Page 2 - Spotlight A+ SPM Additional Mathematics Form 4 & 5

Extra Features of This Book

CHAPTER
1 Functions

Important Learning Standards
sMART SCOPE • Explain function using graphical representations and notations. Page 3 CONCEPT MAP
1.1 Functions • Determine domain and range of a function. 5
• Determine the image of a function when the object is given and 6 The entire content of the
vice versa.
Contains Learning
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• Describe the outcome of composition of two functions. 9 chapter is summarised in
Standards (LS) that need to • Determine the composite functions. 9
1.2 Composite • Determine the image of composite functions when the object is 10 the form of a concept map.
given and vice versa.
Functions
be achieved in each chapter. • Determine a related function when the composite function and 11
another function are given.
• Solve problems involving composite functions. 11
• Describe inverse of a function. 13 Form
4 Additional Mathematics Chapter 7 Coordinate Geometry
1.3 Inverse Functions • Make and verify conjectures related to properties of inverse 13
functions.
Form • Determine the inverse functions. 15
5 Additional Mathematics Chapter 3 Integration
3.1 Integration as the Inverse of Example 1
Differentiation (a) If f(x) = 9x 2 + 5x and f(x) = 18x + 5, find Coordinate Geometry
∫ (18x + 5) dx.
Explaining the relation between • Function / Fungsi • Range / Julat
• Relation / Hubungan and dy = g(x), find ∫ g(x) dx.
differentiation and integration (b) Given y = 4x(1 – x) 3 dx • Domain / Domain
5
• Arrow diagram / Gambar rajah anak panah • Codomain / Kodomain Divisor of a
(c) Given dy ( 3 – x ) = h(x), find ∫ h(x) dx.
CHAP. 1. Given y = x 3 + 7, thus dy = 3x 2 . • Absolute value function / Fungsi nilai mutlak • Object / Objek Line Segment Area of Polygon The Straight Line Equation of Locus
dx
dx
3 Conversely, if given dy = 3x 2 , thus y = x 3 + 7 will • Function notation / Tatatanda fungsi • Image / Imej
Solution:
• Absolute value function graph
• Composite function / Fungsi gubahan
Graf fungsi nilai mutlak
be obtained. dx (a) Given f(x) = 9x 2 + 5x • Inverse function / Fungsi songsangan
f (x) = 18x + 5
2. The reverse process of this differentiation is • Vertical line test / Ujian garis mencancang • Horizontal line test / Ujian garis mengufuk m n Parallel line Perpendicular line
Hence, ∫ 18x + 5 dx = f(x)

called integration. A(x 1 , y 1 ) P(x, y) B(x 2 , y 2 )
∫ 18x + 5 dx = 9x 2 + 5x
3. If d [f(x) = f (x), thus the integral f (x) with Thus, ∫ (18x + 5) dx is 9x 2 + 5x. 1 nx 1 + mx 2 , ny 1 + my 2  Gradient of two Product of gradient
dx
respect to x is ∫ f (x) dx = f(x). (b) Given y = 4x(1 – x) 3 P(x, y) =  m + n m + n lines are the same, for two lines is –1,
m 1 m 2 = –1
m 1 = m 2
dy = g(x)
dx
Differentiation dx d [f(x)] = f (x) Hence, ∫ g(x) dx = y
∫ g(x) dx = 4x(1 – x) 3 CHAP 7 Triangle Quadrilateral Form 5 n-sides of polygon
Integration ∫ f (x) dx = f(x) Thus, ∫ g(x) dx is 4x(1 – x) 3 . Chapter 1 Circular Measure Additional Mathematics
5
(c) Given d ( 3 – x ) = h(x) 1.1 Radian y Example 1 y CHAP. Area = 1 x 1 x 2 … x n x 1 
dx
A(x 1 , y 1 )
B(x 2 , y 2 )
Hence, ∫ h(x) dx = y Convert the angle in the unit of radian to the B(x 2 , y 2 ) 1 2  y 1 y 2 … y n y 1
Comparison between ∫ h(x) dx = 5 Relating angle measurement in radian and degree. [Use π = 3.142]
C(x 3 , y 3 )
differentiation and integration 3 – x degree (a) 1.15 radian (b) 5π radian
bit.ly/2K4y3b0 Thus, ∫ h(x) dx is 5 . 1. In circular measures, the angle can be measured Solution: x 6 D(x 4 , y 4 ) C(x 3 , y 3 )
3 – x
(a)
Try question 1 to 5 in Formative Zone 3.1 in 2 units, which are A(x 1 , y 1 ) π rad = 180° x
(a) degree (°) and minute (ʹ).
1.15 rad = 1.15 × 180°

