Page 3 - Spotlight A+ Form 4 & 5 Mathematics KSSM

Form
Chapter 1 Quadratic Functions and Equations in One Variable Mathematics 4
1.1 CHAP. 1 alternative method
1. 9y 2 + 16 x 2 – 2y 2 + 4 6. Diagram shows the graph y = x 2 . On the
diagram, draw the graphs y = x 2 + 4 and
7
—– – 20 1 — k 2 5 y = x 2 – 3.
v 2
Identify quadratic expressions in one variable y
from the list of expressions above. C1 8
2. Determine whether each of the following 6 Provides alternative solutions to
expressions is a quadratic expression in one
formative zone variable. Give your justification. C2 y = x 2 4 2 x certain questions.
(b) 1
—– – 6
(a) 7c 2 + 3
y 2
(c) a 2 + 4b 2 + 9
(d) 3p 2 – 10p + 5
3. Given the quadratic function y = –x 2 + 4. C2 –2 –1 –2 0 1 2
(a) Explain why the quadratic function is a Make generalisation about the effect of
many-to-one relation.
(b) Sketch the graph y = –x 2 + 4 by describing change for the values of c on the graph
y = x 2 + c.
C4
Questions to test 4. On one diagram, draw the graph y = ax 2 for 7. Diagram shows the graphs y = x 2 , y = x 2 – 6x
the characteristics of the quadratic function.
a = 1 — and a = – 1 — where –3 < x < 3. Hence, and y = x 2 + 6x. Form 4 Mathematics Chapter 1 Quadratic Functions and Equations in One Variable
2
2
students’ understanding at determine the relation between the graphs y = x 2 + 6x y y = x 2 y = x 2 – 6x CHAP.
y = 1 —x 2 and y = – 1 —x 2 .
Alternative Method
2 2 C4 0 x 1 30 × 20 – 1 —(30 – x)(20 – x) – 1 —x(20) = 388 y = ax 2 + bx + c
2
2
6
the end of each subtopic. 5. Draw the graphs y = 3 —x 2 and y = 4x 2 on the Make a generalisation about the effect of 600 – 1 —(600 – 50x + x 2 ) – 10x = 388 x = 1, y = 0: 25a – 5b + c = 0 ............... 
3
–6
–3
a + b + c = 0 ............... 
2
2
x = –5, y = 0:
following diagram.
2
y change for the values of b on the graph 600 – 300 + 25x – 1 —x 2 – 10x = 388 x = –2, y = –18: 4a – 2b + c = –18 ........... 
 – ,
24a – 6b = 0
16 y = x 2 + bx. C4 300 + 15x – 1 —x 2 = 388 4a – b = 0 ................................
2
 – , 3a – 3b = –18
14 8. In the diagram, PQRS is a trapezium. 1 —x 2 – 15x + 88 = 0  – , a – b = –6 ..............................
2
3a = 6
12 S R x –8 –8x x 2 – 30x + 176 = 0 a = 2
(x – 8)(x – 22) = 0
10 (x + 5) cm x –22 –22x x = 8 or x = 22 From , b = 8
©PAN ASIA PUBLICATIONS
From , 2 + 8 + c = 0
8 x 2 +176 –30x c = –10
P Q y = 2x 2 + 8x – 10
6 When x = 22, 20 – x = 20 – 22 = –2. Length of NP
The length of PQ is 4 cm more than RS. Try question 16 in Formative Zone 1.1
4 C3 is a positive quantity. Therefore, x = 8.
2 y = x 2 (a) Form a quadratic function to represent the Try question 15 in Formative Zone 1.1
area of the trapezium PQRS.
–2 –1 0 1 2 x (b) Hence, find a quadratic equation in the Example 17
Hence, complete the following generalisation form ax 2 + bx + c = 0, given the area of BRILLIANT Tips A quadratic function y = ax 2 + bx + c has the
following information.
Form about the effect of change for the values of a the trapezium PQRS is 323 cm 2 . 1. Represent the desired quantity to be found by a • The axis of symmetry is x = 2.
C4
4 Mathematics Chapter 1 Quadratic Functions and Equations in One Variable values of x is the root of the quadratic equation suitable symbol such as x. • The graph cuts the x-axis at x = –1.
9. Determine whether each of the following
on the graph y = ax 2 .
When the value of a increases, the graph

• The graph has a maximum point.
x 2 – 4x – 5 = 0. C2
given information.
y = ax 2
CHAP. 1 10. Determine the roots for each of the following . 15. In the diagram, KLMN is a plot of land in the 2. Form a quadratic equation in terms of x from the Sketch the graph y = ax 2 + bx + c.
