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SECONDARY 1 ALGEBRA
1.
Simplify the following, showing your working clearly. 4 c+3 d ÷ 2+5 d ÷ 10 (a) (b)
w−5 v 7 w−5 v − −1 2 3
2.
(a)
Factorise the following completely. mx+3 mxy (i) (ii)
2 ax−bx−6 ay+3 by
(b) Solve the following equation. 5 ( x−1 )−2 ( 1−x )=4
Answers :
3.
(ai)
_______________
[1]
(aii) _______________
[2]
(b)
[2]
_______________
A book costs $2.50 and a magazine costs $7.00. If Glen wants to buy q
books
and (q+ 3) magazines, write down and simplify an algebraic expression for the amount, in dollars, that he has to pay for the books and magazines.
$
Answer :
[2]
______________
4.
Write down and simplify an algebraic expression for each of the following: (a) The product of 3 x and 5 y (b) Subtract the sum of 3 ab and 5 a+ba from 10−7 ab
Answers :
5.
(a)
Factorise
(a)
_________________
[1]
(b)
_________________
[3]
pq−kq
(b) Hence, find the value of 3320 ×45−2320 × 45 without the use of a calculator.
Answers :
6.
(a)
________________
[1]
(b)
________________
[2]
(a)
____________
[2]
The base of a triangle, y, is 4 times its height. (a) What is its area in terms of y? (b) Find the area of the triangle if the base is 16 cm.
Answers :
cm 2 (b)
____________
cm
7.
(a)
[1]
2
A piece of wire is cut into 2 equal parts. One part is bent to form a triangle with sides 4 x cm, (x−22) cm and 2 x cm while the other part is used to form a rectangle with breadth (x−2) cm and length twice as long as its breadth. (i) Write down and simplify an expression for the (a) perimeter of the triangle (b) perimeter of the rectangle (ii Hence, find the value of x . )
[1] [1] [2]
8.
Solve the following equations (a) 2 ( x+1 ) 2 x +3 −x=1− 3 2
[3]
x+ 7 1 =1 4 2( x−7)
[3]
(b)
9. Express the following word statements in terms of algebraic expressions. (a) Add 21 to the product of 3x and y. (b) Subtract 3c from the quotient of a divided by b. (c) Multiply 4 by two thirds of d.
Ans: (a) __________________ [1] Ans: (b) __________________ [1] Ans: (c) __________________ [1]
3x 2 y 7 10. (a) Subtract (
(b) Evaluate
x y x y
6 x 8 y 13 ) from (
when
x4
).
y 3 and
.
Ans: (a) __________________ [2] Ans: (b) ________________ [2] 11. Simplify
3(4a 9b) (a)
,
10c 3(11c 6d ) (b)
, 55e 18ef 5 11e 3 f g 9f 2 g4
(c)
2
.
Ans: (a) __________________ [1]
Ans: (b) __________________ [2] Ans: (c) __________________ [2]
12.
Write algebraic expressions for each of the following statements. (a) The cost of x litres of petrol at $1.90 per litre. (b)
Half the product of m and the square of v.
Ans wer:
(a) $ ____________________ [1] (b) ______________________ [1]
13. Mrs Fields sold pens at 55 cents each and markers at $2 each. She sold 40 pens and markers altogether and collected a total of $59.70. (a) If Mrs Fields sold x pens, express in terms of x for the number of markers that she has sold. [1] (b) Form an equation in x connecting the cost of pens and that of the markers that were sold by Mrs Fields. (c)
Solve the above equation to find the number of markers.
[2] [4]
14. Factorise the following
3( x 7 y ) 2a( 7 y x) (a) 2 x 3 y 2 8 x 2 y 2 8 xy 3 (b) 3x 2 9 xz 9 xy 27 yz (c)
Ans: (a) __________________ [1] Ans: (b) __________________ [2] Ans: (c) __________________ [2]
15.
If a = –2, t = 3, u = 4, v = –5, evaluate the following.