3.1 (b) unit of radian (in or not in the terms of π). Area = 1 x 1 x 2 y 2 x 3 y 3 x 1  π y 1 180° Area = 1 x 1 x 2 x 3 x 4 x 1 
2  y 1 y 2 y 3 y 4 y 1
2 y 1
2. Radian involves the angle that related with the = 1.15 × 3.142
1. If f(x) = 7 – 3x 4 and f(x) = –12x 3 , find ∫ –12x 3 dx. C1 3. Given d (x 2 – 5x + c) = 2x – 5, find ∫ (2x – 5) dx. radius and circumference of a circle. = 65.89°
Form
C1 dx 3. The diagram below shows the angle in degree (b) π rad = 180°
2. Given dy = 2x – 2 and y = ( x + 1 ) , find Chapter 4 Indices, Surds and Logarithms Additional Mathematics 4 5π rad = 5π × 180°
and minutes and radian while the radius have
2
dx 4. Given d ( 4 + 5x) = g(x), find ∫ g(x) dx. C2 the same length. 6 6 π
x
x 3
dx x 2
x 3)
∫ ( 2x – 2 dx. C1 Calculator 5. Given y = f(x) and dy = 3h(x), find ∫ h(x) dx. C2 Example 25 A constant from a fixed point: A constant ratio from two fixed points:
= 5 × 180°
(x – x 2 ) 2 + (y – y 2 ) 2 = m 2
1. To find the value of antilog 3.26 using a scientific 6 (x – x 1 ) 2 + (y – y 1 ) 2 = r 2 (x – x 1 ) 2 + (y – y 1 ) 2
dx
= 150°
calculator, press SHIFT log 3.26 = . Prove the following. O 57° 17' O 1 rad Alternative Method n 2
(a) log a xy = log a x + log a y
SPOTLIGHT PORTAL Deriving and determining indefinite integral Example 24 (c) ∫ ax n dx = ax n + 1 . (b) y Substitute π =180° into the expression,
2. The screen will be display 1 819.700859
3.2
Indefinite Integral
n + 1 + c such as n ≠ 1 log a x = log a x − log a y
Always equidistant from two fixed points:
5π = 5(180°) = 150°
and c are constants.
6
6
for algebraic functions
Solution:
Find the value of x in each of following equation. Let log In degree and minute In radian Try question 1 in Formative Zone 1.1 (x – x 1 ) 2 + (y – y 1 ) 2 = (x – x 2 ) 2 + (y – y 2 ) 2
2. The following is the steps to find the integral of a a x = m
57° 17ʹ = 1 rad
(b) log 5 x = −2
1. Formula of integration: (a) log 9 27 = x function, ax n with respect to x: log a y = n 4. 1 radian is a measurement of an angle subtended
…❶
x = a m
Step 1: Maintain the value of constant, a.
(c) log 2 (x 2 + 3x) = 2
(d) log x 1 = −6
Scan the QR code to (a) ∫ a dx = ax + c where a and c are (a) log 9 27 = x Step 2: Add 1 to the power or index of x first. (a) y = a n (m + n) log a a = log a xy A CHAP 4 126 Convert Example 2
64
…❷
about the centre of a cirlce such as the arc length
constants.
Step 3: Divide the term with the new index. ❶ × ❷:
a m × a n = xy
is the same as the length of radius of circle.
Solution:
a m + n = xy
(a) 30° into radian unit, in term of π.
(b) ∫ x n dx = x n + 1
n + 1 + c such as n ≠ 1 and c are
Step 4: Add the constant c with the integrals.
(b) log 5 x = −2
(b) 200° into radian unit.
x = 5 −2
m + n = log a xy
browse website or 270 constants. 27 = 9 x For example, ∫ x n dx = x n + 1 n + 1 + c 3.1.1 3.2.1 3.2.2 ❶ ÷ ❷: log a x + log a y = log a xy O r 1 radian B r [Use π = 3.142]
3 3 = 3 2x
= 1
Solution:
2x = 3
a m
5 2
a n = x
(a) 180° = π rad
(b)
r
x = 3
y
video related to the 2 = 1 25 5. Hence, angle subtended about the centre of a 30° = 30° × π 180°
a m − n = x
= π rad
6
y
log a a m − n = log a x
(c) log 2 (x 2 + 3x) = 2
64
y
x 2 + 3x = 2 2 (d) log x 1 = −6 circle, ˙AOB is 1 radian if the arc length of AB is Alternative Method ALTERNATIVE METHOD
subtopics learned. (x + 4)(x − 1) = 0 64 1 = x −6 (m − n) log a a = log a x y 30° = 180° = π rad
equal to the radius of the circle.
Substitute 180° = π into the expression,
x 2 + 3x − 4 = 0
AB = OA = OB = r
6
6
64 −1 = x −6