(a) x = 2
3. Solve the quadratic equation by the method of
spm simulation hots quadratic equations by the method of shape of a trapezium. (b) x = –1 4. Check the roots of the quadratic equation to Solution: y
factorisation.
(c) x = 5
C3
factorisation.
M
determine the value of x that represents the
(a) x 2 – 3x – 4 = 0
(b) 2x 2 + x = 10
quantity in the problem.
(c) (4x + 1)(x – 2) = –5
(d) (3x + 4)(x – 2) = 16 – 3x
questions 11. Sketch the graph for each of the following 15 m N Q x m 9 Diagram shows the graph of a quadratic function. –1 O 2 5 x
Example 16
quadratic functions. C2
(a) y = x 2 + 5
(b) y = 3x 2 + 4
y
P L
(d) y = –2x 2 – 3
(c) y = –x 2 + 1
K x m
12. Sketch the graph for each of the following Given KL = 10 m and LM = 25 m. The shaded –5 O 1 x
quadratic functions. Mark the points where region is a fish pond with an area of 188 m 2 . Try question 17 in Formative Zone 1.1
the graph cuts the x-axis. C2 Calculate the possible values of x. C5 (–2, –18)
(a) y = x 2 – 2x
Provide a complete solutions with 16. Diagram shows the graph of the quadratic Determine the quadratic function in the form BRILLIANT Tips
(b) y = –x 2 – 4x
function y = ax 2 + bx + c.
y = ax 2 + bx + c.
(c) y = 2x 2 – x
(d) y = –4x 2 + 8x y (2, 48) Solution:
the examiner’s comments for the y = a(x – 1)(x + 5) 1. The maximum point lies on the axis of
13. Sketch the graph for each of the following
symmetry.
When x = –2, y = –18,
quadratic functions. Label its maximum point
2. The distances of the points of intersection of the
or minimum point. C2
–18 = a(–2 – 1)(–2 + 5)
(a) y = 3(x – 2) 2 –2 O 6 x –18 = a(–3)(3) graph and the x-axis from the axis of symmetry
are equal.
–18 = –9a
SPM simulation HOTS questions. 17. A quadratic function y = ax 2 + bx + c has the a = 2 x = 1 ⇒ x – 1 is a factor
BRILLIANT Tips
(b) y = –(x + 4) 2
Determine the values of a, b and c.
(c) y = –x 2 + 6x – 9
C5
y = 2(x – 1)(x + 5)
(d) y = 4x 2 + 24x + 36
= 2x 2 + 8x – 10
14. Sketch the graph for each of the following following information. C5 = 2(x 2 + 4x – 5) x = –5 ⇒ x + 5 is a factor
quadratic functions. Mark the points where the
graph cuts the x-axis and the y-axis. C4 • The minimum point is (–3, –2).
• One x-intercept of the graph
(a) y = –(x – 2)(x + 4) y = ax 2 + bx + c is –2.
(b) y = (2x – 5)(x – 1) 8 1.1.8
(c) y = 3x 2 – 17x + 24 (a) Sketch the graph.
(d) y = –4x 2 – x + 18 (b) Determine the values of a, b and c.
Form
4 Mathematics Chapter 3 Logical Reasoning
SPM Simulation HOTS Questions EXAMINER’S
COMMENT
1. An aquarium in the shape of a cuboid has length (x + 9) cm, width x cm and height 50 cm. The volume Paper 1
of the aquarium is 31 500 cm 3 . Calculate the value of x. C3
1. Which of the following sentences is a statement? 6. Which of the following shows a false statement
Examiner’s Comments: C1 A What is the value of 100? C2 is changed to a true statement by using the
Volume = 31 500 CHAP. B The square of 8 is 64. word “no” or “not”?
(x + 9)(x)(50) = 31 500 50 cm 3 C Multiply both sides of the inequality –y  –2 A The symbol π is a rational number.
x 2 + 9x = 630 by –1. The symbol π is not a rational number.
x 2 + 9x – 630 = 0 D Factorise the quadratic expression x 2 + 4x – 5. B x 2 = 8 is a quadratic equation.
(x – 21)(x + 30) = 0 x 2 = 8 is not a quadratic equation.
x = 21 or x = –30 x cm 2. I The lowest common multiple of 3 and 12 C 20 000 has one significant figure.