(a)
vu t ut
(b)
1 2 at 2
Answer:
(a) ______________________ [1] (b) ______________________ [2]
16. Solve the following equations.
5 x (3 5 x) 4 x 6 (a)
.
5 (b)
2 x 1 3x 4 4 5
Ans: (a) __________________
[2]
.
Ans:
__________________
[3]
17. (a) Solve the following equations.
(i)
3 7 x 2 3x
[2]
3( y 7) 4 35 y (ii)
[3] k
(b) Given that
3l 2m 1
, find m when k = 9 and l = 30.
[3]
18.
(a) Factorise 4x2 – 16x completely. (b) Factorise 5p2 – 4qr + 5pr – 4pq by grouping.
Answer:
(a) ______________________ [1] (b) ______________________ [2]
19.
(a)
(b)
Subtract
Simplify
3x 2 5 x 1
x 1 x 2 3 4
from
7x 3 x 2 4x 6
.
Answer:
(a) ______________________ [2] (b) ______________________ [3]
3(2 x 1) 2( x 3) 20.
(a) (b)
Expand and simplify Simplify
(c)
Simplify
.
8a 3 b 4 a b 2a
2
4( x 1) 3
, leaving your answer in the simplest form.
.
, express your answer as a single fraction.
[2] [3]
[2]
21. Simplify the following: 2a 3b 6ab (a) 15 xz 3y (b)
Answer:
(a)
[2]
(b)
[1]
(a)
[1]
(b)
[2]
22. Simplify 9 x 5 xy 2 xy 4 x (a) 3 p 2 q 5 p (b)
Answer:
7 y 2x 3 23. (a)
(b)
Subtract
from 5u 3u u 6 4 2 Simplify .
10 y 3x 4 .
Answer:
24. Given that
25. Simplify
(a)
[2]
(b)
[2]
a 12 b 3 c 1 ac 2b , and , find the value of
5x x 3 3 5
Answer:
[2]
Answer:
[2]
.
27. Solve the following equations:
(a)
(b)
2x 5 9
4 7 x 3
Answer:
(a)
x=
[2]
(b)
x=
[3]
28. Factorise 36 x 2 48 x (a) 6 x x 4 3 x 4 (b) px qx 2qy 2 py (c)
Answer:
(a)
[1]
29.
(b)
[2]
(c)
[3]
Simplify (a) (b)
6 x 7
[1]
2 6 x 5 3 3x 2
[2] .
30. Simplify the following expressions. 6a (3a 4a ) (a) , 3 ( x 3 y ) 5 (3x 2 y ) 1 (b) , 3 p 2 q 4 p 5q 3 4 (c) .
Answer (a)
………………………..…. [1]
(b)
………………….……..… [2]
(c)
…………...……………… [2]
31. James went to a supermarket which sells 4 apples at $x and 3 honeydews at $
8 x 4
. He bought 8 apples and 3 honeydews. (a) Giving your answers in dollars, write down in terms of x, the cost of each (i)
apple
(ii) honeydew (b) Write down and simplify, in terms of x, an expression for the total amount of
[1] [1] [2]
money that James has to pay for the apples and honeydews that he bought. Give your answer in dollars. (c) James paid the shopkeeper $20 and received a change of $4. Form an equation, in terms of x, and find the value of x.
[3]
32. (a)
(b)
Solve
3x 2 20 4
[3] .
3 k 5 4
The diagram below shows a rectangle PQRS in which PQ = 2 1 k 1 k 3 2 3 PS = cm and QR = cm. Find the value of k. 2 k 1 cm 3
P
S
Q
R
3 k 5 cm 4
1 k 3 cm 2
[3] cm,
33 .
(12 x 3 5 x 2 3 x 6) Subtract
(10 x 4 14 x 3 4 x 2 5 x 3) from
.
Answer 34 .
