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Algebraic Methods and Proof Practice Problems with Solutions - A Level Year 13

A Level Year 13 | Math | Algebraic Methods and Proof
Name: _________________________ Date: _________________________ Score: _______ / 40

Dividing x³ − 2x + 3 by (x − 1) leaves a remainder of 2, while f(x) = x³ − x² − 4x + 4 breaks apart with (x − 1), and these two results frame the factor and remainder work here. A proof that √2 is irrational completes the key's algebra.

Key Takeaways

Section A: Multiple Choice Questions (1 Mark Each)

Choose the correct option for each question.
1. The remainder when x³ − 2x + 3 is divided by (x − 1) is:
(a) 2(b) 1(c) 0(d) 3
2. Which of the following is a factor of f(x) = x³ − x² − 4x + 4?
(a) (x − 1)(b) (x + 1)(c) (x − 4)(d) (x + 4)
3. When f(x) is divided by (x − a), the remainder is:
(a) f(a)(b) f(−a)(c) f(x) ÷ a(d) 0
4. A proof by contradiction that √2 is irrational begins by assuming:
(a) √2 = a ÷ b where a and b are integers with no common factor(b) √2 is irrational(c) √2 is greater than 1(d) √2 is a whole number
5. Which number is a counterexample to the statement: all prime numbers are odd?
(a) 2(b) 1(c) 3(d) 9
6. In the proof that √2 is irrational, from √2 = a ÷ b you obtain a² = 2b², so a² is even and a is even. Writing a = 2k gives b² = 2k², so b is even too. Why is this a contradiction?
(a) a and b were assumed to have no common factor(b) a and b must both be odd(c) √2 must equal 2(d) 2 is not a prime number

Section B: Short Answer Type Questions (2 Marks Each)

Show all steps clearly.
7. Show that (x − 3) is a factor of f(x) = x³ − 2x² − 5x + 6.
8. Find the remainder when x³ + 2x² − x + 1 is divided by (x − 1).
9. Express (x + 5) ÷ ((x + 1)(x + 2)) in partial fractions.
10. Prove that n² + n is even for every integer n.
11. Divide x³ − 7x + 6 by (x − 1) and write down the resulting quadratic factor.
12. Give a counterexample to the statement: x² > x for all positive x.
13. Factorise x³ − 6x² + 11x − 6 completely.
14. Use proof by contradiction to show that there is no greatest positive integer.

Section C: Numericals & Word Problems (3 Marks Each)

Apply the concepts to solve the problems. Show all working.
15. A cuboid has volume V = x³ + 2x² − x − 2 and one of its edges has length x + 2. Show that (x + 2) is a factor of V and factorise V completely.
16. Prove that the product of any three consecutive integers is divisible by 6.
17. Prove that the sum of any two odd integers is even.
18. When the polynomial f(x) = x³ + ax + 3 is divided by (x − 1), the remainder is 5. Find the value of a.
19. Prove by contradiction that there is no smallest positive rational number.
20. When f(x) = x³ + kx − 2 is divided by (x − 2), the remainder is 10. Find the value of k.

Answer Key

1. a) 2
2. a) (x − 1)
3. a) f(a)
4. a) √2 = a ÷ b where a and b are integers with no common factor
5. a) 2
6. a) a and b were assumed to have no common factor
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

What is the remainder when x³ − 2x + 3 is divided by (x − 1)?

2, option (a) in this worksheet (Q1).

Which of the following is a factor of f(x) = x³ − x² − 4x + 4?

(x − 1), option (a) in this worksheet (Q2).

How is the irrationality of √2 proved?

By assuming √2 = a ÷ b with a and b sharing no common factor, as this worksheet shows.

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