x = −4 or x = 1
m − n = log a x
(2 6 ) −1 = x −6 6. The relationship between the measurement of (b) 180° = π rad
y
2 −6 = x −6 angle in radian with degree is 200° = 200° × π
x = 2 log a x − log a y = log a x 2π rad = 360° 180°
y
Try question 5 in Formative Zone 4.3 π rad = 180° 200° = 200° × 3.142
180°
7. The conversion of angle measurement in the degree = 3.49 rad Provide alternative
Example 26 Try question 2 and 3 in Formative Zone 1.1
to the radian and vice versa are as follow:
Proving laws of logarithms Prove that log a x n = n log a x.
Calculator
1. The following is the laws of logarithms. Solution: × 180° π Recheck the answer in Example 2(b) by using solutions to certain
• Product rule
CALCULATOR • Quotient rule Let m = log a x (a m ) n = x n Radian Degree calculator,
log a xy = log a x + log a y

a m = x
2. The screen will display 3.490658504 questions.
1. Press 2 0 0 × SHIFT EXP ÷ 1 8 0 =
× π

180°
log a x = log a x – log a y
log a a mn = log a x n
(mn) log a a = log a x n
y (log a x)(n log a a) = log a x n 4 Form
• Power rule n log a x = log a x n Additional Mathematics Chapter 5 Progressions 217

log a x p = p log a x 1.1.1 Example 30
Explains how to use 2. From the laws of logarithms, the following can Example 27 Find the sum of the first 8 terms of the For Example 31, the sum from 5th term to 9th term
be derived:
following geometric progression.
• log a 1 = 0 Given that log 3 2 = 0.631 and log 3 5 = 1.465, is S 9 – S 4 . S 9
find the value for each of the following. 2 , …
(a) 14, 2, 7
• log a a = 1
a scientific calculator • log a a r = r (a) log 3 10 (b) 1.6, 3.2, 6.4, … T 1 + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + T 8 + T 9
S 4
(b) log 3 2.5
• log a 1 = –log a b (c) log 3 45 Solution: S 9 – S 4
b
in mathematics • log 1 b = –log a b (d) log 3 6 5 (a) a = 14, r = 2 = 1 7 |r|  1
a
14
4.3.1 4.3.2 Sum of the first 8 terms, S 8 Example 32
69
1 8
calculations. = 14  1 –  7   The term of a geometric progression is given
by T n = 2 1 + n . Find
(a) the first term and the common ratio,
Form CHAP = 16.33 1 – 1 7 (b) the sum of the first 8 terms,
of the geometric progression.
5 Additional Mathematics Chapter 6 Trigonometric Functions 5
Solution:
(b) a = 1.6, r = 3.2 = 2 |r|  1 (a) First term, T 1 = 2 1 + 1
Example 19 Example 20 1.6 = 4
Given f(x) = 4 cos 2x for 0 < x < 2π. Sketch the graph of the following trigonometric 2nd term, T 2 = 2 1 + 2

