A polynomial such as f(x) = x³ + 4x² + x − 6 is divisible by (x − 1), and x² − 5x + 6 = 0 solves to x = 2 or x = 3, hence factorisation and function analysis carry this worksheet. Later answers include a range of f(x) ≥ 2; the key is on this page.
Key Takeaways
(x − 1) divides f(x) = x³ + 4x² + x − 6 without a remainder (Q1).
x² − 5x + 6 = 0 has solutions x = 2 and x = 3 (Q2).
The answers given in the key include the range value f(x) ≥ 2.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.Which of the following is a factor of f(x) = x³ + 4x² + x − 6?
(a) (x − 1)(b) (x − 2)(c) (x + 4)(d) (x − 4)
2.The solutions of the equation x² − 5x + 6 = 0 are:
(a) x = 2 and x = 3(b) x = 1 and x = 6(c) x = −2 and x = 3(d) x = 2 and x = −3
3.For the function f(x) = x² + 2, where x is any real number, the range of f is:
(a) f(x) ≥ 0(b) f(x) ≥ 2(c) f(x) ≤ 2(d) all real values
4.Given f(x) = 3x + 1 and g(x) = x², the value of f(g(2)) is:
(a) 13(b) 49(c) 37(d) 18
5.The equation x² + 4x + k = 0 has exactly one solution when:
(a) k = 4(b) k = −4(c) k = 8(d) k = 2
6.If f(x) = 2x − 1, then f⁻¹(x) =
(a) (x + 1) ÷ 2(b) (x − 1) ÷ 2(c) 2x + 1(d) x + 2
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.Solve 3x − 7 = 2x + 5 for x.
8.Write x² + 6x + 11 in the form (x + a)² + b.
9.Show that (x − 2) is a factor of f(x) = x³ − 3x² − 4x + 12.
10.Solve the inequality x² − 9 > 0.
11.Find the range of f(x) = 5 − 2x on the domain 0 ≤ x ≤ 3.
12.Solve 2x² + 3x − 2 = 0.
13.Given g(x) = x² + 1, simplify g(x + h) − g(x).
14.Find the values of k for which x² + kx + 9 = 0 has exactly one solution.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.A rectangle has perimeter 40 cm and length x cm. Express its area A in terms of x and find the maximum possible area.
16.A ball is thrown so that its height after t seconds is h = 20t − 5t². Find the maximum height reached.
17.Two numbers have a sum of 20 and a product of 96. Form a quadratic equation in x and solve it to find the two numbers.
18.A taxi firm charges £3 per journey plus £2 per mile, so the cost in pounds is C = 2d + 3, where d is the distance in miles. Find how far a customer can travel for £15.
19.A right-angled triangle has legs of length x cm and (x + 2) cm and a hypotenuse of 10 cm. Use Pythagoras to form a quadratic equation and find x.
20.The function f(x) = x² is defined on the domain 0 ≤ x ≤ 10. State the range of f and find the value of f(5) − f(2).
Answer Key
1. a) (x − 1)
2. a) x = 2 and x = 3
3. b) f(x) ≥ 2
4. a) 13
5. a) k = 4
6. a) (x + 1) ÷ 2
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
Is (x − 1) a factor of x³ + 4x² + x − 6?
Yes — it is option (a) in this worksheet (Q1).
What are the solutions of x² − 5x + 6 = 0?
x = 2 and x = 3, which is option (a) in this worksheet (Q2).
What is the range of the function on this sheet?
The answer key records f(x) ≥ 2.
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