AS Induction revision Material
Linear Equations:
Solutions:
Section A
Section B
Section C
Section D
1.
x= 4
7. x= 1
2. x=6
3. x=2
4. x= -12
5. x= 3.5
8. x=0.1
9. x=0.2 10. x= 4 11. x= −
2. x=9
3. x=
6.
5
12.
6
x=-6.5
x=−
Section E
1.
x= 2
7. x= 3
8. x=-3
Simultaneous Equations:
9
5
4. x= 2
9. x=-9 10. x=-
2
3
5. x= -2
11. x=
12
13
6.
12.
x=-3
x=−
1
6
3
4
Solve the simultaneous equations
5x + y = 11
3x – y = 9
2.
Solve the simultaneous equations.
5a + 3b = 9
2a – 3b = 12
3.
4 apples and 1 banana cost £1.70. 2 apples and 1 banana cost 90p. Work out how much one apple costs
and how much one banana costs each?
4.
Solve
x + 2y = 4
3x – 4y = 7
5.
Solve the simultaneous equations
3x + 7y = 26
4x + 5y = 13
6.
Solve the simultaneous equations
6x – 2y = 33
4x + 3y = 9
7.
By eliminating y, find the solutions to the simultaneous equations
x2 + y2 = 25
y=x–7
8.
By eliminating y, find the solutions to the simultaneous equations
x2 + y2 = 25
y = 2x – 2
9.
Solve the simultaneous equations
x= 64- 7y
x= 28- 3y
10.
Three bananas and two pears cost 95p. Five bananas and three pears cost £1.51.
Find the cost of ten bananas and ten pears.
Solutions:
1.
5
3
2
2
x= , y=-
2. a=3, b=-2
3. a=40, b=10
4. x=3, y=
1
2
5. x=-3, y=5
9
6. x= , y= -3
2
7. x= 3, y=-4
x=4, y=-3
7
24
5
5
8. x=- , y= -
9. x=1, y=9
10. 10b = £1.70 10p = £2.20
Fractions:
Evaluate the following, giving you answer in its simplest form:
1
4
7
10
2 1
+
3 2
2 1
−
3 2
2 1
×
3 2
2 1
÷
3 2
Algebraic Fractions:
Simplifying
Addition and Subtraction
Multiplication and Division
2
5
8
11
4 1
+
5 6
5 2
−
6 3
3 1
×
5 3
4 1
÷
5 3
3
6
9
12
3 3
+
8 4
3 1
−
4 5
2 3
×
3 5
1 2
÷
8 5
Fractions Solutions:
Numerical fractions
1
4
7
10
7
6
1
6
1
3
4
3
Simplifying
Addition and subtraction
Multiplication and Division
2
5
8
11
29
30
1
6
1
5
12
5
3
6
9
12
9
8
11
20
2
5
5
16
Quadratics:
1.
Section A: Expanding Brackets
3. 3x 2  10 x  8
x  2 x  3
4. 6 x 2  17 x  12
2.
 x  1 x  4
3.
 2x  1 2 x 1
1. 4 x 2  9
4.
3x 1 4x  3
2. x 2  8 x  15
Section B: Difference of two squares
Section F: a mixture
3. 6 x 2  x  1
1. x 2  4
4. x 2  10 x  24
2. x 2  2500
5. 1  x 2
3.
 4x 
2
 y2
4. 9 y 2  100
Section C: Common Factor
6. x 2  11x  28
7. x 2 y 2  2 xy  1
8. x 2  5 x  24
1. x 2  3x
9. 9 x 2  6 x  1
2. 2 x 2  6 xy
10. x 2  7 x  6
3. 15a 2b  9ab 2
11. 3  2x  x 2
4. 5a 2  20a
12. 5 x 2  61x  12
Section D: Coefficient of x 2 is 1
13. 25 x 2  16
1. x 2  5 x  6
14. 9 x 2  30 x  25
2. x 2  4 x  3
15. 6 x 2  11xy  4 y 2
3. x 2  3x  10
16. 7 x 2  5 x  150
4. x 2  2 x  24
17. 36a 2  49 x 2
Section E: Coefficient of x 2 is not 1
18. 40 x 2  17 x  12
1. 2 x 2  x  1
19. 49  84 x  36 x 2
2. 3 x 2  8 x  4
20. 3  5 x  2 x 2
Solutions:
1.
