functions quadratics polynomials skill builder
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CfE Higher Maths. Unit Assessment Skills Builder
Layout and content of the Unit Assessment will be different. This is not meant to be a carbon copy of the Unit Assessment. This
booklet is an opportunity to practice all of the essential skills required to pass the Unit Assessment.
This booklet should be used to identify any areas for improvement before you sit the Unit assessment for the first time.
Unit Assessment standard Sub-skills
H4LD 76
Relationships
and Calculus
RC1.1 Applying algebraic
skills to solve equations factorising a cubic polynomial expression with unitary x
3 coefficient
solving cubic polynomial equations with unitary x3 coefficient
given the nature of the roots of an equation, use the discriminant to find an unknown
RC#2.1 Interpreting a
situation where mathematics
can be used and identifying a
valid strategy
For candidates undertaking the Course, Assessment Standard 2.1 should be achieved on at least two
occasions from across the Course.
H4LC 76
Expressions
and Functions
EF1.3 Applying algebraic and
trigonometric skills to
functions
identifying and sketching related algebraic functions
identifying and sketching related trigonometric functions
determining composite and inverse functions (knowledge and use of the terms domain and
range is expected)
EF#2.1 Interpreting a
situation where mathematics
can be used and identifying a
valid strategy
For candidates undertaking the Course, Assessment Standard 2.1 should be achieved on at least two
occasions from across the Course.
EF#2.2 Explaining a solution
and, where appropriate,
relating it to context
For candidates undertaking the Course, Assessment Standard 2.2 should be achieved on at least two
occasions from across the Course.
Mathematics Higher
Revision Materials
Functions, Quadratics & Polynomials Skills Builder
CfE Higher Maths. Unit Assessment Skills Builder
RC1.1 Applying algebraic skills to solve equations
RC#2.1 Interpreting a situation where mathematics can be used and identifying a valid strategy
Sub-skills
factorising a cubic polynomial expression with unitary x3 coefficient (extended to non-unitary)
solving cubic polynomial equations with unitary x3 coefficient (extended to non-unitary)
Q1 a) Show that (π₯ β 1) is a factor of π(π₯) = 2π₯3 β 5π₯2 β 2π₯ + 5 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 β π₯2 β 2π₯ + 3 = 4π₯2 β 2
Q2 a) Show that (π₯ β 3) is a factor of π(π₯) = 3π₯3 β 17π₯2 + 29π₯ β 15 and hence factorise π(π₯) fully.
b) Hence solve the equation 3π₯3 β 15π₯2 + 29π₯ β 14 = 2π₯2 + 1
Q3 a) Show that (π₯ β 1) is a factor of π(π₯) = 2π₯3 β 3π₯2 β 5π₯ + 6 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 β 2π₯2 + 3π₯ + 13 = π₯2 + 8π₯ + 7
Q4 a) Show that (π₯ + 3) is a factor of π(π₯) = π₯3 + 4π₯2 β 17π₯ β 60 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 + 4π₯2 β 17π₯ β 61 = π₯3 β 1
Q5 a) Show that (π₯ β 2) is a factor of π(π₯) = 2π₯3 β 9π₯2 + 13π₯ β 6 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 β 13π₯2 + 13π₯ β 1 = 5 β 4π₯2
Q6 a) Show that (π₯ + 3) is a factor of π(π₯) = 3π₯3 + 20π₯2 + 29π₯ β 12 and hence factorise π(π₯) fully.
b) Hence solve the equation 3π₯3 + 22π₯2 + 29π₯ β 19 = 2π₯2 β 7
Q7 a) Show that (π₯ + 1) is a factor of π(π₯) = 2π₯3 β 8π₯2 + 2π₯ + 12 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 β 11π₯2 + π₯ + 13 = 1 β π₯ β 3π₯2
Q8 a) Show that (π₯ + 3) is a factor of π(π₯) = 2π₯3 + π₯2 β 12π₯ + 9 and hence factorise π(π₯) fully.
