1 2x + x
x =2
). This is
due to the symmetry of the
parabola about its turning
point
SH
AP
E
(a)
a > 0
concave up (happy face)
a < 0
concave down (sad face)
As the value of a increases from 0
the graph is compressed and gets
narrower.
Example:
x
y
f(x)=½x2
As the value of a decreases from 0 the graph
is compressed and gets narrower.
Example:
x
y
f(x)=-2x2
g(x)=-x2
h(x)=-½x2
VERTICAL
SHIFT
(q or c)
The value of q or c results in a vertical shift. If the value of q or c is greater than 0 the
graph will be shifted up. If q or c is less than 0 the graph will be shifted down.
DOMAIN
and
RANGE
The domain of a parabola is:
x ϵ ℝ or x ϵ ( ; )
The range of the parabola is:
y ≥ q or y ϵ[q; ) if a > 0
y ≤ q or y ϵ( ; q] if a < 0
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THE HYPERBOLA
In Grade 11 you were introduced to the hyperbola in the form of y =a
+ qx + p
.
It was shown that a determines the shape of a hyperbola as well as quadrants which the hyperbola lies in.
The value of q resulted in a vertical shift (up or down) and the line y = q is the vertical asymptote. The value
of p resulted in a horizontal shift (left or right) and the line x = p is the vertical asymptote. The properties of
a hyperbola that you learnt in Grade 11 are summarised below:
a > 0 a < 0
SHAPE
(a)
axis of
symmetry
The hyperbola is symmetrical about the lines:
y = x + c and y = - x + c (use coordinates of the point of intersection of the
asymptotes, (-p ;q) to calculate value of c)
OR
y = x + p + q and y = - x – p + q
DOMAIN
And
RANGE
The domain of the hyperbola is:
x ϵ ℝ; x - p or x ϵ ( ; - p) (-p; )
The range of the hyperbola is:
y ϵ ℝ; y q or x ϵ ( ; q) (q; )
THE EXPONENTIAL FUNCTION
In Grade 11 you were introduced to exponential function in the form of:
y = a.bx+p + q where b > 0 and b 0
It was shown that the value of a determine the shape of the graph as well as on which side of the asymptote
the exponential function lie. Together with a, the value of b determines whether the function is increasing or
decreasing.
The value of q resulted in a vertical translation (up or down) and the line y = q is the horizontal asymptote.
The value of p resulted in a horizontal translation. The properties of the exponential function that you learnt
in Grade 11 are summarised below:
SH
AP
E (
a a
nd b
)
a > 0 and b>1 a < 0 and b > 1
The graph lies above the horizontal asymptote
and is an increasing function.
The graph lies below the horizontal
asymptote and is a decreasing function.
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A > 0 and 0 < b < 1 a < 0 and 0 < b < 1
The graph lies above the horizontal asymptote
and is a decreasing function.
The graph lies below the horizontal
asymptote and is an increasing function
DO
MA
IN A
ND
RA
NG
E
The domain of the exponential function is:
x ϵ ℝ or ( ; )
The range of the exponential function is:
y > q or y ϵ (q ; ) if a > 0
y < q or y ϵ ( ;q) if a < 0
DEFINITION OF A FUNCTION
A function is defined as a relationship between values, where each input value maps to one output value. In
other words, for an equation to be called a function, there can only be one y-value for a particular x-value.
There are two types of functions: 1. One-to-one functions
A function where there is single y-value for a particular x-value.
One-to-one function
2. Many-to-one functions
A function cannot have more than one y-value to each x-value (One-to-many). However, a
function can have more than one x-value for a particular y-value. These are known as
many-to-one functions.
Many-to-one function One-to-many (not a function)
NB: A: Use a vertical line test to check if the relation is a function or not.
B: Use horizontal line test to check if the function is one-to-one or many-to-one.
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INVERSE OF A FUNCTION
You obtain the inverse of a function by swopping the x and y values.
The domain becomes the range and the range becomes the domain.
A one-to-one function becomes a one-to-one inverse function, but a many-to-one function becomes a
one-to-many relation which is not a function.
For a many-to-one function (quadratic function/ parabola), you must restrict domain if you want to
make its inverse a function.
By swopping x and y values we get (x ; y) (y ; x). This means the graph is reflected about the line
y = x. A graph and its inverse are therefore always symmetrical about the line y = x.
You use f -1 to represent the inverse of f(x). Be careful not to mistake the -1 in f -1 for an exponent.
f -1(x) does not mean the reciprocal 1
f(x)
The inverse of an exponential function is a logarithmic function and a visa versa.
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BASELINE ASSESSMENT
QUESTION 1 KZN JUNE 2019 GR11
The hyperbola, f, is defined by 3
( ) 12
f xx
.
1.1 Write down the equations of the asymptotes of f. (2)
1.2 Write down the y-intercept of f. (1)
1.3 Determine the x-intercept of f. (2)
1.4 Sketch the graph of f , showing all intercepts with the axes and any asymptotes. (3)
1.5 Write down the domain of h, if ( ) ( 2).h x f x (2)
[10]
QUESTION 2 KZN JUNE 2019 GR11
2.1 The following are graphs defined by 2 .y ax bx c
A B C
Match each of the following statements below with associated graph(s) above.
You must only write down the letter of the corresponding graph(s).