Section A: Expanding Brackets
3. 3x 2  10 x  8
 3x  4 x  2
x  2 x  3
4. 6 x 2  17 x  12
3x  4 2 x  3
2 x 2  3x
2.
 x  1 x  4
x 2  3x  4
3.
 2x  1 2 x 1
4x2 1
1. 4 x 2  9
 2x  3 2 x  3
4.
3x 1 4x  3
12 x 2  13x  3
2. x 2  8 x  15
 x  5 x  3
3. 6 x 2  x  1
3x 1 2x 1
Section B: Difference of two squares
Section F: a mixture
1. x 2  4
 x  2 x  2
4. x 2  10 x  24
 x 12 x  2
2. x 2  2500
 x  50 x  50
5. 1  x 2
1  x 1  x 
 4x 
 4x  y  4x  y 
6. x 2  11x  28
 x  7  x  4
3 y 103 y 10
7. x 2 y 2  2 xy  1
 xy  1
8. x 2  5 x  24
 x  8 x  3
3.
2
 y2
4. 9 y 2  100
Section C: Common Factor
2
1. x 2  3x
x  x  3
9. 9 x 2  6 x  1
 3x  1
2. 2 x 2  6 xy
2x  x  3 y 
10. x 2  7 x  6
 x  6 x  1
3. 15a 2b  9ab 2
3ab  5a  3b 
11. 3  2x  x 2
3  x 1  x 
4. 5a 2  20a
5a  a  4
12. 5 x 2  61x  12
5x 1 x 12
13. 25 x 2  16
5x  45x  4
Section D: Coefficient of x 2 is 1
2
1. x 2  5 x  6
 x  2 x  3
14. 9 x 2  30 x  25
 3x  5
2. x 2  4 x  3
 x  3 x 1
15. 6 x 2  11xy  4 y 2
 2x  y 3x  4 y 
3. x 2  3x  10
 x  5 x  2
16. 7 x 2  5 x  150
 7 x  30 x  5
4. x 2  2 x  24
 x  6 x  4
17. 36a 2  49 x 2
 6a  7 x  6a  7 x 
18. 40 x 2  17 x  12
8x  35x  4
Section E: Coefficient of x 2 is not 1
2
1. 2 x 2  x  1
 2x  1 x 1
19. 49  84 x  36 x 2
 7  6x 
2. 3 x 2  8 x  4
3x  2 x  2
20. 3  5 x  2 x 2
1  2x 3  x 
2
Solving Quadratics:
Solving Quadratics- Using the quadratic formula: (Using Decimals)
Solving Quadratics- Using the quadratic formula: (Using surds)
Solutions:
Using Decimals:
Using Surds:
Indices:
1. Simplify:
2. Simplify:
3. Simplify:
4. Multiply out:
5. Simplify:
Solutions:
1. a. 𝑥 7
j. 𝑤 8
2.
a. 𝑎4 𝑏 4
3.
a. 2𝑎5
b. 𝑦 7
k. 𝑝7
c. 𝑝4
l. 𝑥 6
b. 𝑥 3 𝑦 3
b. 3𝑥 6
d. 𝑥 4
c.
1
3𝑎2
c. 6𝑦 4
e. 𝑥 3
d. 8𝑎3
d. 4𝑥
f. 𝑦 4
e. 81𝑦 4
e. 5𝑦
h. 4𝑥 6
4. a. 𝑥 5 + 𝑥 7
5. a. 𝑥 5
b. 𝑡 5
b. 𝑦 5 − 𝑦 3
c. 𝑤 3
g. 𝑛6
c. 𝑧 6 − 𝑧 2
h. 𝑧 3
i. 𝑥 7
f. 𝑚5 𝑛10
f. 2𝑧 2
g. 2𝑦 −1 or
2
𝑦