Since x . 0, x = 21. (x + 9) cm C2 is 12. 20 000 does not have one significant figure. summative zone
II The factors of 14 are 2 and 7. D Right-angled triangles have an angle of 90°.
III Write the formula for the area of Right-angled triangles do not have an angle
parallelogram ABCD. of 90°.
IV The sum of two odd numbers is an even
10 number. 7. Which of the following compound statements is
true?
Which of the following are statements? C4 A 5 – 6 = 1 or –8 + 3 = –11
A I, II and III B 0 ÷ 1 = 0 or 1 ÷ 0 = 0 Questions of various
B I, II and IV C 4 × (–2) = –6 or (–3) × (–1) = –3
C I, III and IV D 3 2 = 6 or 2 3 = 9
D II, III and IV levels of thinking skill
SPM MODEL PAPER 3. Which of the following statements is true? 8. C4 I 0.0028 = 2.8 × 10 –2 or
7.03 × 10 4 = 70 300
A 17 × 10 –3 = 0.017
C4
B 38 000 = 3.8 × 10 3 II 1 : 2 = 2 : 3 or 40 : 56 = 5 : 8
3
Paper 1 / Kertas 1 C 2 2 + 3 2 = 5 2 III (x – 3)(2x + 1) = 2x 2 – 5x – 3 are provided to evaluate
Instruction: Answer all questions. Time: 1 hour 30 minutes D 7  3 or x 2 – 4x + 4 = (x – 2) 2
Arahan: Jawab semua soalan. Masa: 1 jam 30 minit 10 5 IV 30 ÷ 0.01 = 300 or 0.006 × 1 000 = 60
4. Determine the statement that is true. Determine the false compound statements. the understanding of
1. Round off 3.04856 correct to three significant
A All quadrilaterals have four sides of the same
5. Express 243 7 as a number in base three.
C4
spm model paper figures. C 3.048 Ungkapkan 243 7 sebagai satu nombor dalam asas tiga. B All quadratic equations have two positive A I and III each chapter.
length.
B II and III
Bundarkan 3.04856 betul kepada tiga angka bererti.
A 10210 3
C 12010 3
B 11210 3
C II and IV
A 3.04
roots.
D 12110 3
B 3.05
D III and IV
D 3.049
D Some cuboids have a square base.
A 101111 2
560 × 10 5
2. ————— = 6. 1011101 2 – 100110 2 = C 110111 2 C Some proper fractions are greater than 1. 9. Determine the true compound statement.
3 = 0.6 and 0.7 = 7
B 110011 2
C4
0.007
D 111011 2
A 8 × 10 6 7. In Diagram 3, QT is a tangent to the circle PQR 5. Which of the following is true? A 5 10
C4
Statement
SPM format questions B 8 × 10 7 with centre O at Q. PORT is a straight line. A All obtuse angles lie Truth value B 6 : 12 = 1 : 2 and 3 : 2 = 9 : 4
C 8 × 10 8
C 3 –1 = 1 and 49 = 7 –2
False
Dalam Rajah 3, QT ialah tangen kepada bulatan PQR
3
D 8 × 10 9
dengan pusat O pada Q. PORT ialah satu garis lurus.
between 90° and 180°.
D (–1) 2 = –1 and –4 = –2
3. Diagram 1 shows a cylindrical water pipe with
according to the latest SPM Q B All triangles have the True
radius 2 m.
same area.
Rajah 1 menunjukkan sebatang paip air yang
berbentuk silinder dengan jejari 2 m. 27° C Some even numbers are True
P O R x° T prime numbers.
2021 assessment format D Some multiples of 3 are False
7 m Diagram 3/ Rajah 3 SPM MODEL PAPER divisible by 5.
Diagram 1/ Rajah 1
cover all the chapters in Calculate the volume, in cm 3 , of the water pipe. Find the value of x. 58
Hitung isi padu, dalam cm 3 , bagi paip air itu.
Cari nilai x.
3 Use/ Guna π = 22 4 —– 7 A 27 C 46
D 63
B 36
Forms 4 and 5. A 1.76 × 10 7 C 1.76 × 10 8 8. In Diagram 4, T is the midpoint of PQ. It is
D 8.8 × 10 8
B 8.8 × 10 7
4. In Diagram 2, PQRSTU is a regular hexagon. given that cos x° = 3 — 5 and sin y° = 7 — 8 . answers
PQH and PRL are straight lines such that Dalam Rajah 4, T ialah titik tengah bagi PQ. Diberi
PH = PL. bahawa kos x° = 3 — dan sin y° = 7 —.
