IGCSE Further Pure Mathematics Revision Questions
Year 10 (KS3) Revision Set
Chapter 1: Surds and Logarithmic Functions
Easy Questions
1. Simplify √50.
2. Write 3 + √12 in the form a + b√c, where a, b and c are integers and c has no perfect square
factors.
3. Express log₂(32) as a single number.
4. Write log₃(27) as a single number.
5. Solve the equation 2ˣ = 16.
6. If log₄(x) = 3, find the value of x.
Medium Questions
7. Rationalise the denominator of 5/(3√2).
8. Simplify (2√3 - √5)(√3 + 2√5).
9. If log₂(x) = 5 and log₂(y) = 3, find the value of log₂(xy).
10. Use the laws of logarithms to express log₅(125) in terms of log₅(5).
11. Solve the equation 3ˣ = 27.
12. Change log₃(x) to base 5. Give your answer in terms of log₅(x) and log₅(3).
Hard Questions
13. Simplify (√6 + √8)/(√6 - √2) giving your answer in the form a + b√c, where a, b and c are integers.
14. Solve the equation 2ˣ⁺¹ - 5(2ˣ) + 6 = 0.
15. If f(x) = 3ˣ and g(x) = log₃(x), find f(g(9)).
16. Use the laws of logarithms to solve log₄(x) + log₄(x+3) = 1.
17. Sketch the graph of y = 2ˣ and y = log₂(x) on the same axes. Mark clearly where these graphs
intersect.
18. Solve the equation log₃(2x-1) - log₃(x+2) = 2.
Chapter 2: The Quadratic Function
Easy Questions
1. Factorise x² + 7x + 12.
2. Solve the equation x² - 5x + 6 = 0 by factorisation.
3. Find the value of the discriminant for the quadratic equation 2x² - 5x + 2 = 0 and state what this
tells you about the roots.
4. Using the quadratic formula, solve 3x² - 4x - 2 = 0. Give your answer in exact form.
5. Complete the square for the expression x² - 6x + 8.
6. If the roots of the quadratic equation x² + bx + c = 0 are 3 and -5, find the values of b and c.
Medium Questions
7. Factorise 2x² + 7x - 15.
8. Solve the equation 3x² + 10x - 8 = 0 using the quadratic formula.
9. Find the roots of the equation x² - 6x + 10 = 0. Give your answer in the form a ± bi, where a and b
are integers.
10. The quadratic equation 2x² + kx - 8 = 0 has equal roots. Find the value of k.
11. Express x² - 4x - 12 in the form (x - p)² + q, where p and q are constants.
12. If α and β are the roots of the equation x² - 5x + k = 0, and α + β = 5, find the value of k and the
product αβ.
Hard Questions
13. Factorise 4x² - 12xy + 9y².
14. The quadratic equation 3x² + px + q = 0 has roots 2 and -3. Find the values of p and q.
15. For what values of k will the equation kx² + 6x + 9 = 0 have: a) two distinct real roots b) exactly one
real root c) no real roots?
16. If α and β are the roots of x² - 6x + 10 = 0, find the value of α² + β².
17. Form a quadratic equation whose roots are (2 + √3) and (2 - √3).
18. Solve the equation 2/(x-1) + 3/(x+2) = 4.
Chapter 3: Inequalities and Identities
Easy Questions
1. Solve the inequality 3x - 2 > 7.
2. Solve the inequality -2 < 3x + 4 ≤ 10.
3. Solve the system of equations: x + y = 5 x - y = 1
4. Factorise x³ - 8.
5. Find the remainder when x³ + 2x² - 5x + 3 is divided by (x - 2).
6. Solve the simultaneous equations: 2x + y = 7 x² + y = 13
Medium Questions
7. Solve the inequality x² - 5x + 6 > 0.
8. Solve the inequality x² - 4x - 12 ≤ 0.
9. Solve the system of equations: 2x + 3y = 16 y = x² - 1
10. Use the factor theorem to determine whether (x - 3) is a factor of x³ - 5x² + 3x + 9.
11. Find the quotient and remainder when 2x³ - 3x² + 4x - 1 is divided by (x - 2).
12. Graph the solution to the inequality system: y ≥ x + 1 y < 2x - 3
Hard Questions
13. Solve the inequality (x-2)/(x+1) > 3.
14. Solve the inequality |2x - 5| ≤ 3.
15. Solve the system: x² + y² = 5 x + y = 3
16. If P(x) = x³ - 6x² + ax + b leaves a remainder of 10 when divided by (x - 1) and a remainder of 6
when divided by (x - 2), find the values of a and b.
17. Using the remainder theorem, find the remainder when P(x) = 2x⁴ - 3x³ + 4x - 5 is divided by (x 2).
18. Factorise x⁴ - 16 completely.
Answers
Chapter 1: Surds and Logarithmic Functions
Easy Questions
1. 5√2
2. 3 + 2√3
3. 5
4. 3
5. x = 4
6. x = 64
Medium Questions
7. 5√2/6
8. 6√3 - 2√5 - 5
9. log₂(xy) = 8
10. log₅(125) = 3
11. x = 3
12. log₅(x) = log₅(x)/log₅(3)
Hard Questions
13. 5 + 2√3
14. x = 1 or x = 2
15. f(g(9)) = 9
16. x = 1
17. [Graph would show intersection at (1,1)]
18. x = 5
Chapter 2: The Quadratic Function
Easy Questions
1. (x + 3)(x + 4)
2. x = 2 or x = 3
3. Discriminant = 25 - 16 = 9 > 0, two real distinct roots
4. x = 2 ± √10/3
5. x² - 6x + 8 = (x - 3)² - 1
6. b = 2, c = -15
Medium Questions
7. (2x - 3)(x + 5)
8. x = (-5 ± √73)/3
9. x = 3 ± i
10. k = 8
11. (x - 2)² - 16
12. k = 5, αβ = 5
Hard Questions
13. (2x - 3y)²
14. p = 3, q = -18
15. a) k < 2, b) k = 2, c) k > 2
16. α² + β² = 36 - 20 = 16
17. x² - 4x + 1 = 0
18. x = 2
Chapter 3: Inequalities and Identities
Easy Questions
1. x > 3
2. -2 < 3x + 4 ≤ 10 means -6/3 < x ≤ 6/3, so -2 < x ≤ 2
3. x = 3, y = 2
4. x³ - 8 = (x - 2)(x² + 2x + 4)
5. Remainder = 11
6. x = 3, y = 1 or x = -1, y = 9
Medium Questions
7. x < 2 or x > 3
8. 2 - 2√4 ≤ x ≤ 2 + 2√4, which is -2 ≤ x ≤ 6
9. x = 2, y = 3
10. Yes, (x - 3) is a factor
11. Quotient = 2x² + x + 6, Remainder = 11
12. [Graph would show region above y = x + 1 and below y = 2x - 3]
Hard Questions
13. x < -1 or x > 5/2
14. 1 ≤ x ≤ 4
15. x = 2, y = 1 or x = 1, y = 2
16. a = 13, b = -8
17. Remainder = 27
18. x⁴ - 16 = (x² + 4)(x + 2)(x - 2)