2.1.1 042 acb (1)
2.1.2 042 acb (1)
2.1.3 042 acb (1)
2.1.4 2 equal roots (1)
2.1.5 Non real roots (1)
2.1.6 2 unequal roots (1)
2.1.7 Concave down (1)
2.1.8 Concave up (1)
2.1.9 a > 0; b < 0 & c > 0 (1)
O
x
x
y
O
x
x
y
O
x x
y
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2.2 The sketch below shows the graphs of xy k t and 2 .y ax bx c
The x-intercepts of h are (0 ; 0) and (0 ; 4). Q(1 ; –3) is a point on h.
The horizontal line through P, the turning point of h, is the asymptote of g.
The x-intercept of g lies on the axis of symmetry of h.
2.2.1 Show that 1a and 4.b
(3)
2.2.2 Determine the coordinates of P, the turning point of h.
(2)
2.2.3 Write down the range of h.
(1)
2.2.4 Write down the value of t.
(1)
2.2.5 Hence, calculate the value of .k
(2)
2.2.6 For which value(s) of r will the roots of 2 4x x r be non-real?
(2)
[11]
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QUESTIONS FROM PAST EXAMINATION PAPERS
QUESTION 1 KZN SEPT 2019
Given 12
4)(
xxf
1.1 Write down the equations of the vertical and horizontal asymptotes of f. (2)
1.2 Determine the intercepts of the graph of f with the axes. (3)
1.3 Draw the graph of f. Show all intercepts with the axes as well as the asymptotes of the
graph.
(4)
[9]
QUESTION 2 KZN SEPT 2019
In the diagram, the graphs of 65)( 2 xxxf and 1)( xxg are drawn below.
The graph of f intersects the x – axis at B and C and the y – axis at A.
The graph of g intersects the graph of f at B and S. PQR is perpendicular to the x – axis
with points P and Q on f and g respectively. M is the turning point of f.
2.1 Write down the co-ordinates of A. (1)
2.2 S is the reflection of A about the axis of symmetry of f. Calculate the coordinates of S. (2)
2.3 Calculate the coordinates of B and C. (3)
2.4 If PQ = 5 units, calculate the length of OR. (4)
2.5 Calculate the:
2.5.1 Coordinates of M. (4)
2.5.2 Maximum length of PQ between B and S. (4)
[18]
P
f
A
y
g
S
Q
x C B
R O
M
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TEACHER ACTIVITIES
QUESTION 3 KZN SEPT 2019
In the diagram, the graph of xxg 5log)( is drawn.
3.1 Write down the equation of 𝑔−1, the inverse of g, in the form y = … (2)
3.2 Write down the range of 𝑔−1. (1)
3.3 Calculate the value(s) of x for which .4)( xg (4)
[7]
QUESTION 4
Given 12
4)(
xxf
4.1 Write down the equations of the vertical and horizontal asymptotes of f. (2)
4.2 Determine the intercepts of the graph of f with the axes. (3)
4.3 Determine the coordinates of the image of the x intercept if it is reflected about the point
of intersection of the asymptotes .
(3)
[8]
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QUESTION 5 FREE STATE 2019
In the diagram, the graph of xxf alog)( is drawn. B
2;
9
25is a point on f.
5.1 Determine the value of a. (2)
5.2 Determine the value(s) of x for which .0)( xf (2)
5.3 Write down the equation of 1f , the inverse of f , in the form y = … (2)
5.4 B// is the reflection of B on the graph .5
3)(
x
xg
Write down the coordinates of B//.
(2)
5.5 Determine for which value(s) of x will 9
25)(1 xf .
(2)
[10]
QUESTION 6
Given 2
6)(
x
xxf
6.1 Express )(xf in the form of q
px
axf
)(
(2)
6.2 Determine the equations of the axis of symmetry. (4)
6.3 Draw the graph of f. Show all intercepts with the axes as well as the asymptotes of the
graph.
(4)
[10]
B
y
x O
f
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QUESTION 7 FREE STATE 2019
In the diagram below, the graph of 34
2)(
xxg is drawn. The graph f passes through A, the point of
intersection of the asymptotes of g, and cuts the x-axis and the y-axis at L and R respectively. K is the
y-intercept of g.
7.1 Determine the equation of f in the form cmxy . (3)
7.2 Write down the equation of the asymptotes of 2xg + 1. (2)
7.3 Calculate the length of KR. (3)
7.4 The graph of h, where h is the reflection of f in the line y = –7, passes through the point
S(– 4 ; p). Calculate the area of ARS. (4)
[12]
QUESTION 8
Given qpx
axf
)(
a > 0
Range; yϵR; y ≠ -2
Domain; xϵR; x ≠ -3
f(0) < 0
Use the given information above to sketch the graph of f [4]
y
A
L O
f
g
𝑥
R
K
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QUESTION 9 FREE STATE 2019
In the diagram below, the graphs of 16)( 2 bxaxxf and 2412 xxg are drawn. The graph of g
is a tangent to the graph of f at B. A and B are the x-intercepts of f and C, the turning point.
9.1 Calculate the coordinates of B. (2)
9.2 Determine the values of a and b. (6)
9.3 If it is given that ,1642)( 2 xxxf determine:
9.3.1 The range of f (5)
9.3.2 The value(s) of x for which f / (x). g(x) > 0 (2)
[15]
QUESTION 10
Given qpx
axf
)(
a < 0
p < 0
q > 0
Use the given information above to sketch the graph of f. [4]
y
A B
C
f
g
x O
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QUESTION 11 NORTH WEST 2019
The graphs of 21
( ) 2 9 and ( )2
ag x x h x q
x p
are sketched below.
