Download Add Maths Gym Sec 34.indb

Preface
This book is prepared for students embarking on the study of Additional Mathematics.
Topical Approach
Examinable topics for Upper Secondary Mathematics are discussed in detail so students
can focus on building their foundation in the subject.
Learning Objectives
These show clearly the purpose and extent of coverage for each topic.
Useful Notes
Each chapter begins with a quick recap and presentation of the main focus and content
with direct explanations to formulae and concepts. Worked Examples show best
methods and sometimes, alternate ways of working out typical problems.
Practice Questions
Over 400 questions are provided so students learn to apply mathematical concepts
confidently. Challenging Exercises are included for further application and learning.
Worked Solutions
Step-by-step solutions are included so students can learn independently. They also
serve as a quick assessment of the work done.
The Editorial Team
Contents
Chapter 1
Quadratic Equations And Inequalities ............................................................................. 1
Learning Objectives
• conditions for a quadratic equation to have
(i) two real roots
(ii) two equal roots
(iii) no real roots
and related conditions for a given line to
(i) intersect a given curve
(ii) be a tangent to a given curve
(iii) not intersect a given curve
• solution of quadratic inequalities, and the representative of the solution set on the number line
• conditions for ax2 + bx + c to be always positive (or always negative)
• relationships between the roots and coefficients of the quadratic equation ax2 + bx + c = 0
Chapter 2
Indices And Surds ............................................................................................................. 20
Learning Objectives
• four operations on indices and surds
• rationalising the denominator
• solving equations involving indices and surds
Chapter 3
Polynomials ....................................................................................................................... 33
Learning Objectives
• multiplication and division of polynomials
• use of remainder and factor theorems
• factorisation of polynomials
• solving cubic equations
Chapter 4
Simultaneous Equations In Two Unknowns .................................................................. 48
Learning Objectives
• solving simultaneous equations with at least one linear equation, by substitution.
• expressing a pair of linear equations in matrix form and solving the equations by inverse matrix
method.
Chapter 5
Partial Fractions ............................................................................................................... 63
Learning Objectives
Include cases where the denominator is no more complicated than:
• (ax + b) (cx + d)
• (ax + b) (cx + d)2
• (ax + b) (x2 + c2)
Chapter 6
Binomial Expansions ........................................................................................................ 75
Learning Objectives
Include:
• use of the Binomial Theorem for positive integer n
• use of the notations n! and ( nr )
• use of the general term ( nr ) an – r br, 0 < r ñ n
Chapter 7
Exponential, Logarithmic And Modulus Functions ...................................................... 91
Learning Objectives
• functions ax , ex , log a x , 1n x and their graphs
• laws of logarithms
• equivalence of y = ax and x = log a y
• change of base of logarithms
• function xxx and graph of xf(x)x, where xf(x)x is linear quadratic or trigonometric
• solving simple equations involving exponential, logarithmic and modulus functions
Chapter 8
Trigonometric Functions, Identities And Equations ................................................... 107
Learning Objectives
• six trigonometric functions for angles of any magnitude (in degrees or radians)
• principal values of sin–1x, cos–1x, tan–1x
• exact values of the trigonometric functions for special angles (30º, 45º, 60º) or __6 ,__4 ,__3
• amplitude, periodicity and symmetrices related to the sine and cosine functions
• graphs of y = a sin (bx) + c, y = a sin ( __b ) + c, y = a cos (bx) + c, y = a cos ( __b ) + c and
y = a tan (bx), where a and b are positive integers and c is an integer
x
x
• use of the following
sin A
cos A
____
2
2
2
2
2
2
* ____
cos A = tan A, sin A = cot A, sin A + cos A = 1, sec A = 1 + tan A, cosec A = 1 + cot A
* the expansions of sin (A ± B), cos (A ± B) and tan (A ± B)
* the formulae for sin 2A, cos 2A and tan 2A
* the formulae for sin A ± sin B and cos A ± cos B
* the expression for a cos θ + b sin θ in the form R cos(θ ± α) or R sin(θ ± α)
• simplification of trigonometric expressions
• solution of simple trigonometric equations in a given interval
• proofs of simple trigonometric identities
Chapter 9
Coordinate Geometry In Two Dimensions ................................................................... 123
Learning Objectives
• condition for two lines to be parallel or perpendicular
• midpoint of line segment
• finding the area of rectilinear figure given its vertices
• graphs of equations
* y = axn, where n is a simple rational number
* y2 = kx
• coordinate geometry of the circle with the equation (x – a)2 + (y – b)2 = r2 and
x2 + y2 + 2gx + 2 fy + c = 0
• transformation of given relationships, including y = axn and y = kbx, to linear form to determine
the unknown constants from the straight line graph
Chapter 10