b) Hence solve the equation 2π₯3 + 6π₯2 β 13π₯ + 9 = 5π₯2 β π₯
Q9 a) Show that (π₯ β 2) is a factor of π(π₯) = π₯3 β 2π₯2 β π₯ + 2 and hence factorise π(π₯) fully.
b) Hence solve the equation π₯3 β 2π₯2 + 8π₯ + 1 = 9π₯ β 1
Q10 a) Show that (π₯ β 4) is a factor of π(π₯) = 3π₯3 β 25π₯2 + 56π₯ β 16 and hence factorise π(π₯) fully.
b) Hence solve the equation 3π₯3 + 5π₯2 + 27π₯ β 15 = 30π₯2 β 29π₯ + 1
CfE Higher Maths. Unit Assessment Skills Builder
RC1.1 Applying algebraic skills to solve equations
Sub-skills
given the nature of the roots of an equation, use the discriminant to find an unknown
Q11 Each of the following functions has 2 real and distinct roots.
Calculate the range of values for π so that the function will maintain 2 real and distinct roots.
a) π(π₯) = ππ₯2 + 6π₯ + 8
b) π(π₯) = 3π₯2 + ππ₯ + 12
c) π(π₯) = 4π₯2 β 2π₯ + π
d) π(π₯) = ππ₯2 + 7π₯ + 12
e) π(π₯) = β2π₯2 + ππ₯ + 2
f) π(π₯) = β4π₯2 + π₯ + π
g) π(π₯) = ππ₯2 β 5π₯ + 6
h) π(π₯) = 3π₯2 + ππ₯ + 8
i) π(π₯) = βπ₯2 + π₯ + π
j) π(π₯) = ππ₯2 β 6π₯ + 8
Q12 For what values of π does the equation π₯2 β 2π₯ + π = 0 have:
a) equal roots b) real and distinct roots c) non real roots
Q13 Find π given that π₯2 + ππ₯ + π₯ + 9 = 0 has equal roots.
Q14 Find the values of π, π πππ π if each of the equations have equal roots
a) ππ₯2 + 4π₯ + 2 = 0 b) 3π₯2 + ππ₯ + 3 = 0 c) π₯2 + 6π₯ + π = 0
Q15 For what value of π‘ does π‘π₯2 + 6π₯ + π‘ = 0 have equal roots?
Q16 Find the value of π if π₯2 + (π β 3)π₯ + 1 = 0 has equal roots.
Q17 Find the range of values of π for which 5π₯2 β 3ππ₯ + 5 = 0 has two real and distinct roots.
Q18 For what value of π does the graph π¦ = ππ₯2 β 3ππ₯ + 9 touch the π₯ axis.
Q19 Find the values for n which ensure that the following equations have equal roots:
a) π₯2+1
π₯= π b)
(π₯β2)2
π₯2+2= π
Q20 find π if (2π β 2)π₯2 + 24π₯ + π = 0 has:
a) equal roots b) real and distinct roots
CfE Higher Maths. Unit Assessment Skills Builder
EF1.3 Applying algebraic and trigonometric skills to functions.
Sub-skills
identifying and sketching related algebraic functions
Q21 The graph of π¦ = π(π₯) is shown opposite.
a) Sketch π¦ = π(π₯ β 2)
b) Sketch π¦ = π(π₯) + 5
c) Sketch π¦ = π(π₯ β 1) + 2
d) Sketch π¦ = π(βπ₯)
e) Sketch π¦ = β2π(π₯ β 4)
Q22 The graph of π¦ = π(π₯) is shown opposite.
a) Sketch π¦ = π(π₯ + 3)
b) Sketch π¦ = π(π₯) β 4
c) Sketch π¦ = π(π₯ β 2) β 1
d) Sketch π¦ = π(βπ₯)
e) Sketch π¦ = 2π(π₯ + 1)
Q23 The graph of π¦ = β(π₯) is shown opposite.
a) Sketch π¦ = β(π₯ β 1)
b) Sketch π¦ = β(π₯) + 2
c) Sketch π¦ = β(π₯ + 1) + 4
d) Sketch π¦ = ββ(π₯)
e) Sketch π¦ = 2β(π₯)
Q24 The graph of π¦ = π(π₯) is shown opposite.
a) Sketch π¦ = π(π₯ β 3)
b) Sketch π¦ = π(π₯) β 2
c) Sketch π¦ = π(π₯ β 1) + 3
d) Sketch π¦ = βπ(π₯)
e) Sketch π¦ = 4π(π₯ β 3)
CfE Higher Maths. Unit Assessment Skills Builder
EF1.3 Applying algebraic and trigonometric skills to functions.