The axis of symmetry of graphg is the vertical asymptote of graph h. The line f is an axis
of symmetry of graph h. B is the y-intercept of h, g and f .
y h
g
x
B
f
11.1 Write down the coordinates of C, the turning point of g.
(2)
11.2 Determine the coordinates of B.
(2)
11.3 Write down the equation of f.
(2)
11.4 Determine the equation of h.
(5)
11.5 Write down the equations of the vertical and horizontal asymptotes of
( ) 3 ( ) 2k x h x .
(2)
11.6 Determine the x-intercept of h.
(3)
11.7 For which values of x will:
4.7.1
/ ( )0
( )
g x
h x
(3)
4.7.2 1 ( 1) 2f x
(4)
11.8 Calculate the value(s) of k for which g(x) = f (x) +k has two unequal positive roots.
(6)
[29]
h
C
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QUESTION 12 NORTH WEST 2019
QUESTION 13
Given .42.2)( 1 xxg
13.1 Determine the x and y intercepts of g (2)
13.2 Determine the domain and range of g (2)
13.3 Draw the graph of g Show all intercepts with the axes as well as the asymptotes of the
graph.
(4)
[8]
12.1 Consider the function 5
( )6
x
f x
12.1.1
Write down the equation of h, the reflection of f in the y-axis.
(1)
12.1.2 Write down the equation of 1 ( )f x
in the form y = ...
(2)
12.1.3 For which value(s) of x will1 ( ) 0f x ?
(2)
12.2 The function defined as 2( )f x ax bx c has the following properties:
/ ( 2,5) 0f
(1) 0f
2 4 0b ac
( 2,5) 6f
Draw a neat sketch graph of f. Clearly show all x-intercepts and turning point.
(4)
[9]
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QUESTION 14 DBE NOV 2019
Below are the graphs of 3)( 2 bxxxf and px
axg
)( .
f has a turning point at C and passes through the x-axis at (1 ; 0).
D is the y-intercept of both f and g. The graphs f and g also intersect each other at E
and J.
The vertical asymptote of g passes through the x-intercept of f.
14.1 Write down the value of p. (1)
14.2 Show that a = 3 and b = 2. (3)
14.3 Calculate the coordinates of C. (4)
14.4 Write down the range of f. (2)
14.5 Determine the equation of the line through C that makes an angle of 45 with the
positive x-axis. Write your answer in the form y = …
(3)
14.6 Is the straight line, determined in QUESTION 4.5, a tangent to f? Explain your answer.
(2)
14.7 The function h(x) = f (m – x) + q has only one x-intercept at x = 0. Determine the
values of m and q.
(4)
[19]
E
J f
g
y
x O 1
D
C .
.
. g
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QUESTION 15 DBE NOV 2019
Sketched below is the graph of xkxf )( ; k > 0. The point (4 ; 16) lies on f.
15.1 Determine the value of k. (2)
15.2 The graph of g is obtained by reflecting graph f about the line y = x. Determine the
equation of g in the form y = …
(2)
15.3 Sketch the graph g. Indicate on your graph the coordinates of two points on g. (4)
15.4 Use your graph to determine the value(s) of x for which:
15.4.1 f (x) × g(x) > 0 (2)
15.4.2 g(x) 1 (2)
15.5 If h(x) = f (–x), calculate the value of x for which 4
15h(x)f(x)
(4)
[16]
x
y
f
O
(4 ; 16)
1
.
.
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QUESTION 16 DBE MAY/JUNE 2019
16.1 Given: 32
1
xxf
.
16.1.1 Determine the equations of the asymptotes of f. (2)
16.1.2 Write down the y-intercept of f. (1)
16.1.3 Calculate the x-intercept of f. (2)
16.1.4 Sketch the graph of f . Clearly label ALL intercepts with the axes and any
asymptotes.
(3)
16.2 Sketched below are the graphs of cbxaxxk 2 and 42 xxh .
Graph k has a turning point at (–1 ; 18). S is the x-intercept of h and k.
Graphs h and k also intersect at T.
16.2.1 Calculate the coordinates of S. (2)
16.2.2 Determine the equation of k in the form qpxay 2
(3)
16.2.3 If 1642 2 xxxk , determine the coordinates of T. (5)
16.2.4 Determine the value(s) of x for which )()( xhxk . (2)
16.2.5 It is further given that k is the graph of xg /.
(a) For which values of x will the graph of g be concave up? (2)
(b) Sketch the graph of g, showing clearly the x-values of the turning
points and the point of inflection.
(3)
[25]
(–1 ; 18)
T k
h
y
x O S
.
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QUESTION 17 KZN JUNE 2019
2
1;0A;)(Given c
bx
axf is the y – intercept of the graph. The asymptotes to the graph intersects
the x – axis at 1 and the y – axis at 2 respectively.
