A Level Induction Test
You may NOT use a calculator
If ax2 + bx + c = 0 then x =
1.
2.
3.
4.
5.
6.
 b  b 2  4ac
2a
Expand and simplify
(a) (2x + 3)(2x – 1) (2)
Factorise
(a) x2 – 7x (2)
Simplify
4x3 y
(a)
8x 2 y 3
(b) (a + 3)2
(b) y2 – 64 (2)
(2)
(b)
(2)
(c) 4x(3x – 2) – x(2x + 5) (2)
(c) 2x2 + 5x – 3 (2)
3x  2 4 x  1
+
3
6
(2)
Solve the following equations
h  1 3h
(a)
+
= 4 (3)
(b) x2 – 8x = 0 (3)
4
5
Write each of the following as single powers of x and / y
x5
1
(a) 4
(1)
(b) (x2y)3
(1)
(c)  2
x
x
(d) 6t2 – 13t + 5 (2)
(c) p2 + 4p = 12 (3)
(1)
Work out the values of the following, giving your answers as fractions
1
(a) 4-2 (1)
7.
8.
9.
(b) 100 (1)
 8 3
(c)  
 27 
Solve the simultaneous equations
(2)
3x – 5y = -11
5x – 2y = 7
Rearrange the following equations to make x the subject
1
(a) v2 = u2 + 2ax
(2)
(b) V = πx2h
(2)
3
(3)
(c) y =
Solve 5x2 – x – 1 = 0 giving your solutions in surd form (3)
x2
x  1 (3)
10.
Solve this pair of simultaneous equations
x + 2y = 9
x2 + y2 = 29 (4)
11.
Complete the square on the following expressions
a) x2 + 4x + 10
12.
2
2
1
3
4
3
4
b) 2 + 3
(3)
b) 5 −
4
7
5
(1)
2
𝑥−1
(3)
(2)
(3)
(3)
b) √80
(2)
Expand and simplify
a) (2 + √3)(2 − √3)
b) (√5 − 2√3)(3 + 3√3)
18.
1
Simplify fully, writing each in the form k√𝑚 where k and m are both integers:
a) √175
17.
(1)
(1)
Work out the gradient (m) and the y intercept (c) of the following straight line graphs:
a) y = 5x + 3
b) 2y + 6x = 10
c) 5x – 3y + 7 = 0
16.
b) 2 7
Simplify:
a) 𝑥 + 𝑥+1
15.
3
(1)
Add these fractions:
a) +
14.
(2)
Write these mixed numbers as improper fractions
a) 4 5
13.
b) x2 – 6x - 5
(2)
(3)
(3)
Rationalise the denominator
a)
5
√2
3
(2)
b) 4√3
(2)
c) 1−√2
(3)
6
(2)