Sub-skills
identifying and sketching related algebraic functions
Q25 For each of the following graphs identify the value of π.
a) b) c) d)
e) f) g) h)
EF1.3 Applying algebraic and trigonometric skills to functions.
Sub-skills
determining composite and inverse functions (knowledge and use of the terms domain and range is expected)
Q26 Write down the equation of the inverse functions for the following.
a) π(π₯) = 2π₯ b) π(π₯) = 3π₯ c) π(π₯) = 4π₯ d) π(π₯) = 7π₯
e) π(π₯) = ππ₯ f) π(π₯) = 10π₯ g) π(π₯) = ππ₯ h) π(π₯) = 5π₯
i) π(π₯) = (1
2)
π₯ j) π(π₯) = 9π₯ k) π(π₯) = (2
3)
π₯ l) π(π₯) = (1
5)
π₯
CfE Higher Maths. Unit Assessment Skills Builder
EF1.3 Applying algebraic and trigonometric skills to functions
Sub-skills
determining composite and inverse functions
(knowledge and use of the terms domain and range is expected)
Q27 Determine the inverse function for each of the following, assuming an appropriate domain and range.
a) π(π₯) = 5π₯ β 2 b) π(π₯) = 3 β 4π₯ c) π(π₯) = 2π₯ + 7
d) π(π₯) =π₯β2
4 e) π(π₯) =
3
2β 5π₯ f) π(π₯) = 2(3π₯ β 1)
g) π(π₯) = 5π₯2 h) π(π₯) = 3(π₯ β 2)2 i) π(π₯) = 2(3π₯ + 1)2
EF1.3 Applying algebraic and trigonometric skills to functions
EF#2.2 Explaining a solution and, where appropriate, relating it to context
Sub-skills
determining composite and inverse functions
(knowledge and use of the terms domain and range is expected)
Q28 In each of the examples below, functions π and π are defined on suitable domains.
For each question, obtain expressions for π(π(π₯)) πππ π(π(π₯)) in their simplest form.
a) π(π₯) = π₯ β 2 π(π₯) = π₯2
b) π(π₯) = 2π₯ + 1 π(π₯) = π₯2 β 2π₯ + 1
c) π(π₯) = 5π₯ β 3 π(π₯) =π₯+3
5 , explain the relationship between π(π₯)πππ π(π₯).
d) π(π₯) = 4 β 3π₯ π(π₯) =4βπ₯
3 , explain the relationship between π(π₯)πππ π(π₯).
e) π(π₯) = π₯2 + 1 π(π₯) = (π₯ + 1)2
f) π(π₯) =2
3π₯ β 4 π(π₯) =
3
2π₯ + 6 , explain the relationship between π(π₯)πππ π(π₯).
g) π(π₯) = 3 β 2π₯2 π(π₯) = π₯ β 1
h) π(π₯) = 3π₯ + 5 π(π₯) =2
π₯+1
i) π(π₯) =π₯β2
π₯+1 π(π₯) =
1
π₯β2
j) π(π₯) = βπ₯ + 4 π(π₯) = π₯2 β 4 , explain the relationship between π(π₯)πππ π(π₯).