17.1 Write down the equations of the vertical and horizontal asymptotes of f. (2)
17.2 Calculate the value of a. (3)
17.3 Determine the coordinates of A/, the image of A, if it is reflected about ).2;1( (4)
17.4 Determine the equation of g if ).3()( xfxg (2)
[11]
x
y
A
1
2
f
O
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QUESTION 18 DBE NOV 2018
In the diagram below, the graph of 2)( axxf is drawn in the interval .x 0
The graph of 1f is also drawn. P(–6 ; –12) is a point on f and R is a point on .1f
18.1 Is 1f a function? Motivate your answer. (2)
18.2 If R is the reflection of P in the line xy , write down the coordinates of R. (1)
18.3 Calculate the value of .a (2)
18.4 Write down the equation of 1f in the form ...y (3)
[8]
x
y
f
R .
P(–6 ; –12)
O
.
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ANALYTICAL GEOMETRY
Weighting 37 – 43 marks (±40)
Sub-topics/Clarification 1. EQUATION OF A CIRCLE: (𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 = 𝑟2
Use the same formula no matter the centre is at the origin
A point lie inside, outside or on the circle
Completing the square to find centre
2. INTERSECTION OF CIRCLES
Intersect at one point: the distance between the centres is equal to
the sum of their radii
Intersection of two circles: the distance between the centres is
less than the sum of their radii
Never intersect. The distance between the centres is greater than
the sum of their radii
3. STRAIGHT LINES AND TANGENTS TO CIRCLES
Radius/diameter is perpendicular to the tangent.
In a tangent the product of the gradient and radius/diameter is
negative one.
Point or points of intersection of straight lines and the circle.
The two tangents from the common point outside the circle are
equal.
Related
concepts/terms/vocabulary Gradient of a line: 𝑚 =
𝑦2−𝑦1
𝑥2−𝑥1
Inclination : tanα = m
Midpoint: When given two points 𝑚 = 𝑥1
𝑤ℎ𝑒𝑛 𝑔𝑖𝑣𝑒𝑛 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑎𝑛𝑑 𝑎 𝑝𝑜𝑖𝑛𝑡 𝑢𝑠𝑒 ∶ 𝑥 =𝑥1 + 𝑥2
2; 𝑦
=𝑦1 + 𝑦2
2
Equation of a straight line : 𝑦 = 𝑚𝑥 + 𝑐 𝑜𝑟 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
Parallel lines: equal gradients
Perpendicular lines: 𝑚1 × 𝑚2 = −1
Collinear points gradients between the points are equal.
Equation of a tangent: 𝑦 = 𝑚𝑥 + 𝑐 𝑜𝑟 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)
Prior-knowledge/
Background knowledge Revision of grades 10 and 11:
Grade 10 and 11 Analytical Geometry.
Properties and definitions of geometric shapes.
Different types of triangles.
4. Definition: circle, Radius, Tangent, Centre
5. Distance formula
6. Interpretation of circles (centre theorem; rad-tan theorem; tan-chord
theorem)
7. Finding equations of the straight lines and tangents/normal and tangent
8. Intersecting circles
Resources Chalkboard Instruments
Mathematical instruments
Scientific Calculator
Activities
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23
Exercises on different maths textbooks can be used
Maths Handbook and Study Guide by Kevin Smith (page 221- 224).
See the activities below.
Give leaners past examination papers
Methodology Revise grades 10 and 11 analytical Geometry. Then give learners the
revision exercises below
Use the distance formula to derive the standard form of equation of a
circle:
(x – a)2 + (y – b)2 = r2 , where (a ; b) is the centre point of
circle and r is the length of the radius. Then do a couple of examples from any CAPS approved textbooks
Integration with Trigonometry and Euclidean Geometry
Assessment
o The practice exercises below can be taken from any relevant
textbooks.
Classwork
Homework
Assignment
Test
Errors/misconceptions Incorrect use of gradient formula
FROM ATP:
HINTS:
Should be able to determine the:
DATES CURRICULUM STATEMENT
31/02 – 03/04
(4 days)
1. The equation 222 )()( rbyax defines a circle with radius r and
centre ).;( ba
NOTE: Include circles that touch internally and externally
06/04 – 08/04
(3 days) 2. Determination of the equation of the tangent to given circle.
Distance formula: 2
12
2
12 yyxxd
Gradient formula: 12
12
xx
yym
Relationship between the Parallel lines: 21 mm
Relationship between the Perpendicular lines 121 mm
Midpoint of a line segment:
2;
2
1212 yyxxM
Equation of a straight line: cmxy OR 11 xxmyy
Angle of inclination: tanm
Completing a square Writing cbxax 2 in the form of qpxa 2)(
The relationship between tangent and a radius Tangent is perpendicular to the radius at point of contact.
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N(4; 1)
K(−1; 4)
L(3; 6)
𝜃 O
𝑥
𝑦
R
P
M
BASELINE ASSESSMENT
QUESTION 1
In the diagram, K (–1 ; 4) ; L(3 ; 6); M and N(4 ; 1) are vertices of a parallelogram. R is the
midpoint of LN. P is the x -intercept of the line MN produced.
1.1 Calculate:
1.1.1 gradient of KL . (2)
1.1.2 coordinates of R. (3)
1.1.3 coordinates of M. (4)
1.2 Determine the equation of NM in the form y = mx + c. (3)
1.3 Calculate the:
1.3.1 coordinates of P. (2)
1.3.2 size of Ɵ , the inclination of PM. (4)
[20]
Properties of polygons Quadrilaterals and triangles
Centre and the radius of a circle Centre: 0;0 OR Centre: );ba and radiusr
The equation of a circle:
with centre at the origin 0;0 : 222 ryx
AND
with centre ba; : 222rbyax
The equation of the tangent to a circle: Using 1 rt mm and a given point.