Proofs In Plane Geometry .............................................................................................. 144
Learning Objectives
• symmetry and angle properties of triangles, special quadrilaterals and circles
• mid-point theorem and intercept theorem for triangles
• tangent-chord theorem (alternate segment theorem)
• use of above properties and theorems
Chapter 11
Differentiation And Integration ..................................................................................... 163
Learning Objectives
• derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point
• derivative as rate of change
2
[ ( )]
dy ___
d y
dy
d __
__
• use of standard notations f '(x), f '(x), __
dx , dx 2 = dx dx
• derivatives of xn, for any rational n, sin x, cos x, tan x, ex, and ln x, together with constant multiples,
sums and differences
• derivatives of composite functions
• derivatives of products and quotients of functions
• increasing and decreasing functions
• stationary points (maximum and minimum turning points and stationary points of inflexion)
• use of second derivative test to discriminate between maxima and minima
• applying differentiation to gradients, tangents and normals, connected rates of change and maxima
and minima problems
• integration as the reverse of differentiation
• integration of xn for any rational n, sin x, cos x, sec2 x and ex, together with constant multiples,
sums and differences
• integration of (ax + b)n for any rational n, sin(ax + b), cos(ax + b) and e(ax + b)
• definite integral as area under a curve
• evaluation of definite integrals
• finding the area of a region bounded by a curve and lines parallel to the coordinate axes
• finding areas of regions below the x-axis
• application of differentiation and integration to problems involving displacement, velocity and
acceleration of a particle moving in a straight line with variable or constant acceleration
Exclude:
• differentiation of functions defined implicity and parametrically
• finding the area of a region between a curve and an oblique line, or between two curves
• use of formulae for motion with constant acceleration
Chapter 12
Challenging Exercise ...................................................................................................... 177
Solutions .............................................................................................................................S1
Calculators should be used when necessary. If the degree of accuracy is not specified
in any question, and if the answer is not exact, give the answer to three significant
figures. Answers in degrees should be given to one decimal place. For , use either the
calculator value or 3.142, unless the question requires the answer in terms of .
Chapter
QUADRATIC EQUATIONS
AND INEQUALITIES
1
Learning
Objectives
•
•
•
•
conditions for a quadratic equation to have
(i) two real roots
(ii) two equal roots
(iii) no real roots
and related conditions for a given line to
(i) intersect a given curve
(ii) be a tangent to a given curve
(iii) not intersect a given curve
solution of quadratic inequalities, and the representative of
the solution set on the number line
conditions for ax2 + bx + c to be always positive (or always
negative)
relationships between the roots and coefficients of the
quadratic equation ax2 + bx + c = 0
USEFUL NOTES
Nature of Roots of
Quadratic Equations
For any quadratic equation
y = ax2 + _______
bx + c = 0,
– b ± √b2 – 4ac
x = ___________
.
2a
b2 – 4ac is the discriminant of the equation.
1. If b2 – 4ac < 0, the equation has no real roots. It has complex
roots.
If b2 – 4ac ú 0, the equation has real roots.
(i) If b2 – 4ac = 0, the equation has equal (repeated, identical)
___
b
roots which are equal to – .
2a
(ii) If b2 – 4ac > 0, the roots are unequal (different, distinct).
(a) If a, b and c are rational and b2 – 4ac is non-zero
and a perfect square, then the roots are rational.
(b) If b2 – 4ac is not a perfect square but positive, then
the roots are irrational.
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Chapter 1
2. The x-coordinates of the points of intersection of the straight
line y = mx + d and the quadratic curve y = ax2 + bx + c can
be obtained from:
ax2 + bx + c = mx + d
Hence ax2 + (b – m) x + (c – d) = 0
Discriminant = (b – m)2 – 4a(c – d)
If the discriminant is negative, the straight line and the curve
have no common point.
If the discriminant is zero, the straight line touches the curve
at one and only one point. It is a tangent to the curve.
If the discriminant is positive, the straight line intersects the
curve at two distinct points.
3. If y = ax2 + bx + c is either positive or negative for all values
of x, then the equation has no real roots and b2 – 4ac < 0.
Quadratic Inequalities
1. y = ax2 + bx + c
[
( ) ] + c – a ( __2ab )
b
b
= a x2 + __a x + __
2a
(
)
2
2
2
4ac – b2
= a x + __
+ ______
2a
4a
b
4ac – b2
(i) If a is positive, the minimum value of y is ______
.
4a
.
The corresponding value of x is – __
2a
b
4ac – b2
(ii) If a is negative, the maximum value of y is ______
.
4a
.
The corresponding value of x is – __
2a
b
2. y = a(x – α) (x – β) where a > 0
(i)
x
y > 0 for all real values of x.
α and β are complex numbers.
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Chapter 1
(ii)
x
α
α=β
y > 0 if x ≠ α
y = 0 if x = α
(iii)
α
β
x
y ú 0 if x ñ α or β ñ x
y > 0 if x < α or β < x
y = 0 if x = α or x = b
y < 0 if α < x < β
y ñ 0 if α ñ x ñ β
The converse is also true.