Sum of the first 8 terms, S 8
(a) State the period of the graph function y = f(x). functions in the given range. = 8
Hence, state the number of cycle of the graph (a) y = sin x + 1 for 0 < x < 2π = 1.6(2 8 – 1) Common ratio, r = 8 = 2
2 – 1
in the given range. (b) y = –2 cos x for 0 < x < 2π 4
(b) State the amplitude of the graph. (c) y = | tan x | for 0 < x < 2π = 408 Thus, the first term and the common ratio
(c) Write the coordinates of the maximum and the (d) y = | cos 2x | + 1 for 0 < x < 2π Try question 21 in Formative Zone 5.2 are 4 and 2 respectively.
minimum points.
(d) Sketch the graph of y = f(x). Solution: (b) Sum of the first 8 terms, S 8
(e) Using the same axes, sketch the graph of (a) y = sin x + 1 for 0 < x < 2π = 4(2 8 – 1) BRILLIANT TIPS
2 – 1
function y = –|4 cos 2x| for 0 < x < 2π. 1 Sketch the basic graph, y = sin x. Example 31 = 1 020
EXAMPLE Compare f(x) = 4 cos 2x with the basic cosine 2 The graph moves 1 unit upward, such that Try question 25 to 27 in Formative Zone 5.2
Solution:
It is given that –9, 27, –81, … is a geometric
0
progression. Find the sum from 5th term to 9th
translation ( ) .
function, f(x) = a cos bx + c.
1
term of the geometric progression.
(a) Period 2π = π or 180°. Number of cycle, b = 2.
2
Example 33
(b) Amplitude, a = 4 y Solution: The sum of a geometric progression is given by Useful tips to help
(c) Maximum point: (0, 4), (π, 4) and (2π, 4).
CHAP. Minimum point: ( π , – 4) and ( 3π , – 4) 2 y = sin x + 1 a = −9, r = − 27 = –3 S n = 3(2 n ) − 3. Find
9
2
2
(a) the sum of the first 5 terms,
Examples with 6 (d) To sketch graph function y = 4 cos 2x: 1 O π 3π S 4 = −9(−3) 4 − 1) (b) the 7th term of the geometric progression.
students solve
Number of class = 2 × 2 × 2 = 8

x
−3 − 1
Solution:
2π
Size of class interval = 2π = π 4 –1 π – 2 y = sin x –– 2 = 180 (a) S 5 = 3(2 5 ) − 3
8 = 93
complete solutions x 0 π 2 π 3π 2 2π –2 S 9 = −9((−3) 9 − 1) (b) S 6 = 3(2 6 ) − 3 problems in the related
−3 − 1
= 189
y 4 – 4 4 – 4 4 (b) y = –2 cos x for 0 < x < 2π = − 44 289 S 7 = 3(2 7 ) − 3
Thus, the graph function of y = 4 cos 2x: 1 Sketch graph of y = cos x. The sum from 5th term to 9th term = 381
to enhance students' y 2 Reflect the graph at 1 on x-axis to make 7th term, T 7 = S 7 − S 6 subtopics.
= S 9 − S 4
= − 44 289 − 180

381 − 189
the graph of y = – cos x.
=

= 192
4 y = 4 cos 2x = − 44 469 Try question 28 to 30 in Formative Zone 5.2
Try question 22 to 24 in Formative Zone 5.2
understanding of the 2 O π 3π 2π x 2 y
–2 π – 2 –– 2 1 y = – cos x 94 5.2.3
chapters learned. –4 O π 3π –– 2π x
(e) Steps in sketching the graph of y = –|4 cos 2x| –1 π – 2 y = cos x 2
1 y = |4 cos 2x| is a reflection of graph on –2
negative side of x-axis.
2 y = –|4 cos 2x| is a reflection of graph at 1 3 The value of a is –2. The maximum value is
on x-axis. (π, 2) and the minimum value is (0, –2) and

y (2π, –2).
4 y

2 2 y = –2 cos x
x 1
O π 3π 2π
–– y = – cos x
–2 π – 2 2 x
O π 3π 2π
–4 π – 2 –– 2
y = –|4 cos 2x| –1
–2
Try question 3 in Formative Zone 6.3
354 6.3.1
iv
Prelims Spotlight Add Math Tg4&5.indd 4 23/04/2021 2:10 PM