Year 12 Skills Test
Name ..........................................................
Calculators are not permitted
Score .................... / 60
1.
Simplify
(i)
(4x3y)3
.....................................
(2)
(ii) (5t–3)–2
.....................................
(2)
2.
Simplify fully
(a) 6(3x + 4) – 8(4x – 5)
.................................................
(2)
(b)
n2 – 1
2
´
n +1 n – 2
.................................................
(3)
3.
y
5
4
3
×
B
×C
2
1
–5
–4
–3
–2
–1 O
1
2
3
4
5
x
–1
–2
×
A
–3
In the diagram
A is the point (0, –2),
B is the point (–4, 2),
C is the point (0, 3).
Find an equation of the line that passes through C and is parallel to AB.
.....................................
(4)
4.
Solve
3x + y = 8
4x + 2y = 9
x = ....................................
y = ....................................
(3 )
5.
Solve the simultaneous equations
4x + 2y = 8
2x – 5y = 10
x = ................... , y = ..................
(3)
6.
Expand and simplify
(x + 3y)( x – 4y)
x2 + xy –7y2 x2 – xy –12y2
A
B
x + xy –12y
x2 + xy –12y2 x2 – 7xy –12y2
C
D
E
(1)
7.
(a) Factorise
9x2 – 6x + 1
……………………………
(2)
(b) Simplify
6x 2 + 7 x - 3
9x 2 - 6x + 1
……………………………
(3)
8.
By eliminating y, find the solutions to the simultaneous equations
y – 2x = 3
x2 + y2 = 18
x =……………………. y =…………………….
or x =……………………. y =…………………….
(7)
9.
Given that x2 + 6x - 5 = (x + p)2 + q for all values of x,
find the value of
(i)
p,
(ii) q.
p = ...................................
q = ...................................
(3)
10. (a) Complete this table of values for y = x3 + x – 2
x
–2
y
–12
–1
0
1
2
0
(3)
(b) On the grid, draw the graph of y = x3 + x – 2
y
10
8
6
4
2
–2
–1
O
–2
1
2
x
–4
–6
–8
–10
–12
(2)
11. (a) Rationalise the denominator of
!
#$!
√
....................................
(1)
(b) Expand (2 + 3 )(1 + 3 )
Give your answer in the form a + b 3 where a
and b are integers.
....................................
(2)
12. (a) Work out 2 7 × 4 1
8
3
....................
(2)
(b) Work out 3 1 ÷ 2 4
2
5
....................
(2)
13.
y
3
2
1
–2
–1
O
1
3
2
x
–1
The line with equation 6y + 5x = 15 is drawn on the grid above.
(a) Rearrange the equation 6y + 5x = 15 to make y the subject.
y = .........................
(2)
(b) The point (–21, k) lies on the line.
Find the value of k.
k = .........................
(2)
14.
5( 2 x + 1)
= 4x + 7
Solve
3
x = ...............................
(3)
15.
𝑓 (𝑥 ) = 4𝑘𝑥 % + (4𝑘 + 2)𝑥 + 1
(1) Find the discriminant of f(x) in terms of k & Prove that f(x) has two distinct real roots
for all non-zero values of k
(2) Explain why f(x) cannot have two distinct real roots when k=0
(6)
Challenge
1 1 1
+ =
u v f
1
2
u=2 ,v=3
1
3
(a) Find the value of f.
.....................................
(b) Rearrange
1 1 1
+ =
u v f
to make u the subject of the formula.
Give your answer in its simplest form.
.....................................