MATHS EXAMINATION PRESENTATION GRADE 11
GRADE 11
PAPER 1
ALGEBRA
EQUATIONS:
1.
Solve for 𝑥:
(a)
2.
−3(𝑥 + 2)(𝑥 − 5) = 0
(b)
2(𝑥 + 1)3 − 16 = 0
Solve for 𝑥 in terms of 𝑘. Leave your answer in its simplest form.
(a)
(𝑥 − 𝑘 − 3)2 = 25
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(b)
𝑥 2 − 2𝑘𝑥 + 𝑘 = 0
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MATHS EXAMINATION PRESENTATION GRADE 11
3.
EQUATIONS WITH SQUARE ROOTS:
Solve for 𝑥:
(a)
𝑥 = 2 − √2𝑥 − 5
(c)
4√𝑥 − 2 − 1 = 𝑥
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(b)
(√𝑥 − 1 − 3)(2 + √𝑥 + 1) = 0
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MATHS EXAMINATION PRESENTATION GRADE 11
4.
COMPLETING THE SQUARE:
(a)
Solve for 𝑥 by completing the square.
Leave your answer in terms of 𝑚.
2𝑥 2 − 2𝑥 = 𝑚
5.
(b)
Solve for 𝑥 in terms of 𝑝 by completing the square.
Leave your answer in terms of 𝑝
2𝑥 2 − 6𝑝𝑥 = 20𝑝2
NATURE OF ROOTS:
(a)
Find the value of 𝑝 if 2𝑥 − 8 + 𝑥 2 = 𝑝 has equal roots.
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MATHS EXAMINATION PRESENTATION GRADE 11
(b)
The sketch shows the graph of: 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
For each of the following, answer TRUE or FALSE:
(c)
(i)
𝑓(0) = 0
(ii)
𝑏2 − 4𝑎𝑐 > 0
(iii)
𝑎<0
(iv)
𝑏<0
If the roots of the equation 𝑘𝑥 2 + 2𝑘𝑥 + 3 = 0 are real and unequal, find the value of 𝑘.
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MATHS EXAMINATION PRESENTATION GRADE 11
3 ±√−𝑘 − 4
(d)
The roots of a quadratic equation are 𝑥 =
(e)
Prove that the roots of (𝑡 + 1)𝑥 2 + (2𝑡 + 3)𝑥 + 2 = 0
are real and rational where 𝑡 ∈ ℚ
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2
.
For which values of k are the roots non-real?
(f)
Show that the roots of 𝑥 2 + 2𝑏𝑥 + 3𝑏2 = 0 are non-real,
if 𝑏 ∈ ℝ and 𝑏 ≠ 0
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MATHS EXAMINATION PRESENTATION GRADE 11
6.
INEQUALITIES:
Solve for 𝑥:
(a)
𝑥 2 + 8𝑥 > 15
(b)
(d)
−(𝑥 + 2)2 < 1
(e)
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4𝑥 2 < 49
−4
5 − 2𝑥
> 0
(c)
𝑥 2 ≥ 4𝑥
(f)
𝑥2 + 3
𝑥−1
<0
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MATHS EXAMINATION PRESENTATION GRADE 11
7.
ROOT QUESTIONS:
(a)
If one root of the equation ax 2 − 9 x + 10 = 0 is 2,
find a and the other root.
(c)
The roots of a quadratic equation are given as 5 − √2 and 5 + √2 .
(b)
If 2 and −4 are the roots of the equation 𝑥 2 + 𝑏𝑥 + 𝑐 = 0 ,
determine the values of 𝑏 and 𝑐.
Determine the equation in the form 𝑥 2 + 𝑏𝑥 + 𝑐 = 0.
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MATHS EXAMINATION PRESENTATION GRADE 11
8.
SIMULTANEOUS EQUATIONS:
Solve for 𝑥 and 𝑦 simultaneously
(a)
2𝑥 + 𝑦 = 5 and 4𝑥 2 − 𝑦 2 = 0
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(b)
𝑦 +𝑥 = 1
and
𝑥 2 − 𝑥𝑦 − 𝑥 + 𝑦 = 0
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MATHS EXAMINATION PRESENTATION GRADE 11
9.