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THE CIRCLE
Equation of a circle
222 )()( rbyax
Where: a is the x -coordinate of the circle centre
b is the y -coordinate of the circle centre
r is the length of the circle’s radius (by distance formula)
If the centre is at the origin (0;0), the equation reduces to: 222 ryx
EXAMPLES
1. Write down the equation of the circle with centre (-2;-1) and radius 4
2. Determine the equation of the circle with centre M(-3;1) and A(2;-2) a point on the circle.
3. Given the equation of the circle with centre O is 02024 22 yyxx .
Determine the coordinates of O and the radius of the circle
SOLUTIONS
1. 222 )()( rbyax
Substitute the centre (-2;-1) and radius 4
222 4))1(())2(( yx
16)1()2( 22 yx 222 )()( rbyax
2. Find the radius of the circle 222 ))2(1()23( r
9252 r
342 r
Equation of the circle
34)1())3(( 22 yx
34)1()3( 22 yx
3. 2024 22 yyxx
1420)12()44( 222 yyxx
25)1()2( 22 yx
The circle has a centre at O (2 ; -1) and a radius of 5 units
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ACTIVITY
Write down the equation of a circle with
a) centre at (0;0) and radius 3
b) centre at );( qp and radius a
c) centre at (3 ; -2) and passes through (3 ; 4)
Determine the centre and the radius of the circles defined by:
a) 082422 yxyx
b) 03622 xyx
c) 024222 yxyx
INTERSECTION OF CIRCLES
Two circles can intersect one another at two points, one point or not at all
1. Circles intersecting at two points
2. Circles intersecting at one point(touching)
21 rrd
3. Circles that do not intersect
21 rrd
r
RA
B
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4. Circles that touch internally
rRd
EXAMPLE
Given two circles 422 yx and 9)3()2( 22 yx , Determine:
a) The circle centres and lengths of the radii of each circle
b) Whether or not the circles intersect each other
SOLUTION
a) Centres (0;0) and (2;-3)
21 r and 32 r
220302 d
13
21 rrd the two circles do intersect (at two points)
ACTIVITY
Given two circles with equations 0161012 22 yyxx and 02422 yxyx ,
prove that circles touch each other.
r
R
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TANGENTS TO CIRCLES
A radius is any line that connects the centre
of a circle and any point on the
circumference.
A diameter is any line that joins two points
on the circumference and passes through the
centre. (Diameter=2radii)
A tangent is any line that touches a curve (in
this case a circle) externally at only one
point.
The radius is perpendicular to the tangent at
the point of contact. (tan rad)
1tan gentradius mm
THE EQUATION OF A TANGENT
EXAMPLES
1. Determine the equation of a tangent which touches the circle 17)3()2( 22 yx at the
point (1;1)
2. Write down the equations of the tangents to the circle 36)3( 22 yx which are parallel
to the x -axis
SOLUTIONS
1. 4
12
13
radiusm
4
1
4
11tan
radius
gentm
m
Equation of a tangent: )( 11 xxmyy
)1(4
11 xy
4
3
4
1 xy
2.
Centre(0 ; 3)
Radius = 6 units
9y and 3y
(0 ;3)
y
x
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ACTIVITY
1.Determine the equation of the tangent which touches the circle with centre at the origin, at
the point (4 ; 2)
2.Determine the length of a tangent from the point R(-2 ; 2) to the point of contact P on the
circle 0128 22 yxx
DBE/2017
In the diagram, N is the centre of the circle. M(-3 ; -2) and P(1 ; 4) are points on the circle.
MNP is the diameter of the circle. Tangents drawn to circle N from point R, outside the
circle meet the circle at S and M respectively.
1. Determine the coordinates of N.
N is the Midpoint of the diameter MP
N
2
24;
2
)3(1
N )1;1(
2. Determine the equation of the circle in the form 222 )()( rbyax
2212)1(3 radius
= 13 132 r
Centre = (-1 ; 1) equation 13)1()1( 22 yx
3. Determine the equation of the tangent RM in the form cmxy
2
3
)3(1
)2(1
NMm
3
2RMm
Equation of RM ))3((3
2)2(
xy
43
2 xy
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4. If it is given that the line joining the S to M is perpendicular to the x -axis, determine the
coordinates of S.
RSNM is a kite (adjacent sides are equal)/ tangents from same point
RN bisects SM
Vertical distance from -2 to 1 is 3 to get to S from 1 the distance will also be 3 (diagonals of
a kite)
S is (-3 ; 4)
5. Determine the coordinates of R, the common external point from which both tangents to
the circle are drawn
2
3
)3(1
41
NSm
3
2RSm
))3((3
24 xy
63
2 xy
63
24
3
2
xx
304 x
2
15x
2
7y S
2
7;
2
15
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N(4; 1)
K(−1; 4)
L(3; 6)
𝜃 O
𝑥
𝑦
R
P
M
PRACTICE EXERCISES
QUESTION 1 KZN SEPT 2019
In the diagram, K (–1 ; 4) ; L(3 ; 6); M and N(4 ; 1) are vertices of a parallelogram. R is the
midpoint of LN. P is the x -intercept of the line MN produced.