α+β
y is a minimum when x = _____
.
2
3. y = a(x – α) (x – β) where a < 0.
(i)
x
y < 0 for all real values of x.
α and β are complex numbers.
(ii)
α
α=β
y < 0 if x ≠ α
y = 0 if x = α
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Chapter 1
x
(iii)
β
α
x
y ñ 0 if x ñ α or β ñ x
y < 0 if x < α or β < x
y = 0 if x = α or x = β
y > 0 if α < x < β
y ú 0 if α ñ x ñ β
The converse is also true.
α+β
y is a maximum when x = _____
.
2
Conditions for a
quadratic expression
to be always positive or
always negative
If b2 – 4ac is negative, the equation ax2 + bx + c = 0 has no real
roots. This means that the graph of y = ax2 + bx + c is entirely
above the x-axis if a is positive or entirely below the x-axis if a is
negative. Therefore, if b2 – 4ac is negative, ax2 + bx + c is always
positive if a is positive and is always negative if a is negative.
Roots and Coefficients
of Quadratic
Equations
1. Let α and β be the roots of the quadratic equation
ax2 + bx + c = 0, where a, b and c are real numbers.
_______
_______
– b + √b2 – 4ac
– b – √b2 – 4ac
__________
and
β
=
.
Let α = ___________
2a
2a
Sum of the roots = α + β_______
_______
– b + √b2 – 4ac __________
– b – √b2 – 4ac
___________
+
=
2a
2a
2b
= – __
2a
= – __a
= negative value of the coefficient of x
divided by the coefficient of x2
b
Product of the roots = αβ
_______
_______
– b + √b2 – 4ac __________
– b – √b2 – 4ac
= ___________
×
2a
2a
_______
2
(– b) – ( √b2 – 4ac )
______________
2
=
(2a)2
b2 – b2 + 4ac
= _________
2
c 4a
__
=a
= the constant term divided by the
coefficient of x2
If a = 1, the sum and product of roots are – b and c
respectively.
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Chapter 1
2. ax2 + bx + c = 0 can also be expressed as x2 + __a x + __a = 0, or
x2 – (α + β) x + αβ = 0 where α and β are the roots of the
quadratic equation.
b
c
3. A function of α and β is said to be symmetric if it remains
unchanged when α and β are interchanged.
α+β
α + β + αβ and ____
are symmetric functions of α and β.
αβ
β+α
They are the same as β + α + βα and ____
respectively.
βα
α(α – β)
is not a symmetric function of α and β because
But _______
β
β(β – α)
it is not the same as _______
unless α = β.
α
4. Any symmetric function of α and β can be expressed in
terms (α + β) and αβ. Hence, by using point 1, it can also be
expressed as function of a, b and c.
EXAMPLE
(i) (α + β)3 = α3 + 3α2β + 3β2α + β3
= α3 + β3 + 3αβ(α + β)
( – __ba )
3
= (α3 + β3) + 3( __a ) ( – __a )
c
b3
a
b
3bc
a
α3 + β3 = – __3 + ___
2
3abc – b3
a
= _______
3
β α +β
α __
(ii) __
+ = _____
β α
αβ
2
2
(α + β)2 – 2αβ
= ___________
αβ
( – __a ) – 2( __a )
_________
b
=
2
c
c
__
a
b2 – 2ac
= ______
ac
5. The sum, difference, product and quotient of any two
symmetrical functions are also a symmetrical functions.
6. If p and q are the roots of x 2 – (p + q)x + pq = 0, and
(p + q) and pq are symmetric functions of α and β, then the
coefficients of this equation can be expressed in terms of
α + β and αβ and eventually in terms of a, b and c.
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Chapter 1
EXAMPLE
If α and β are the roots of 2x2 – 3x + 5 = 0,
find the equation whose roots are 2α + β and
α + 2β.
3
Solution: α + β = __2
5
αβ = __2
Sum of the roots of the new equation
= (2α + β) + (α + 2β)
= 3(α + β)
3
= 3__2
9
= __2
Product of the roots of the new equation
= (2α + β)(α + 2β)
= 2α2 + 4αβ + αβ + 2β2
= 2α2 + 4αβ + 2β2 + αβ
= 2(α2 + 2αβ + β2) + αβ
= 2(α + β)2 + αβ
( )
3 2 5
= 2 __2 + __2
=7
The required equation is
9
x2 – __2 x + 7 = 0
or 2x2 – 9x + 14 = 0.
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Chapter 1
Practice 1
__
1. Solve the equation 3x + 5√3 x – 6 = 0.
2
2. The sides of a right angled triangle are x – 1, x + 2 and x + 3 m. Find the perimeter and
area.
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Chapter 1