EXPONENTIAL EQUATIONS:
TYPE 1:
𝑥 is in the exponent position
Solve for 𝑥:
(a)
3𝑥 − 1 = 0
(b)
92𝑥 + 3 − 27𝑥 + 5 = 0
(c)
2𝑥 − 5 = − 2𝑥 − 2
(d)
9𝑥 − 2. 3𝑥 + 2 + 81 = 0
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MATHS EXAMINATION PRESENTATION GRADE 11
(e)
5𝑥 + 5 = 2 . 52− 𝑥
NOTE:
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5−𝑥 =
1
5𝑥
(f)
1
2𝑥
+ 5. 2− x + 21 − 𝑥 = 8
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MATHS EXAMINATION PRESENTATION GRADE 11
TYPE 2:
𝑥 is in the base position
1
(a)
1
𝑥4 = 2
(b)
1
𝑥2 = 3
1
(c)
𝑥 2 = −2
(f)
3𝑥 − 3 = 27
1
NOTE: 𝑥 2 = − 2 is NOT A SOLUTION because 𝑥 2 ≥ 0 by definition
(d)
27𝑥 3 − 1 = 0
(g)
𝑥 − 2𝑥 2 + 1 = 0
2
(e)
2
2𝑥 3 − 32 = 0
1
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1
(h)
1
𝑥 2 − 3𝑥 4 + 2 = 0
1
NOTE:
1
1
1
1
𝑥4 . 𝑥 4 = 𝑥4 + 4 = 𝑥2
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MATHS EXAMINATION PRESENTATION GRADE 11
10.
NUMBER PATTERNS:
ARITHMETIC / LINEAR
GEOMETRIC
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
𝑇𝑛 = 𝑎𝑟 𝑛−1
𝑇2 − 𝑇1 = 𝑇3 − 𝑇2
𝑇2
𝑇3
=
𝑇1
𝑇2
QUADRATIC
𝑻𝒏 = 𝒂𝒏𝟐 + 𝒃𝒏 + 𝒄
𝟐𝒏𝒅 level differences are constant
(a)
Write down the next two terms of the sequence:
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32 34
;
; 12 ; ___ ; ___
3
3
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MATHS EXAMINATION PRESENTATION GRADE 11
(b)
The heights above ground level of steps in a staircase form an
arithmetic sequence.
The heights of the 3 rd and 7th steps are 52 cm and 78 cm above the
ground respectively.
Determine the height above ground level of the 43 rd step.
(c)
Consider the following arithmetic sequence:
(i)
Determine the value of 𝑥
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(𝑥 + 5) ; (37 − 𝑥) ; (𝑥 + 13) . . .
(ii)
Determine the general term.
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MATHS EXAMINATION PRESENTATION GRADE 11
(d)
Given 𝑇6 = 𝑥 − 4 ; 𝑇7 = 𝑥 + 2 and 𝑇8 = 3𝑥 + 1 are three consecutive terms of a geometric sequence.
Find:
(i)
(e)
the value of 𝑥, where 𝑥 ∈ ℕ
(ii)
−320 ; 160 ; −80 ; . . . ; −
Consider the geometric sequence:
Determine the value of 𝑛 if 𝑇𝑛 = −
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the common ratio
5
64
5
16
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MATHS EXAMINATION PRESENTATION GRADE 11
(f)
PATTERN 1
PATTERN 2
PATTERN 3
(1)
How many white tiles are added each time to make a new pattern?
(2)
How many tiles will there be in PATTERN 7?
(3)
How many WHITE tiles will there be in PATTERN 10?
(4)
Sam’s formula for the pattern is 𝑛 + 2.
What does the 𝑛 stand for?
(5)
Sam builds a pattern using 30 tiles altogether.
How many white tiles does he use?
(6)
Sam builds a pattern using 40 tiles.
What PATTERN number did he use?
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MATHS EXAMINATION PRESENTATION GRADE 11
(g)
Given:
1 ; 𝑥 ; 𝑦 ; 16 ; 24
The last four terms of the above pattern form a quadratic sequence. [ 𝑥, 𝑦 ∈ ℕ]
𝑦
It is further given that = 𝑥 . Determine the values of 𝑥 and 𝑦.
𝑥
(h)
The second term of a quadratic sequence is equal to 1. The third term is equal to −6 and the fifth term is equal to −14.
(a)
Determine the second difference of the sequence
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(b)
Hence, determine the first term of the sequence.
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MATHS EXAMINATION PRESENTATION GRADE 11
(i)
A certain quadratic sequence has the following characteristics:
•
•
•
•
𝑇1 = 𝑝
𝑇2 = 18
𝑇4 = 4𝑇1
𝑇3 − 𝑇2 = 10
Determine the value of 𝑝. Show ALL your working details.