1.1 Calculate:
1.1.1 distance of KL . (2)
1.1.2 coordinates of R. (3)
1.1.3 coordinates of M. (4)
1.2 Determine the equation of NM in the form y = mx + c. (3)
1.3 Calculate the:
1.3.1 coordinates of P. (2)
1.3.2 size of Ɵ , the inclination of PM. (4)
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QUESTION 2 FREE STATE 2019
In the diagram below, ),4;1(A )4;3( B , );2( C k and );( D yx are the vertices of a
rectangle. AB and DC cuts the x axis at G and H respectively. GD is drawn. CHG .
2.1 Calculate the gradient of BG. (2)
2.2 Determine the equation of AB in the form .cmxy (2)
2.3 Calculate the:
2.3.1 Value of k (𝑦-coordinate of C). (4)
2.3.2 Coordinates of D. (3)
2.3.3 Size of . (3)
2.3.4 Area of DHG. (7)
[21]
A (1; 4)
D (x; y)
O G H
B (–3; –4)
C (2; k)
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QUESTION 3 FREE STATE 2019
In the diagram below, the circle centred at 1;3E passes through point 5;5P .
3.1 Determine the equation of:
3.1.1 The circle in the form 0CBA22 yxyx . (4)
3.1.2 The tangent to the circle at 5-;5P in the form cxy m . (5)
3.2 A smaller circle is drawn inside the circle. Line EP is a diameter of the small circle.
Determine the:
3.2.1 Coordinates of the centre of the smaller circle. (3)
3.2.2 Length of the radius. (3)
3.3 Hence, or otherwise, determine whether point 3;9C lies inside or outside the
circle centre at E. (3)
[18]
●
O
●
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QUESTION 4 SCE: DBE/May-June 2018
In the diagram, ABCD is a quadrilateral having vertices A(7; 1), B(-2; 9), C(-3; -4) and D(8;-11). M
is the midpoint of BD.
4.1 Calculate the gradient of AC. (2)
4.2 Determine :
4.2.1 The equation of AC in the form cmxy (2)
4.2.2 Whether M lies on AC. (4)
4.3 Prove that BDAC. (3)
4.4 Calculate:
4.4.1 , the inclination of BD (2)
4.4.2 The size of angle of DBC ˆ (3)
4.4.3 The length of AC (2)
4.4.4 The area of ABCD (5)
[23]
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QUESTION 5 SCE: DBE/ May-June 2018
In the diagram, a circle having centre at the origin passes through P (4; -6). PO is the diameter of a
smaller circle having centre at M. The diameter RS of the larger circle is a tangent to the smaller
circle at O.
5.1 Calculate the coordinate of M. (2)
5.2 Determine the equation of:
5.2.1 The large circle (2)
5.2.2 The small circle in the form 022 EDyCxyx (3)
5.2.3 The equation of RS in the form cmxy (3)
5.3 Determine the length of cord NR, where N is the reflection of R in the y-axis. (4)
5.4 The circle with centre at M is reflected about the x-axis to form another circle centred at K.
Calculate the length of the common chord of these two circles
(3)
[17]
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QUESTION 6 NSC: DEB/Feb.-Mar. 2018
In the diagram, P, Q (-7; -2), R and S (3; 6) are vertices of quadrilateral. R is a point on the x-axis.
QR is produced to N such that QR = 2RN. SN is drawn. 057,71ˆ OTP and NRS ˆ .
Determine:
6.1 The equation of SR (1)
6.2 The gradient of QP to the nearest integer (2)
6.3 The equation of QP in the form cmxy (2)
6.4 The length of QR. Leave your answer in surd form (2)
6.5 090tan (3)
6.6 The area of RSN , without using a calculator (6)
[16]
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QUESTION 7 NSC: DEB/Feb.-Mar. 2018
In the diagram, PKT is a common tangent to both circles at K (a; b). the centres of both circles lie
on the line xy2
1 . The equation of the circle centred at O is 18022 yx . The radius of the
circle is three times that of the circle centred at M.
7.1 Write down the length of OK in surd form (1)
7.2 Show that K is the point (-12; -6) (4)
7.3 Determine:
7.3.1 The equation of the common tangent, PKT, in the form cmxy (3)
7.3.2 The coordinate of M (6)
7.3.3 The equation of the smaller circle in the form 222rbyax (2)
7.4 For which value(s) of r will another circle, with equation 222 ryx , intersect the circle
centred at M at two distinct points?
(3)
7.5 Another circle 0240163222 yxyx , is drawn. Prove by calculation that this
circle does not cut the circle with centre M(-16;-8).
(5)
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APPLICATION OF CALCULUS
Sub-topics/Clarification Optimization and rate of change
Related
concepts/terms/vocabulary
Maxima and minima.
Rate of change /stationary points.
Prior-knowledge/
Background knowledge
Measurements (perimeter, area, Total Surface Area, volumes etc).
Rules of differentiation.
Resources Worksheet, textbooks, study guide, tin, cans, chalk box.
Methodology Revise with learners prior knowledge done in grade 10 and 11
(measurement)
Remind them formulae (Perimeter, Area, TSA, and Volume)
Look at the terminology that will be used in the sub-topic
Drill learners on the concept of maxima and minima integrating with
graphs (stationery points)
Drill learners on the concept of rate of change and other terms they
will be using in the sub-topic for example stationery point is a
turning point; gradient of a tangent is a derivative
Practical activity can be performed by learners in terms of
optimization, e.g.: using can, boxes and their nets etc.