CfE Higher Maths. Unit Assessment Skills Builder
EF1.3 Applying algebraic and trigonometric skills to functions
Sub-skills
identifying and sketching related trigonometric functions
Q29 Sketch each of the following trigonometric functions, where 0 β€ π₯ β€ 2π
a) π¦ = sin π₯ b) π¦ = cos π₯
c) π¦ = 2 sin π₯ d) π¦ = 3cos π₯
e) π¦ = sin π₯ + 3 f) π¦ = cos π₯ β 4
g) π¦ = sin 2π₯ h) π¦ = cos (1
2π₯)
i) π¦ = β sin π₯ j) π¦ = β cos π₯
k) π¦ = 2 sin 3π₯ β 1 l) π¦ = 2 β cos 3π₯
EF1.3 Applying algebraic and trigonometric skills to functions
EF#2.1 Interpreting a situation where mathematics can be used and identifying a valid strategy
Sub-skills
identifying and sketching related trigonometric functions
Q30 Use the information provided in the table below to identify the equation of the curves.
All equations are of the form π¦ = π sin ππ₯ + π ππ π¦ = π cos ππ₯ + π.
Type of curve Domain Maximum occurs at Minimum occurs at
a) sine 0 β€ π₯ β€ π ( π
4, 1 ) (
3π
4, β1 )
b) cosine 0 β€ π₯ β€π
2 ( 0, 1 ) (
π
2, 1 ) (
π
4, β1 )
c) sine 0 β€ π₯ β€ 2π ( π
2, 3 ) (
3π
2, 1 )
d) cosine 0 β€ π₯ β€ π ( 0, 3 ) ( π, 3 ) ( π
2, β3 )
e) sine 0 β€ π₯ β€ 2π ( 3π
2, 5 ) (
π
2, 3 )
f) cosine 0 β€ π₯ β€ 2π ( π, β1 ) ( 0, β3 ) ( 2π, β3 )
g) sine 0 β€ π₯ β€ π ( π
4, 5 ) (
3π
4, 1 )
h) cosine 0 β€ π₯ β€ π ( 0, 2) ( π, 2 ) ( π
2, β4 )
i) sine 0 β€ π₯ β€2π
3 (
π
6, 4 ) (
π
2, β2 )
j) cosine 0 β€ π₯ β€π
2 (
π
4, 1 ) ( 0, β3 ) (
π
2, β3 )
Q31 Use your answers from Q30 to sketch the graphs of the trigonometric functions.
CfE Higher Maths. Unit Assessment Skills Builder
ANSWERS
Q1 a) π(π₯) = (π₯ + 1)(π₯ β 1)(2π₯ β 5) b) π₯ = β1, 1, 5
2
Q2 a) π(π₯) = (π₯ β 3)(π₯ β 1)(3π₯ β 5) b) π₯ = 3, 1, 5
3
Q3 a) π(π₯) = (π₯ β 1)(π₯ β 2)(2π₯ + 3) b) π₯ = 1, 2, β3
2
Q4 a) π(π₯) = (π₯ + 3)(π₯ + 5)(π₯ β 4) b) π₯ = β3, β 5, 4
Q5 a) π(π₯) = (π₯ β 2)(π₯ β 1)(2π₯ β 3) b) π₯ = 2, 1, 3
2
Q6 a) π(π₯) = (π₯ + 3)(π₯ + 4)(3π₯ β 1) b) π₯ = β3, β 4, 1
3
Q7 a) π(π₯) = 2(π₯ + 1)(π₯ β 3)(π₯ β 2) b) π₯ = β1, 3, 2
Q8 a) π(π₯) = (π₯ + 3)(π₯ β 1)(2π₯ β 3) b) π₯ = β3, 1, 3
2
Q9 a) π(π₯) = (π₯ β 2)(π₯ β 1)(π₯ + 1) b) π₯ = β1, 1, 2