…………………………….[2]
Derek bought some egg tarts and buns from a bakery. The number of egg tarts bought is x. The number of buns bought is 5 more than the number of egg tarts. x (a) Write down, in terms of , an expression for the number of buns Derek bought.
(b) Each egg tart and bun costs 80 cents and 60 cents respectively. x Find, in its simplest form in terms of , an expression for the total cost of the egg tarts and buns Derek bought. (c) If Derek paid a total of $12.80 for all the egg tarts and buns purchased, find the number of egg tarts and the number buns he bought respectively.
35 .
Answer (a)
……………...……… buns [1]
(b)
$ …………………………. [1]
(c)
….… egg tarts, .…… buns [2]
Answer (a)
………………………...…. [1]
(b)
…………………………… [2]
Factorise each of the following. 16 x 2 y 8 xy 2 (a) , 2 pc 8 pd 16qd 4qc (b) .
5 x 3(2 x 1) 2( x 1) 14 36.
Solve
.
[2]
E E E B B
B
C
C C
37 .
(a) Benson thinks of a number, multiplies the number by 5 and subtracts the result from 68. The answer is the same if he adds 4 to the number and multiplies the result by 3. Find the number. [2]
D D
A
A A
5x 2 2 y 2 x 2 6 y 2 (b) Simplify
completely.
[2]
8 y 5x 9 2x y (c) Given that
38.
39.
23x y , find the value of
.
[2]
Given that x = 2 and y = –3, find the value of (a) 5x – 4y (b) xy2
Answer: (a) ____________________
[2]
(b) ____________________
[2]
(a) Simplify the expression 2k – 3(2 – k) + 4
(b) Solve the following equations: 4 5 p (i) , 5t t 1 3 2 (ii) .
Answer: (a) ____________________
3(2a 1) 2(2a 3)
40. (a) Simplify
(b) Factorize
7 x xy 21xz 3 yz
[2]
(b) (i) p =_______________
[1]
(b) (ii) t =_______________
[3] [2] [2]
41.
Solve the following
(a)
(b)
42.
4 x 3 x 1 2 x 1 16
[3]
4k k 1 1 1 3 2 4
[3]
Alice is x years old. Tom, her brother, is 9 years older than Alice. Their mother is 3 times as old as Alice. Their father is twice as old as Tom. (a) Write down expressions, in terms of x, for (i) Tom’s age, (ii) their mother’s age, (iii) their father’s age,
[1] [1] [1]
(b) The sum of the ages of the four members of the family is 139. (i) Write down the equation satisfied by x. (ii) Solve the equation to find the value of x.
[2] [1]
pq 2 pq 1 q 43.
Factorise completely
.
[3]
a (a b) b(2a b) ab 44.
45.
(i)
Expand and simplify
(ii)
Hence, using suitable substitution for a and b from (i), 2(2 3 ) 3 ( 4 3 ) 2 3 evaluate without the use of calculator.
Simplify
2 x 2x 1 4 6
Given the formula (i) the value of
T
(ii) the values of
[2]
[4]
Q 1 R
find
P 3 Q 4 R 2 when , and ;
P
[2]
.
T P2 46.
.
T 13 Q 24 R 1 when , and .
[2] [3]
47.
Solve for the value of x for the following equations.
(a)
2x 1 1 3 4
;
[3]
8 3 3( x 1) 2( x 1) (b)
.
[4]
48
(a) Simplify 3p – 2(3 + p) + 9
.
(b) Factorise completely 7xy – 14y – 2x + 4
Answer (a) ……………..……………..….....[2] (b) ……………..…………….……..[2]
49
(a)
.
Solve
2p 5 5 3
(b )
Given that
a 11 3 5c b
, find the value of a if c = 12 and b = −4
Answer (a) ……………..……………..….....[2] (b) ……………..…………….……..[2]
50 .
Express the following as a single fraction in its simplest form. (a)
(b)
51.
2(3 y 5) 2y 5
5 x 3 3x 1 6 2
.
.
