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Topic 1 Algebraic
Expressions
SPM Notes
1. In algebra, like terms can be
combined into a single term by
addition or subtraction
a.
3x 5x 8x
b.
4y y 3y
c.
5x2 6x2 x2
d.
4aa3b5b 3a8b
2. Expansion is the process of getting
the product of
(a) a term with an expression,
i.
x(x y) x2 xy
ii.
3r(s t) 3rs 3rt
(b) an expression with another
expression
i.
(y 5)(y 2) y2 3y 10
ii.
(pq)2 (pq)(pq) p2 2pq q2
SPM CLONE QUESTIONS
A. Simplify:
1.
(x 2)(x 3) x(x 2)
2.
m(m2n) (m n)2
3.
2x(x 3y) (3x y)2
4.
(3m n)2 m(m2n)
B. Simplify the following
1.
m2 n2 (m n)2
2.
3x2 4y2 (x 2y)2
x 2y
3.
(3m)2 (2m)(3m)
4.
4 x2 (1 x)2
5.
(3 p)2 p(p 3)
6.
p(p 4) (p2)2
7.
5mn3(1mn)
8.
(2x 3y)(x 2y)
9.
4m(m2n) (4m n)2
10.
(3x 4)(2x 3)
11.
5p2 p(1 p)
12.
(2x 3y)2 x(x 2y)
Topic 2 Algebraic Fraction
SPM NOTES
1. The rules of simplifying algebraic
fractions and numerical fractions are
the same
2. To simplify two algebraic fractions
by addition or subtraction, use the
following steps:
i. Find the LCM of the two different
denominators.
ii. Express the two fractions having the
common denominators
iii. Add or sebtract the two numerators.
iv. Simplify the answer to its simplest
form.
SPM CLONE QUESTIONS
A. Simplify these fractions as a single
fraction in its simplest form.
1.
3x y
xy9 y
3y
2.
3
m 32m 3
m(m 3)
3.
m
3n3m
n
B. Simplify
1.
x
3x 2 1
3x
2.
2m 1
mn6 n
3n
3.
6
y 26 3y
y(y 2)
4.
6 x
3xx 2y
xy
5.
1
3d6 e
15de
6.
m 1
4mn3 4n
12n
7.
1
6e3 e
3e2
8.
m 5
5m2m
m
9.
2
3x4x 1
9x 2
10.
m
4n1m
n
11.
(5n2 mn)2m
n2(m 5n)
12.
p 3
p2p1
p
13.
x 5
x2(3 x)
x2
Topic 3 Linear Equations
SPM NOTES
Basic rules in solving linear equations
a.
x 25
x 52
x 7
b.
x 3 7
x 73
x 4
c.
x
4 3
x 3(4)
x 12
d.
3x 6
x 6
3
x 2
SPM CLONE QUESTIONS
1. Solve each of the following
equations
a.
x 2
3x 5
b.
m
3 4
m
5
c. Given that
1
3n 2 5 , find the
value of n.
d. Given that
3
25x 3(2 x) , find
the value of x.
e. Given that
x 2
5 3x 2, find the
value of x.
2. Find the solution for each question.
i. Given that
m 1
3m 2, find the
value of m
ii. Given that
y
2 3
y
5, find the value
of y
iii. Given that
5p 31
2(p12), find
the value of p
iv. Given that
21
3y 5, find the
value of y
v. Given that
1
3(12m) 3 2(m 1) ,
find the value of m
vi. Given that
2x
35
1
3, find the
value of x
vii. Given that
2y
3 y 2, find the
value of y
viii. Given that
21
3x 4, find the
value of x
ix. Given that
2p 7 4(2 p), find
the value of p
x. Given that
13 3(2 y) 9y 4
find the value of y
xi. Given that
x 1
3 2x 3
xii. Given that
y 32(12y)
3,
calculate the value of y
Topic 4 Simultaneous Linear
Equations
SPM NOTES
To solve simultaneous linear equations
we can use one of this three methode:
a. Elimination method
b. Substitution method
c. Matrices method
SPM CLONE QUESTIONS
A. Solve the following linear equations
1.
2x y 7
3x 2y 11
2.
1
2x 3y 10
5x 6y 8
3.
m2n 1
3m5n 25
4.
2d 2
3e 2
6d 3e 8
5.
x 2y 9
3x y 13
6.
1
2m 2n 11
3m 4n 14
7.
3v 2w 12
6v w 9
8.
h 3k 3
2
3h k 5
9.
m2n 6
2m3n 16
10.
2h k 7
h 1
3k 3
Topic 5 Algebraic Formulae
SPM NOTES
1. The subject of a formula is a
variable that is expressed in terms of
other variables.
(a). In the formula
E mc2 , E is the
subject of the formula, expressed in
terms of m and c.
(b). In the formula
m mp r , m is
not the subject of the formula as m is
found on both sides of the formula.
2. Changing the subject of a formula
involves the rearrangement of the
formula so that a specific variable
becomes the subject of the formula
SPM CLONE QUESTIONS
A. Solve the followings.
1. Given that
x y(4 x) y express
x in terms of y
2. Given that
3
m n
n
4, express n in
terms of m
3. If
x
34x 1
y then
y
4. Given that
G 1
4
x
H , express x in
terms of G and H
B. Solve the followings
1. If
x
y y 3x , then x =
2. If
m 3 n 8m
5, then n =
3. If
m
n 5 m, then m =
4. If
p
q5
2q 3p, then p =
5. If
g 3x 2y
x, then x =
6. If
gh
g 5h 3, then h =
7. If
x 5
y xy , then x =
8. If
Mx
3y
x , then y =
9. If
2pq p
q5 then p =
10. Given that
G 81
H N
,
express H in terms of G and N
11. If
x
45x 1
y, then y =
12. Given that
v w
3 w, express w in
terms of v.