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MATHS EXAMINATION PRESENTATION GRADE 11
11.
FINANCE
SIMPLE INTEREST
COMPOUND INTEREST
Appreciation
F = P(1 + i. n)
F = P(1 + 𝑖)𝑛
Depreciation
F = P(1 − i. n)
F = P(1 − 𝑖)𝑛
HIRE PURCHASE
INFLATION
TIMELINES
𝐹 = Future value (𝐴)
𝑃 = Present Value
𝑖 = interest rate
𝑛 = time period
EFFECTIVE TO NOMINAL AND VICE VERSA
1 + 𝑖𝑒 𝑓𝑓 = (1 +
𝑖𝑛𝑜𝑚 𝑛
𝑛
)
NOMINAL TO NOMINAL
(1 +
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𝑖𝑛𝑜𝑚 𝑛
𝑛
) = (1 +
𝑖𝑛𝑜𝑚
𝑚
)
𝑚
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MATHS EXAMINATION PRESENTATION GRADE 11
(a)
(b)
Gary receives R12 000 to invest for a period of 5 years. He is offered an interest rate of 8,5% p.a., compounded monthly.
(i)
Determine the effective interest rate. Round your answer off to one decimal digit.
(ii)
What is the amount that Jerry will receive at the end of the 5 years?
Billy plans to save R 20 000 for a deposit on a new car. He decided to use a part of his annual bonus to pay three even annual deposits
into a savings account at the beginning of every year.
Calculate how much money he must deposit each year to save R20 000 by the end of 3 years if the interest is calculated at 8%p.a
compounded quarterly.
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MATHS EXAMINATION PRESENTATION GRADE 11
(c)
Convert a nominal interest rate of 25,5% p.a. compounded daily to an equivalent nominal rate that is compounded quarterly.
(d)
R16 725 is deposited into a savings account at an interest rate of 11,25% p.a., compounded monthly for the first 2 and a half years and
12% p.a., compounded quarterly thereafter. After 51 months R5 700 was withdrawn from the account. Using a timeline, work out how
much will be in the account after 7 years?
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MATHS EXAMINATION PRESENTATION GRADE 11
(e)
Siphiwe deposits R 22 000 for a period of eight years into a unit trust fund. During the first five years the interest rate is 15% per annum
compounded monthly. The interest rate than changes to 20%, p.a., compounded semi-annually. Using a time-line, calculate the value of
the investment at the end of the eight-year period.
(f)
Joyce wants to save for an overseas trip in three years’ time. She will need to have saved R 50 000
for her trip. The interest rate during the first will year be 14% p.a., compounded quarterly. For the
remaining two years, the interest rate will be 11% p.a. compounded monthly. What must she now
invest to receive R 50 000 in three years time?
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MATHS EXAMINATION PRESENTATION GRADE 11
(g)
Jodie deposited R25 000 into a savings account. The interest rate given was 8% p.a. compounded quarterly for the first 3 years, and
then 9% p.a. compounded monthly for the remaining time. 5 years after the initial deposit, a withdrawal was made.
The balance at the end of 8 years is R37 000. How much was withdrawn.
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MATHS EXAMINATION PRESENTATION GRADE 11
12.
FUNCTIONS and GRAPHS:
WHAT YOU NEED TO KNOW
• The standard formula for each function and, how to find each equation.
• What is the average gradient.
• What is the maximum or minimum length between two graphs.
• What to do when you are given the distance between two graphs.
• How to find points of intersection between two functions.
• Reading inequalities off the graph:
𝒇(𝒙) ≤ 𝒈(𝒙)
𝑓(𝑥)
𝒇(𝒙). 𝒈(𝒙) < 𝟎 or 𝑔(𝑥) ≤ 0 - take note 𝑔(𝑥) ≠ 0
• Nature of roots in realtion to the graph.
• Domain and Range.
• Equations of the axes of symmetry.
• Equations of the asymptotes (where necessary).
• Reflections about the axes (𝑦 = 0 and 𝑥 = 0)
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MATHS EXAMINATION PRESENTATION GRADE 11
1.
FINDING THE EQUATION OF A PARABOLA and DOMAIN and RANGE
(a)
Given that 𝑓(𝑥) = −2𝑥 2 + 𝑏𝑥 + 𝑐
(i)
Determine the values of 𝑏 and 𝑐.
