CAT Practice
Merit 2

Question One
• Expand and simplify

Question One a
(
2 x
+
1
) ( x
-
3
) b 3
( y
+
5
) -
2
( y
-
8
) c 12
-
2
( x
+
2
)

Answers Question One a
(
2 x
+
1
) ( x
-
3
) =
2 x
2 -
5 x
-
3 b 3
( y
+
5
) -
2
( y
-
8
) = y
+
31 c 12
-
2
( x
+
2
) = -
2 x
+
8

Question Two
• Factorise a x
2 +
9 x
-
36 b x
2 -
14 x
+
49

Answer Question Two
• Factorise a x
2 +
9 x
-
36
= ( x
+
12
) ( x
-
3
) b x
2 -
14 x
+
49
= ( x
-
7
)
2

Question Three a
( )
3 b
( )
2

Answer Question Three a
( )
3 =
8 x
12 b
( )
2 =
16 y
6

Question Four
• Solve for x a x
3 = -
64 b 2 x =
64

Answer Question Four
• Solve for x a x
3 = -
64, x
= -
4 b 2 x =
64, x
=
6

• Simplify
Question Five a b c
4 x
+
3
5 x
8 x 2 -
81
2 x
+
18
24 x
9
8 x
3

Answer Question Five
• Simplify a
4 x
+
3
5 x
=
8
32 x
+
15 x
=
24
47 x
24 b x
2 -
81
2 x
+
18
=
( x
+
9
(
) ( x
2 x
+
9
-
)
9
)
= x
-
9
2 c
24 x
9
8 x
3
=
3 x
6

Question Six
• An operation * is defined by a * b
=
(
10 ab a
+ b
)
2
• Find the value of 3*-4

Answwer Question Six
3*
-
4
=
10
´
3
´ -
4
(
3
-
4
)
2
= -
120

Question Seven
• Solve these equations a 7 x
+
25
=
5
x b 4
( +
2
) =
0 c 3 z
5 =
96

Answer Question Seven
• Solve these equations a 7 x
+
25
=
5
x
Þ
8 x
= -
20
Þ x
= -
2.5
b 4
( +
2
) =
0
Þ y
=
0,
-
2 c 3 z
5 =
96
Þ z
5 =
32
Þ z
=
2

Question Eight
• Factorise fully a
2 +
3 a
-
40

Answer Question Eight
• Factorise fully a
2 +
3 a
-
40
= ( a
+
8
) ( a
-
5
)

Question Nine
• Write in simplest form a
2 +
3 a
-
40 a
2 +
8 a

Answer Question Nine
• Write in simplest form a
2 +
3 a
-
40 a
2 +
8 a
=
=
( a
+
8
(
) ( a a a
+
8
-
)
5
) a
-
5 a

Question Ten
• What is the value of k if:
( )
4
a k
16 a
8

Answer Question Ten
• What is the value of k if:
( )
4
a k
16 a
8 k
4

Question Eleven
• Expand and simplify
(
2 a
+
4
) ( )

Answer Question Eleven
• Expand and simplify
(
2 a
+
4
) ( )
=
2 a
2 +
2 a
-
4

Question Twelve

Question Twelve
36
=
(
15
+
9
2
) h
72
=
24 h
3 cm
= h

Question Thirteen
• Clearly state the range and possible
• values of a if (2a + 8)(a - 2) < 4a + 2

Answer Question Thirteen
• 2 a
• values of a if (2a + 8)(a - 2) < 4a + 2
2 a
2
+
4 a
-
16
-
18
<
0
<
4 a
+
2 a
2 -
9
<
0
( a
+
3
) ( a
-
3
) <
0
-
3
< a
<
3

Question Fourteen
• A rectangular swimming pool is
• 30 metres × 10 metres. Around the pool is a concrete path that is w metres wide. The total area of the pool and surrounding path is 800 m 2 .
• Using the information form an equation and solve it to find the width (w) of the path around the pool.

Question Fourteen
•
(
30
+
2 w
) (
10
+
2 w
) =
800
• 30 metres × 10 metres. Around the pool is a
4 w
2 +
80 w
-
500
=
0 m 2 . w
+
20 w
-
125
=
0
•
( w
+
25
) ( w
-
5
) =
0 solve it to find the width (w) of the path w
=
5 m , w
¹ -
25

Question Fifteen
•
• Last year the Mahobe Football club sold pizzas.
There were 250 pizzas sold for a total raised of $1730.
• Large pizzas sold for $8 each and small ones sold for $5 each.
There were y large pizzas and x small pizzas.
Solve the simultaneous equations below to find the number of each sized pizza that was sold.
• x + y = 250
• 5x + 8y = 1730

Answer Question Fifteen
• Last year the Mahobe Football club sold pizzas.
There were 250 pizzas sold for a total raised of $1730.
•
• Large pizzas sold for $8 each and small ones sold for $5 each.
5 x
+
5 y
=
1250
There were y large pizzas and x small pizzas.
5 x
+
8 y
=
1730 number of each sized pizza that was sold.
3 y
=
480
•
• x + y = 250 large
5x + 8y = 1730 y
=
160, small x
=
90

Question Sixteen
• There are many interesting properties of consecutive numbers.
• Consecutive numbers are numbers such as 21, 22, 23,
24.
For example, choose any 5 consecutive numbers. Take the middle number and multiply it by 5. The answer will be the same as if you summed all 5 of the numbers.
Write an expression that represents five consecutive numbers and use this expression to show that if you multiply the middle number by five you get a result the same as if you summed all five numbers.

Answer Question Sixteen n
=
=
•
• There are many interesting properties of consecutive
+
5
5
(
( numbers.
Consecutive numbers are numbers such as 21, 22, 23,
24.
For example, choose any 5 consecutive numbers. Take n will be the same as if you summed all 5 of the numbers.
n
+
10
2
) numbers and use this expression to show that if you multiply the middle number by five you get a result the same as if you summed all five numbers.
)

Question Seventeen
• In another example take three consecutive whole numbers.
• Square each number and sum the three squares.
• Subtract two from the sum and divide the result by three. Write down an expression to represent any three consecutive numbers. Use this expression to show that if you follow the steps outlined above with any set of three consecutive numbers you will always get as a result the square of the second of the numbers that you first started with.

Answer Question Seventeen
•
• In another example take three consecutive whole numbers. n
2 + ( n
+
1
)
2 + ( n
+
2
)
2 -
2
• Subtract two from the sum and divide the result by three. Write down an expression to represent any three consecutive numbers. Use this
=
3
= n
2 +
2 n
+
1
= ( n
+
1
)
2 outlined above with any set of three consecutive numbers you will always get as a result the square of the second of the numbers that you first started with.

Question Eighteen
• To find Sung’s birth month multiply it by 4.
Add to this product the difference between 12 and his birth month. Subtract from this result twice the sum of 5 and his birth month. If you successfully follow this equation you should end up with 10. What must Sung’s birth month be?

Answer Question Eighteen
•
4
To find Sung’s birth month multiply it by 4. n
+
12
n
-
2
( n
+
5
) = n
+
2
=
10 n
=
8 successfully follow this equation you should end up with 10. What must Sung’s birth month be?