13. Given that
5m 2n
3 mn m ,
express m in terms of n
14. Given that
M 1
5
w
N, express N
in terms of w and M.
Topic 6 Linear Inequalities
SPM NOTES
1. An inequality is a relationship
between two unequal quantities.
2.
Symbol Meaning
> Greater than
< Less than
Greater than or equal to
Less than or equal to
3. Representation of an inequality on a
number line
(a)
x 2
-3 -2 -1 0 1
The empty circle ‘o’ indicates that the
values of x do not include -2
(b)
x 2
- 1 0 1 2 3
The solid circle ‘’ indicates that the
values of x include 2
4. To solve a linear inequality in one
unknown is to find its equivalent
inequalities in the simplest form.
x 1 3
x 3 4
x 7
5. The symbol of inequality is reversed
when both sides of an inequality is
multiplied or divided by a negative
number
(a)
2x 10
x 10
2
x 5
6. Solution for two simultaneous linear
inequalities in one unknown is to find
the common values or equivalent
inequalities that satisfy both the
inequalities.
(a)
x 2
x 0
The common solution for
x 2 and
x 0 is
x 0
(b)
x
3 5
x 5 (3)
x 15
-3 -2 -1 0 1
(b)
x 2
x 0
The common solution for
x 2 and
x 0 is
2 x 0
(c)
x 0
x 2
The common solution for
x 0 and
x 2 is
x 2
d.
x 2
x 0
No common values hence no solution
for
x 2 and
x 0.
SPM CLONE QUESTIONS
A. Solve the followings
1. Given that x is an integer, list all the
values of x which satisfy both the
inequalities
5 x 1 and
x 2x
2
2. Given that
p x q satisfies both
the inequalities
32x
31 and
72x 5, find the value of p and of q.
3. List all the integers which satisfy
both inequalities
x
41 x and
1
2(x 3) x
-3 -2 -1 0 1
-3 -2 -1 0 1
-3 -2 -1 0 1
3. The solution for
3x 4 13x
4 is
B. Solve the followings
1. The smallest integer value of x
which satisfies the inequality
x 6 1
3(2 x) is
2. Given
p x q satisfies both the
inequalities
5x
2 4 and
4(x 3) 30 x , find the value of
p q
3. If
7 32m8m, then
4. If
5y 3 7 2y 9, then
5. Given that
p x 3 satisfies both
the inequalities
2 x 1 and
x
3 3 4 , find the value of p
6. List all the interger values of x
which satisfy both the inequalities
3x 22 and
5 x 3
7. List all the integer values of y which
satisfy the inequality
2y 7 15 3y 9
8. List all the integer value of x which
satisfy both the inequalities
5x 9 3(x 1) and
6x
2 2
9. The diagram represents two
simultaneous inequalities on a number
line
The inequality which represents the
common values of both the inequalities
is
10. List all the integers values of x
which satisfy both the inequalities
2x 117 and
3x
4 9
11. List all the integer values of x
which satisfy both the inequalities
x 6 and
152x 7
12. Find the solution for
x
21 2x 7
-3 -2 -1 0 1
13. List all the integer values of x
which satisfy the inequalities
3x 6 x 7 4x
14. List all the integer value of m
which satisfy both the inequalities
m
51 m and
1
2(m 3) m
15. The solution for
3x 5 23
4x
16. Draw a number line to represent
the solution of the simultaneous linear
inequalities
3x 15 and
2 x 1
Topic 7 Indices
SPM NOTES
Laws of Indices
am an amn
an 1
an
am an amn
(am)n amn
a0 1
a1
n an
a1 a
am
n amn
( an )m
SPM CLONE QUESTIONS
A. Solve it
1. Find the value of
84
3 22
2. Simplify
21
2 (21
2 31
3 )3
3. Simplify
42 71
3
142
3
4. Simplify
(2eb2)3 4e3b
5. Find the value of
(856)1
3 (23 54 )
6. Given that
5x 25
52x , find the value
of x
7. Simplify
(p2)3 p4
8. Given that
6n 36
63n, find the value
of n.
9. Simplify
92 71
3
212
3
10.
5
3x 7 can be written as
A.
5
3x7
B.
5
3x 7
C.
3
5x 7
D.
15x7
11. Simplify
(m6n9)1
3 m1n4
12. Simplify
(x 2y 3)1
x5y 4
13. Simplify
( 64)1
3 1252
3
23
14. Simplify
(4 p2q)3 1
8p2q1
15. Given
1
x n
1
2
x 3
16.
k2
3 can be written as
A.
k 3
B.
k 23
C.
( k3 )2
D.
( k2 )3
17. Simplify
2n2 (4 p2)1
2
(n6p3)1
3
18. If
1
k n
2
k6 , then the value of n
is
19.
(mp2)3 (m1p4)
20. Given that
58 5p 52, therefore
p =
21.
(81m6)1
4 m1
2