(ii)
Determine the co-ordinates of T, the turning point of 𝑓
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(iii)
Give the domain and range of 𝑓.
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MATHS EXAMINATION PRESENTATION GRADE 11
(b)
Given that 𝑔(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
(i)
Determine K, the turning point of 𝑔(𝑥).
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(ii)
Determine the domain and range of 𝑔(𝑥).
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MATHS EXAMINATION PRESENTATION GRADE 11
(c)
1
1
The minimum value of ℎ(𝑥) = 2 𝑥 2 + 𝑘𝑥 − 12 is −12 2 .
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Determine the value of 𝑘 if 𝑘 < 0.
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MATHS EXAMINATION PRESENTATION GRADE 11
2.
FINDING THE EQUATION OF AN EXPONENTIAL FUNCTION and DOMAIN and RANGE
(a)
𝑔(𝑥) = 𝑎. 𝑏 𝑥
(i)
Find the values of 𝑎 and 𝑏.
(i)
Find the values of 𝑏 and 𝑞.
(ii)
Give the domain and range of 𝑔.
(ii)
Give the domain and range of 𝑓.
(c)
𝑓(𝑥) = 𝑏 𝑥+1 + 𝑞
(i)
(ii)
(b)
𝑓(𝑥) = 𝑏 𝑥 + 𝑞
Find the values of 𝑏 and 𝑞.
Give the domain and range of 𝑓.
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MATHS EXAMINATION PRESENTATION GRADE 11
3.
FINDING THE EQUATION OF THE HYPERBOLIC FUNCTION and DOMAIN and RANGE
𝑎
𝑦 =
(i)
Find the values of 𝑎 and 𝑞.
(i)
Find the values of 𝑎, 𝑝 and 𝑞.
(ii)
Give the domain and range.
(ii)
Give the domain and range.
𝑥
+ 𝑞
𝑎
(a)
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(b)
𝑔(𝑥) = 𝑥 − 𝑝 + 𝑞
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MATHS EXAMINATION PRESENTATION GRADE 11
𝑎
(c)
ℎ(𝑥) = 𝑥 − 𝑝 + 𝑞
(i)
Find the values of 𝑎, 𝑝 and 𝑞.
(ii)
Give the domain and range.
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MATHS EXAMINATION PRESENTATION GRADE 11
4.
MAXIMUM and MINIMUM LENGTHS, DISTANCES BETWEEN GRAPHS and AVERAGE GRADIENTS
1.
(a)
𝑓(𝑥) = −
𝑥2
2
+ 𝑥 + 4 and 𝑔(𝑥) = 𝑥 + 2
(i)
Determine the co-ordinates of:
(1)
C and D, the 𝑥 −intercepts of 𝑓.
●
●
●
●
●
●
●
(2)
(b)
●
E, the turning point of the parabola.
Give the range of −𝑓(𝑥) + 8
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MATHS EXAMINATION PRESENTATION GRADE 11
(c)
If P is a point on 𝑓 and Q is a point on 𝑔 such that PQ is parallel to the 𝑦 −axis, find the maximum length of PQ.
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MATHS EXAMINATION PRESENTATION GRADE 11
2.
The sketch shows the graphs of 𝑓(𝑥) = 2𝑥 + 3 and 𝑔(𝑥) = −2𝑥 2 + 14𝑥 − 20.
C is any point on 𝑓 and D is any point on 𝑔 such that CD is parallel to the
𝑦 −axis.
(a)
Write down a simplified expression for the length of CD.
(b)
Find the minimum length of CD and the value of 𝑥 where this minimum occurs.
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MATHS EXAMINATION PRESENTATION GRADE 11
3.
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = 𝑥 − 3
EF = 4 units.
Determine:
(a)
the values of 𝑎 , 𝑏 and 𝑐.
(b)
the length of OS if TR = 10 units.
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MATHS EXAMINATION PRESENTATION GRADE 11
4.
Sketched are the functions 𝑔 and 𝑓(𝑥) = 2𝑥 + 5
Determine:
(a)
the equation of 𝑔 in the form 𝑔(𝑥) = 𝑎(𝑥 + 𝑘)2 + 𝑚
(b)
the average gradient between B and O
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MATHS EXAMINATION PRESENTATION GRADE 11
5.
NATURE OF ROOTS:
𝑥 −intercepts
𝑏 2 − 4𝑎𝑐 = 0
𝑏 2 − 4𝑎𝑐 > 0
𝑏 2 − 4𝑎𝑐 < 0
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MATHS EXAMINATION PRESENTATION GRADE 11
EXERCISE:
1.