Dalam Rajah 2, PQRSTU ialah sebuah heksagon 5 8
sekata. PQH dan PRL ialah garis lurus dengan R
keadaan PH = PL.
T S
L y° Complete answers
P Q
R T
U 64° K 6 cm x°
x° are provided.
P Q H S Diagram 4/ Rajah 4
Diagram 2/ Rajah 2 Find the length, in cm, of PR.
Calculate the value of x. Cari panjang, dalam cm, bagi PR. Scan the QR Code
Hitung nilai x. A 12 C 16
A 121 C 133 B 14 D 20
B 129 D 138 ANSWERS Complete answers
http://bit.ly/3vmOen4
429 429 provided to get
FORM 4
Chapter 1 Quadratic Functions and Equations in One 6. y = x 2 + 4 8 y
Variable the steps to the
KMODEL Spotlight A+ Mathematics F5.indd 429 05/03/2021 2:50 PM 6
Question Bank 1. 9y 2 + 16, 1 k 2 1.1 y = x 2 4 2 solution.
5
2. (a) Yes; One variable, c and the highest power of c x
is 2.
Chapter 1 Quadratic Functions and Equations in One Variable (b) No; The power of y is not a whole number. –2 –1 –2 O 1 2
1. Which of the following characteristics of quadratic function is not correct for f(x) = x 2 + 9? (c) No; Two variables, a and b. y = x 2 – 3
(d) Yes; One variable, p and the highest power of p
A The graph f(x) = x 2 + 9 has a minimum point at (0, 9). is 2. Graph y = x 2 + 4 is the translation of the graph y = x 2
B f is a many to one function. four units upwards, graph y = x 2 – 3 is the translation
C The axis of symmetry for the graph f(x) = x 2 + 9 is x = 0. 3. (a) Two values x = –1 and x = 1 are mapped to one of the graph y = x 2 three units downwards.
D The graph f(x) = x 2 + 9 cuts the x-axis at x = ±3. (b) value y = 3. y 7. The axis of symmetry x = – b of the graph y = x 2 + bx
2. Which of the following graphs is correct about the value of a on the graph of the quadratic function y = ax 2 ? 4 changes with the value of b. 2
A y C y 2 y = –x 2 + 4 8. (a) (x 2 + 12x + 35) cm 2
(b) x 2 + 12x – 288 = 0
y = x 2 y = 4x 2 x 9. (a) No (b) Yes (c) Yes
–3 –2 –1 O 1 2 3
y = x 2
y = 3x 2 –2 –4 10. (a) x = 4 or x = –1 (b) x = – 5 or x = 2
2
question bank O x O x 4. Maximum point (0, 4), axis of symmetry x = 0 11. (a) 4 y (b) 3 y FORM 4 ANSWERS
(c) x = 3 or x = 1
(d) x = 8 or x = –3
y
B y D y 4 y = x 2 1 – 2 5 O x 4 O x
y = x 2 y = x 2 2
Drill questions with –3 –2 –1 –2 O 1 2 y =– x 2 3 1 – 2 x (c) y (d) y
O x O x –4 1 x –3 O x
complete answers can be y = – x 2 1 – 2 y = –5x 2 Graphs y = 1 x 2 and y = – 1 x 2 are congruent and O
2
2
reflected on the x-axis. y
5.
y
obtained by scanning the 3. The diagram below shows a right-angled triangle. 16 12. (a) y (b)
(x + 10) cm (x + 5) cm 14 12 O 2 x –4 O x
QR Code on the cover of 25 cm 10 y = 4x 2
Based on the given information, find the quadratic equation that can be formed. 8 6 y = x 2 3 – 2 (c) y (d) y
the book. A x 2 + 15x – 250 = 0 C 2x 2 + 15x – 500 = 0 4 O x O 2 x
D 2x 2 + 15x – 750 = 0
B x 2 + 30x – 500 = 0
4. Which of the following is not a quadratic equation? 2 y = x 2 1 – 2
O
A p 2 + 8 = 3p C x 2 + 3 = 4 increases steeper 1 2 x
–2
–1
x
B k = 10 – 3k 2 D 13 + 9w – 3w 2 = 0 451
7
5. Given 4 is a root of the quadratic equation ax 2 + 14x – 8 = 0, determine the value of a.
A –4 C 3
B –3 D 4
1
v
Extra Features Spotlight A+ Mate F4.indd 5 15/03/2021 3:14 PM