Answer (a) ……………..……………..….....[2] (b) ……………..…………….……..[3] Simplify the following algebraic expressions.
4 p 6q 7 p 15q 10 p (a)
3 ( x 1) 4 ( x 2) (b)
Answer (a) ___________________ [1] (b) ____________________ [2]
52.
Solve the following equations.
3a 6 27 (a)
(b)
2x 1 x8 3
Answer (a) _____________________ [2]
(b) _____________________ [2] 53.
Express the following as a single fraction in its simplest form.
(a)
(b)
x 2x 5x 3 5 6
4 x 3 2 x 1 5 4
Answer (a) ______________________ [2] (b) ______________________ [3]
P 54.
Given that
abc
3a 2
, find the value of P if
a 2 b 1 c 5 , and .
Answer _______________________ [2] (3 x 2 4 xy 5 y ) 55.
Subtract
(3 x 2 2 xy 6 y ) from the sum of
( 2 x 3 xy 2 y ) and
Answer
56.
Factorise the following algebraic expressions.
4a 6ab 16bc (a) y 2 (2 x 3) z (3 2 x) (b)
36 pq 18 pr 2 sq sr (c)
.
____________________ [3]
Answer (a) _____________________ [1] (b) _____________________ [1] (c) _____________________ [2]
2 x 5x 3 3
57.
Subtract
from the sum of
1 3 3 x x2 x 8 5
2 x 2 x and
1 4
.
Answer: _________________________ [2]
7 xy 2 y 8(3 y 5) 4 xy 20 3(2 y 9)
58.
Expand and simplify
.
Answer: _________________________ [2] 59.
a 2 b 1 c 3 d 4 If , , , and evaluate 3
cb d c 3b c2
2
4c b 3ab d
(b)
Answer: (a) ________________ [2] (b) ________________ [2] 60.
Factorise each of the following: (5 x 2 y ) 7 h( 2 y 5 x)
(a)
4 pr 10 sr 14 pq 35 sq
(b)
Answer: (a) ___________________________ [1] (b) ___________________________ [2] 61.
Simplify the following algebraic fractions, expressing as a single fraction
5( 2 7 x ) 7(3 x 3) 2 x 2 6 24 3
(b)
2 w 3 3w 7 4 5 6
Answer: (a) ____________________ [3] (b) ____________________ [3] 62.
Solve the following equations:
(a)
5 4 7 x 1 3x 2
(b)
7r 4 3(r 1) 2(2r 4) 5 4 3
Answer: (a)
(b) 63 .
H
2b 2 1 5b 1
Given that , calculate the value of H when b = Express your answer as (b) a decimal ,correct to 1 significant figure.
.
Simplify
20m 4 5 p 3 6m 2 p
r
__________ [2]
__________ [3]
. [2]
(a) a fraction ,
64
1 9
x
[1]
[2] .
65
Factorise the following
24kb 144km 36kp
. (a)
[1] ,
[2]
3mg 12 g 4mxz 16 xz (b)
66 .
Express
.
6 x 2 3 4 x 4 3 x 7 2 5
as a single algebraic fraction in its simplest form.
[3]
67
Solve the following equation
. (a) (b)
1 5 2 w 3 9 5w 6
[3] .
5m 12 4 2 m 17 3 m 2
[3] .
68 .
Fiona bought 25 handphone pouches and pencil cases. The cost of each pencil case is $8. Each handphone pouch cost twice as much as each pencil case. She spent a total of $264. Let the number of pencil cases bought be x. (a) Express the number of handphone pouches bought in terms of x. (b) Form an equation in x and solve x. (c) Hence, find the number of handphone pouches and pencil cases Fiona bought respectively.
[1] [3] [2]
69. Express and simplify the following statements algebraically. (a) Subtract s from the product of 3p and –2q. (b) Add the square of a to the quotient of 15a3 divided by 3a. (c) Subtract 2k – 5h from the sum of 6k + 2h – 1 and –7k – h + 5.