Draw rough sketches of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 if
(a)
𝑎 < 0 ; 𝑏 > 0 ; 𝑐 > 0 and 𝑏2 − 4𝑎𝑐 > 0
(b)
𝑎 < 0 ; 𝑏 > 0 ; 𝑐 < 0 and 𝑏2 − 4𝑎𝑐 < 0
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MATHS EXAMINATION PRESENTATION GRADE 11
(c)
𝑎 < 0 ; 𝑏 < 0 and 𝑐 > 0
(d)
4𝑎𝑐 > 𝑏2 and 𝑎 < 𝑏 < 𝑐 < 0
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MATHS EXAMINATION PRESENTATION GRADE 11
2.
Find the equation of the parabola with the following properties:
(a)
Axis of symmetry: 𝑥 = 4 ; 4𝑎𝑐 − 𝑏2 = 0 and 𝑓(8) = 8
(c)
𝑔(𝑥) ≤ 6 ; 𝑔(0) = 𝑔(2) = 4
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(b)
Maximum value of 𝑔(𝑥) = 5 ;
Axis of symmetry is 𝑥 = −1 and 𝑔(0) = 0
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MATHS EXAMINATION PRESENTATION GRADE 11
3.
Given 𝑔(𝑥) = −3(𝑥 − 3)2 − 4
Determine the turning point of:
4.
(a)
𝑦 = 𝑔(𝑥) + 1
(b)
𝑦 = 𝑔(𝑥 − 2)
(c)
𝑦 = 2𝑔(𝑥)
(d)
𝑦 = 𝑔(2𝑥)
(e)
𝑦 = −𝑔(𝑥)
(f)
𝑦 = 𝑔(−𝑥)
4
You are given the function 𝑓(𝑥) = 𝑥+5 − 3
(a)
Write down the equations of the asymptotes of 𝑓(𝑥)
(b)
Determine the equations of the axes of symmetry of 𝑦 = (𝑥 + 3)
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MATHS EXAMINATION PRESENTATION GRADE 11
Given: ℎ(𝑥) = 3𝑥 2 − 2𝑥 − 3
5.
The function is shifted 3 units vertically downwards and 2 units horizontally to the right to form a new function 𝑓.
Write down the new function that defines the function 𝑓 in the form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐.
Draw a sketch graph of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 if it is also given that:
6.
•
•
•
•
The range of 𝑓 is (−∞; 7].
𝑎 ≠ 0.
𝑏 < 0.
One root of 𝑓 is positive and the other root is negative.
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MATHS EXAMINATION PRESENTATION GRADE 11
7.
The graphs of 𝑓(𝑥) = −2𝑥 2 − 4𝑥 + 30 and 𝑔(𝑥) = 2𝑥 + 10 are drawn.
A and B are the 𝑥 −intercepts and C is the 𝑦 −intercept of 𝑓(𝑥).
G is the turning point of 𝑓(𝑥).
(a)
G
C
Determine the coordinates of A and B.
F
H
D
J
E
𝑔(𝑥)
= 2𝑥
K
A
B
f
L
(b)
Give the values of 𝑥 for which 𝑓(𝑥). 𝑔(𝑥) ≤ 0.
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(c)
Determine the maximum length of GH if GH is parallel to the
𝑦 −axis.
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MATHS EXAMINATION PRESENTATION GRADE 11
8.
Consider the sketch, showing the graphs of 𝑔(𝑥) = 𝑥
(a)
𝑘
− 𝑝
+ 𝑞 and 𝑓(𝑥) = 𝑏 𝑥
−𝑚
+ 𝑛 with asymptotes 𝑥 = 1 and 𝑦 = − 2
Determine the equation of 𝑔(𝑥) by finding the values for 𝑘 ; 𝑝 and 𝑞.
F
(b)
Show that 𝑏 = 2 ; 𝑚 = −1 and 𝑛 = − 2.
E
1
C
−2
D
(c)
Determine the axis of symmtry of 𝑔(𝑥 ) which has a positive gradient.
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MATHS EXAMINATION PRESENTATION GRADE 11
(d)
ED is a line parallel to the 𝑦 −axis, with point E on the 𝑥 −axis, point C on 𝑓(𝑥) and point D on 𝑔(𝑥).
9
If the length of CD is units, find the length of OE.
NOTE: 𝑥 ∈ ℤ
4
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