Name: _________KEY______________________
SHOW ALL PROCESS WORK for full marks
1.
Simplify 2y - 3( a + 4y)
2.
Solve the following equations; leave answers as improper fractions when necessary
(a)
Answer
-3a – 10y [2]
3x -8 = -2 3x = 6
Answer (a) x = 2
(b)
[2]
2
x
=x
10 x2 = -20, cannot square root a negative number
Answer (b) x = No solution
(c)
[2]
x -1 2 - 3x
= -1
6
8
LCD = 24, multiply every term by 24
4(x – 1) – 3(2 – 3x) = -1(24)
4x – 4 – 6 +9x = -24
13x = -14
Answer (c) x = -14 / 13
(d)
[3]
2 3 1
+
=
x 2x 3 LCD = 6x, multiply every term by 6x
6(2) + 3(3) = 2x(1)
12 + 9 = 2x
21 = 2x
Answer (d) x = 21 / 2
3.
[3]
Factorise completely
(a) 12x 2 - 3y 2
= 3(4x2 – y2) difference of squares
(b)
(c)
(d)
Answer (a) 3(2x + y)(2x – y)
[2]
Answer (b) (x – 21)(x – 2)
[2]
Answer (c) -(5x – 1)(x – 6)
[2]
x 2 - 23x + 42
-5x 2 + 31x - 6 =-(5x2 – 31x + 6)
4x 2 +12x - x - 3 factor by grouping = 4x(x + 3) - (x + 3)
Answer (d) (x + 3)(4x – 1)
[2]
20 marks
4.
In a set of three numbers, the first is a positive integer, the second is three more than the first and the
third is the square of the second.
(a) The first number in the set is x.
Write the second and third numbers in terms of x.
Answer (a)
(b)
x+3, (x+3)2
[2]
The sum of the three numbers is 77.
(i)
Write down an equation in x.
Answer (b)(i)
(ii)
(iii)
x + (x + 3) + (x + 3)2 = 77
[2]
Show that your equation simplifies to x2 + 8x – 65 = 0
x + x + 3 + x2 +6x + 9 = 77
x2 + 8x + 12 = 77
x2 + 8x – 65 = 0
Answer (b)(ii)
[2]
Answer (b)(iii) x = 5 or x = -13
[2]
Answer (b)(iv) 5, 8, 64
[2]
Solve the equation x2 + 8x – 65 = 0.
(x - 5)(x + 13) = 0
x – 5 = 0 or x + 13 = 0
x = 5 or x = -13
(iv) Write down the three numbers.
5.
3x + 4y = 3
x + 6y = 8
Elimination method:
Solve the equations
3x + 4y = 3
(-)3x +18y = 24
Multiply second equation by 3: -14y = -21
y=
3
2
3
x +6× = 8
2
Substitute y into second equation: x + 9 = 8
x = -1
Answer
x=
y=
-1
3/2
13 marks
[3]
6.
(- 5 )
2´ 8
3
22
7
( 3)
2 3
4
-4.23& 1&
From the list above, write down:
7.
8.
(a)
an integer number,
Answer (a)
(b)
the smallest number,
Answer (b)
(- 5 )
(c)
a rational number,
Answer (c)
22
7
(d)
a square number,
Answer (d)
( 3)
(e)
an irrational number
Answer (e)
(- 5 )
(a)
Factorise ax - bx
(b)
Hence evaluate 1426 ´ 0.6789 - 426 ´ 0.6789
= 0.6789 ( 1426 – 426 )
= 0.6789 × 1000
( 3)
2´ 8
[1]
4
[1]
3
2´ 8
4
( 3)
4
-4.23& 1&
2´ 8
3
2 3
[1]
[1]
[1]
Answer (a) x (a – b)
[1]
Answer (b) 678.9
[1]
Three positive integers are (x -1), x and (x + 1).
When they are multiplied together the answer is 40 times their sum.
(a)
(i)
Write down an equation in x.
Answer (a)(i) x(x – 1)(x + 1) = 40 (x + x – 1 + x +1 )
(ii)
[1]
Show that your equation simplifies to x 3 -121x = 0
x x 2 -1 = 40 ( 3x )
(
)
x 3 - x = 120x
x 3 - x -120x = 0
x 3 -121x = 0
[2]
(b)
Factorise completely x -121x = x ( x2 – 121) difference of squares
3
Answer (b) x (x + 11)(x – 11)
9.
[2]
12 marks
Explain why 3( x - 4) £ 3x + 6 has ‘infinitely many solutions.’
3x – 12 ≤ 3x + 6
-12 ≤ 6 is always true  so we can substitute any value for x and inequality will hold; x does not
effect the inequality
[2]
10. If the solution for the following simultaneous equations is (1, 2), find the values of A and B.
3x + 2y = A
x + By = 7 Substitute (1,2) into each equation as (x, y)
A = 3(1) + 2(2) = 7
(1) + B(2) = 7  1 + 2B = 7  2B = 6  B = 3
Answer
11.
A=
7
B=
3 [3]
2 cm
The diagram shows a square picture in a square frame of side x cm.
The width of the border all round the picture is 2 cm, and the area of the border is 112 cm2.
(a)
(b)
Use this information to form an equation in x.
Answer (a) x2 – (x – 4)2 = 112
[1]
Solve your equation to find the value of x.
x2 – (x2 – 8x + 16) = 112
x2 – x2 +8x – 16 = 112
8x = 128
Answer (b) x = 16
[2]
12. Solve the simultaneous equations:
x 2 + ( 2x - 7) = 61
2
y = 2x - 7
x + y = 61
2
2
x 2 + 4x 2 - 28x + 49 = 61
5x 2 - 28x -12 = 0
Substitute equation (1) into second ( 5x + 2) ( x - 6) = 0
5x + 2 = 0 ¾¾
®x = -
2
5
x - 6 = 0 ¾¾
®x = 6
Answer x = -2 / 5
, y = -39 / 5
x=
6
,y=
5 [3]
11 marks