Theory on the calculus of motion
S(t) represents an equation of motion (height, distance,
displacement) at time t.
/s t represents velocity or speed at time t.
//s t represents acceleration at time t.
Assessment Class activities
Short test
Home work
Errors/Misconceptions/
Problem Areas Learners use equation of Area instead of Total Surface Area
FROM ATP:
DATES CURRICULUM STATEMENT
28/4
(1 day)
1. An intuitive understanding of the limit concept.
2. Use limits to define the derivative of a function 𝑓 at any 𝑥:
𝑓′(𝑥) = limℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ.
Generalise to find the derivative of 𝑓 at any point 𝑥 in the domain of 𝑓, i.e., define the
derivative function 𝑓′(𝑥) of the function𝑓(𝑥).
Understand intuitively that 𝑓′(𝑎) is the gradient of the tangent to the graph of 𝑓 at the point
with 𝑥-co-ordinate 𝑎.
29/4 – 30/4
(2 days)
3. Using the definition (first principles), find the derivative, 𝑓′(𝑥), for
a. 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐;
b. 𝑓(𝑥) = 𝑎𝑥3;
c. 𝑓(𝑥) =𝑎
𝑥; and
d. 𝑓(𝑥) = 𝑐 (𝑎, 𝑏 and 𝑐 are constants).
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HINTS TO TEACHERS: FOCUS ON THE FOLLOWING CHARACTERISTICS:
Definition of terms, e.g.
Displacement
120 10s t t
Velocity
/ 10s t v t
Acceleration
// / 0s t v t a t
Rate
Speed
Maximum
Minimum
Do a thorough revision of the rules of differentiation.
Emphasise sketching and interpretation of cubic functions in real life contexts.
Develop a strategic approach to problems on optimisation, e.g.
Highlight what the question asks and make an expression for what must be optimised.
Make a rough sketch of the situation if possible.
If there two variables in the expression, use simultaneous equations to eliminate one of them.
Write down the derivative of the expression in terms of one of the variables.
Set the derivative equal to zero and solve for one the variables.
The value/s obtained can then be used to determine the maxima or minima.
04/5 – 07/5
(4 days)
4. Use the formula 𝑑
𝑑𝑥(𝑎𝑥𝑛) = 𝑎𝑛𝑥𝑛−1, for any real number 𝑛, together with the rules
a. 𝑑
𝑑𝑥[𝑓(𝑥) ± 𝑔(𝑥)] =
𝑑
𝑑𝑥[𝑓(𝑥)] ±
𝑑
𝑑𝑥[𝑔(𝑥)]; and
b. 𝑑
𝑑𝑥[𝑘𝑓(𝑥)] = 𝑘
𝑑
𝑑𝑥[𝑓(𝑥)] (𝑘 a constant).
08/5
(1 day) 5. Find equations of tangents to graphs of functions.
11/5
(1 day) 6. Apply the Remainder and Factor Theorems to polynomials of degree at most 3.
7. Factorise third degree polynomials.
12/5 – 19/5
(6 days)
8. Introduce the second derivative 𝑓′′(𝑥) =𝑑
𝑑𝑥[𝑓′(𝑥)] of 𝑓(𝑥), and how it determines the
concavity of a function.
9. Sketch graphs of polynomial functions using differentiation to determine the coordinates of
stationary points, and points of inflection (where concavity changes). Also determine the
𝑥-intercepts of the graph, using the factor theorem and other techniques.
20/5 – 26/5
(5 days)
10. Solve practical problems concerning optimisation and rate of change, including calculus of
motion.
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LEARNERS SHOULD:
Be able to use rules of differentiation to determine the first and second derivative.
Be able to sketch graphs of cubic functions and their related derivatives.
Be able to interpret sketched graphs of cubic functions in context.
Be able to relate the point of inflection to the concavity of a graph.
Understand the concept of stationary points of a graph and how it relates to maxima and minima.
Understand the relation between the rate of change and concept of optimisation.
Be able to determine the maximum or minimum values of given expressions.
SUMMARY OF CONCEPTS
CONCEPT CONTEXT
The derivative of a function, 𝑓/(𝑥). An expression relating a change in one variable
with respect to another variable.
When a derivative is equal to zero, 𝑓/(𝑥) = 0. It is at an instant where the change does not happen.
Mensuration formulae.
Only formulae for cones, spheres and pyramids will
be given during exams, but learners should be able
to identity them from an array of formulae.
Calculus of motion.
𝑠(𝑡) represents an equation of motion at time, t.
𝑠/(𝑡) represents speed or velocity at time, t.
𝑠//(𝑡) represents acceleration at time, t.
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PRACTICE EXERCISES
QUESTION 1 KZN SEPT 2019
The sketch below represents the curve of 𝑓(𝑥) = 𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 8. The solutions of the equation
𝑓(𝑥) = 0 are −2 ; 1 and 4.