Q10 a) π(π₯) = (π₯ β 4)(π₯ β 4)(3π₯ β 1) b) π₯ = 4, 1
3
Q11 a) π <9
8 b) π > 12, π < β12
c) π <1
4 d) π <
49
48
e) π΄ππ π£πππ’ππ ππ π f) π > β1
16
g) π <25
24 h) π > 4β6 , π < β4β6
i) π > β1
4 j) π <
9
8
Q12 a) π = 1 b) π β€ 1 c) π > 1
Q13 π = 5 , β7
Q14 a) π = 2 b) π = Β±6 c) π = 9
Q15 π‘ = Β±3
Q16 π = 1 , 5
Q17 π β₯10
3 , π β€ β
10
3
Q18 π = 4, π β 0 ππ π‘βππ π¦ = 9. πβππ ππππ ππππππ‘ π‘ππ’πβ π‘βπ π₯ ππ₯ππ
Q19 a) π = Β±2 b) π = 0 , 3
Q20 a) π = 9, β8 b) π < β8 , π > 9
CfE Higher Maths. Unit Assessment Skills Builder
Q21 Q22
CfE Higher Maths. Unit Assessment Skills Builder
Q23 Q24
CfE Higher Maths. Unit Assessment Skills Builder
Q25 a) π = 3 b) π = 2
c) π = 7 d) ππ = 3
e) π = 4 f) π = 2
g) π =1
2 h) π =
1
3
Q26 a) π = log2 π₯ b) π = log3 π₯
c) π = log4 π₯ d) π = log7 π₯
e) π = logπ π₯ f) π = log10 π₯
g) π = logπ π₯ ππ ln π₯ h) π = log5 π₯
i) π = log12
π₯ j) π = log9 π₯
k) π = log23
π₯ l) π = log15
π₯
Q27 a) πβ1(π₯) =π₯+2
5 b) πβ1(π₯) =
3βπ₯
4 c) πβ1(π₯) =
π₯β7
2
d) πβ1(π₯) = 4π₯ + 2 e) πβ1(π₯) =3β2π₯
10 f) πβ1(π₯) =
π₯+2
6
g) πβ1(π₯) = βπ₯
5 h) πβ1(π₯) = β
π₯
3+ 2 i) πβ1(π₯) =
β2(βπ₯β2)
6
Q28 a) π(π(π₯)) = π₯2 β 2 π(π(π₯)) = (π₯ β 2)2
b) π(π(π₯)) = 2π₯2 β 4π₯ + 3 π(π(π₯)) = 4π₯2
c) π(π(π₯)) = π₯ π(π(π₯)) = π₯
π(π₯) = πβ1(π₯) πππ πβ1(π₯) = π(π₯) (πππ£πππ π ππ πππβ ππ‘βππ)
d) π(π(π₯)) = π₯ π(π(π₯)) = π₯
π(π₯) = πβ1(π₯) πππ πβ1(π₯) = π(π₯) (πππ£πππ π ππ πππβ ππ‘βππ)
e) π(π(π₯)) = (π₯ + 1)4 + 1 π(π(π₯)) = (π₯2 + 2)2
f) π(π(π₯)) = π₯ π(π(π₯)) = π₯
π(π₯) = πβ1(π₯) πππ πβ1(π₯) = π(π₯) (πππ£πππ π ππ πππβ ππ‘βππ)
g) π(π(π₯)) = 1 + 4π₯ β 2π₯2 π(π(π₯)) = 2(1 + π₯)(1 β π₯)
h) π(π(π₯)) =6
π₯+1+ 5 π(π(π₯)) =
2
3(π₯+2)
i) π(π(π₯)) =5β2π₯
π₯β1 π(π(π₯)) = β
(π₯+1)
(π₯+4)
j) π(π(π₯)) = π₯ π(π(π₯)) = π₯
π(π₯) = πβ1(π₯) πππ πβ1(π₯) = π(π₯) (πππ£πππ π ππ πππβ ππ‘βππ)
CfE Higher Maths. Unit Assessment Skills Builder
Q29
CfE Higher Maths. Unit Assessment Skills Builder
Q30 a) π¦ = π ππ 2π₯ b) π¦ = cos 4π₯
c) π¦ = π ππ π₯ + 2 d) π¦ = 3 cos 2π₯
e) π¦ = βπ ππ π₯ + 4 f) π¦ = βcos π₯ β 2
g) π¦ = 2π ππ 2π₯ + 3 h) π¦ = 3 cos 2π₯ β 1
i) π¦ = 3π ππ 3π₯ + 1 j) π¦ = β2cos 4π₯ β 1
Q31