Ans: a) ________________________ [1] b) ________________________ [2] c) ________________________ [2]
70. (a) Given that x = 2 and y = 1, evaluate x2+ 0.2y – S
(b) Given the formula
m 1 3n 1
x 4
.
, find m if n = 3, S = 10.
Ans: a) ________________________ [1] b) ________________________ [2] 71. Simplify the following expressions. (a)
8a 9 2a 6a 2a
.
4(3 x 2 y ) 5 y 6 x (b)
.
2 p q p 2q 3 5 (c)
.
Ans: a) ________________________ [2] b) ________________________ [2] c) ________________________ [3] 72. Factorise the following expressions.
3( x 1) 2 y ( x 1) (a)
.
16 w 8wy 2 x xy (b)
.
Ans: a) ________________________ [1] b) ________________________ [2]
73. Solve the following equations.
27 (a)
9 x .
2 4(5 x 1) ( x 10) 3 (b)
(c)
.
5 2 3x 1 x 2
.
Ans: a) ________________________ [1] b) ________________________ [2] c) ________________________ [2]
74. Simplify the following. a. b.
2x y 4 x 2 y 3a 2b a b 4a 2b
(a) __________________ [1] (c) __________________ [3]
75.
Factorise
24ax– 40ay 12az a. b.
7a 2 b 2 c 28a 2 bc 2 21a 2 bc 6 y 2 15 yz 8 yz 20 z 2
c.
(a) __________________ [1] (b) __________________ [2] (c) __________________ [2] 76.
Solve for x in the following equations. a. 3x – 4 = 11
7( x 4) 2( x 4) b.
(a) __________________ [1] (b) __________________ [3]
77.
Simplify
3ab c
2
5ac 15a 2 b 6b 8b 5 c
__________________ [2]
78.
Express
4y 3 3( y 5) 9 2 8
as a single fraction.
__________________ [2]
5m 7 17 3(2 m ) 79.
(a) Solve the equation
.
2 x y 3z 1 y 3x 2y (b) If
, find x when y = 4 and z = 1.
(a) ___________________ [2] (b) ___________________ [3] 80. Solve the following equations.
(a)
(b)
7 3 5x 4 4x 9 7 2x 1
(a) ___________________ [2] (b) ___________________ [2]
81.
Factorise 3xy – 4wz – 6xz + 2wy.
____________________ [2] 82.
Jacie is x years old. Jaden, her brother, is 11 years older than Jacie. Their mother is three times as old as Jacie. Jaden is half their father’s age. a. Write down expressions, in terms of x for i. Jaden’s age, ii. their father’s age. b. The sum of the ages of the four members of the family is 173. i. Show that the sum of the ages of the four members of the family can be written as 7x + 33 = 173. ii. Solve the equation to find the value of x. iii. Hence find the mother’s age when Jaden was born.
years old[1] (a) i. ___________________
(a) ii. ___________________ [1] years old[1] (b) i. Working, as shown.
(b) ii. ___________________ [1] (b) iii. ___________________ [1] 83.
Express as a single fraction in its simplest form (a)
(b)
x 3x 1 5 7
,
3 x 4 3(1 x ) 3 6
.
Ans:
84.
85.
(a)
[2]
(b)
[2]
Ans: (a)
[1]
(b)
[2]
Factorise completely (a)
33ab + 9b ,
(b)
14ab – 14a + 5b – 5 .
Simplify (a)
7m – 3n + 167m – 9n ,
(b)
3 15 x 25 4 x 28 5
.
86.
Ans: (a)
[2]
(b)
[2]
Simplify 2 3a 4b 3 b 3 2a b (a) , x 1 2 x 1 3 x 1 2 3 4 (b) .
Answer (a) …………………….… [2] (b) ………………………. [2] 87. Solve (a) (b)
(c)
3 7 2 x 57
[2]
, 2x 7 3x 8 5
, 2( x 3) x 2 5 3 4 6
[2]
[3] .
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