1.1 Calculate the values of b, c and d. (4)
1.2 Calculate the x-coordinate of B, the maximum turning point of f. (4)
1.3 Determine an equation for the tangent to the graph of f at 𝑥 = −1. (4)
1.4 Sketch the graph of 𝑓//(x). Clearly indicate the x- and y-intercepts on your sketch. (3)
1.5 For which value/s of x is f(x) concave upwards? (2)
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QUESTION 2 DBE NOV 2012
The sketch below shows the graph of 𝑝(𝑥) where 𝑝(𝑥) = 𝑥3 + 𝑏𝑥2 + 24𝑥 + 𝑐. 𝐴(2 ; 0) is an
x-intercept of both 𝑝(𝑥) and 𝑝/(𝑥). C is the other x-intercept of 𝑝/(𝑥).
2.1 Show that the numerical value of 𝑏 = −9. Clearly show all your calculations. (3)
2.2 Calculate the coordinates of C. (3)
2.3 For what value(s) of x will 𝑝(𝑥) be increasing? (3)
2.4 Calculate the value(s) of x for which the graph of p is concave up. (2)
2.5 Sketch the graph of 𝑝(𝑥). Clearly indicate the intercepts with the axes, turning points and
the point of inflection.
(4)
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QUESTION 3 DBE NOV 2011
The function by 𝑓(𝑥) = 2𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is sketched below.
The turning points of the graph of f are 𝑇(2 ; −9) and 𝑆(5 ; 18).
3.1 Show that 𝑎 = 21, 𝑏 = −60 and 𝑐 = 43. (7)
3.2 Determine an equation of the tangent to the graph of f at 𝑥 = 1 (5)
3.3 Determine the x-value at which the graph of f has a point of inflection. (2)
[14]
QUESTION 4 NW SEPT 2018
A shopkeeper finds that a number of people visiting his shop at any moment during the ten (10)
hours that the shop is open, is represented by N(𝑡) = 𝑡3 − 12𝑡2 + 36𝑡 + 8, where 𝑁(𝑡) is the
number of people in the shop, t hours after the shop opened.
4.1 How many people are in the shop when it opens? (1)
4.2 At what stage is the number of people in the shop increasing? (5)
4.3 At which stage is it the best time for the shopkeeper to take a break and leave his assistant
alone in the shop?
(1)
[7]
QUESTION 5 DBE NOV 2012
A particle moves along a straight line. The distance, s (in metres) of the particle from a fixed point
on the line at time t seconds (𝑡 ≥ 0) is given by 𝑠(𝑡) = 2𝑡2 − 18𝑡 + 45.
5.1 Calculate the particle’s initial velocity. (Velocity is the rate of change of distance.) (3)
5.2 Determine the rate at which the velocity of the particle is changing at t seconds. (1)
5.3 After how many seconds will the particle be closest to the fixed point? (2)
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𝐶 =1200
𝑥+ 600𝑥2
QUESTION 6 KZN SEPT 2018
The depth of water (in metres) left in the dam, t hours after the sluice gate was opened to allow the
flow of water to drain from the dam is given by the equation 𝐷(𝑡) = 28 −1
9𝑡2 −
1
27𝑡3.
6.1 Calculate the average rate of change in the depth of the water after the first two hours. (4)
6.2 Determine the rate at which the level of the water is decreasing after 16 hours. (4)
[8]
QUESTION 7 DBE FEB/MARCH 2013
A rectangular box is constructed in such a way that the length (l) of the base is three times as
Long as it’s width. The material used to construct the top and the bottom of the box costs R100
per square metre. The material used to construct the sides of the box costs R50 per square metre.
The box must have a volume equal to 9 𝑚3. Let the width of the box be x metres.
7.1 Determine an expression for the height (h) of the box in terms of x. (3)
7.2
Show that the cost to construct the box can be expressed as
(3)
7.3 Calculate the width of the box (that is the value of x) if the cost is to be a minimum. (4)
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r
h
QUESTION 8 LIMPOPO SEPT 2019
PQRS is a rectangle with P on the curve ℎ(𝑥) = 𝑥2 and with the x-axis and the line 𝑥 = 6 as
boundaries.
8.1 Show that the area of rectangle 𝑃𝑄𝑅𝑆 can be expressed as 𝐴 = 6𝑥2 − 𝑥3 (3)
8.2 Determine the largest possible area of rectangle 𝑃𝑄𝑅𝑆. Show all your calculations. (4)
[7]
QUESTION 9 DBE FEB/MARCH 2016
A soft drink can has a volume of 340𝑐𝑚3,
a height of h cm and a radius of r cm.
9.1 Express h in terms of r. (2)
9.2 Show that the surface area of the can is given by 𝐴(𝑟) = 2𝜋𝑟2 + 680𝑟−1. (2)
9.3 Determine the radius of the can that will ensure that the surface area is a minimum. (4)
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QUESTION 10 MIND ACTION SERIES
The diagram represents a right circular cone of height, h cm, and a base radius of r cm. If the sum of
the height and base radius is 12 cm.
10.1 Express r in terms of h. (2)
10.2 Express the volume of the cone in terms of h. (4)
10.3 Determine the maximum volume of the cone. (4)
[10]
11 During an experiment, the volume of water in a dam is found to
be 260 8 3V t t t , where V t represents the volume of
water in
the dam, in kilolitres, t days after the experiment has begun.
There is a suspected leak in the dam wall.
a) Calculate the rate of change of volume of water after 3 days.
b) When did the volume of water start decreasing?
c) When will the dam be empty?
(3)
(3)
(4)