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Unit I: Number System — Test 2

Reasoning, proof and modelling · 8 questions · 20 practice marks · 40 minutes.
MCQs are scored automatically; written answers use step-wise self/teacher review.

Section A
Question 1 (1 Mark)
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Full question paper and step-wise model answers

Practice marks are not official examination weightage. Written answers require self- or teacher assessment.

Question 1 · 1 mark

The sum of two rational numbers is always:

  1. Irrational
  2. Rational
  3. An integer
  4. Positive
Model answer and reasoning

(b) Rational

  1. Write a/b + c/d = (ad + bc)/(bd).
  2. The numerator and denominator are integers and bd ≠ 0.

Question 2 · 1 mark

In lowest terms, which fraction has a terminating decimal?

  1. 7/12
  2. 5/18
  3. 13/40
  4. 2/21
Model answer and reasoning

(c) 13/40

  1. A reduced denominator with only factors 2 and 5 gives a terminating decimal.
  2. 40 = 2³ × 5; 13/40 = 0.325.

Question 3 · 1 mark

Which pair shows that the sum of two irrational numbers can be rational?

  1. √2 and −√2
  2. 1/2 and 1/3
  3. √2 and 2
  4. 3 and 4
Model answer and reasoning

(a) √2 and −√2

  1. √2 and −√2 are both irrational.
  2. Their sum is 0, which is rational.

Question 4 · 4 marks

Prove density of rational numbers using the midpoint of two rational numbers a < b. Explain how to generate infinitely many different rationals between them.

Model answer and reasoning

(a + b)/2 is rational and lies strictly between a and b; repeat midpoint selection.

  1. Sums of rationals and division by 2 preserve rationality.
  2. a < b implies 2a < a + b < 2b, so a < (a + b)/2 < b.
  3. Set r₁ = (a + b)/2 and rₙ₊₁ = (a + rₙ)/2.
  4. Each rₙ₊₁ lies strictly between a and rₙ, so the sequence produces infinitely many distinct intermediate rationals.

Question 5 · 4 marks

Prove √3 is irrational. Justify the implication that if 3 divides p², then 3 divides p.

Model answer and reasoning

√3 is irrational.

  1. Squares of integers with remainders 0, 1, 2 on division by 3 have remainders 0, 1, 1 respectively.
  2. Assume √3 = p/q in lowest terms. Then p² = 3q²; the remainder argument gives p = 3k.
  3. Substitution gives q² = 3k², so 3 also divides q.
  4. This contradicts coprimality. Therefore √3 is irrational.

Question 6 · 3 marks

Describe how to locate √5 on the number line using a right triangle. Explain why the construction is valid.

Model answer and reasoning

Construct a right triangle with legs 2 and 1; transfer its hypotenuse to the positive number line.

  1. Mark O = 0 and A = 2. At A draw AB perpendicular to OA with AB = 1.
  2. Pythagoras gives OB² = 2² + 1² = 5, so OB = √5.
  3. With centre O and radius OB draw an arc meeting the positive number line. Its intersection represents √5.

Question 7 · 3 marks

Use a remainder-tracking algorithm to generate the decimal expansion of 2/7. Identify the recurring block.

Model answer and reasoning

2/7 = 0.285714285714… .

  1. Divide 20 by 7: digit 2, remainder 6. Continue with ten times each remainder.
  2. The subsequent digits/remainders are 8/4, 5/5, 7/1, 1/3, 4/2.
  3. The original remainder 2 returns after six digits; the block 285714 repeats.

Question 8 · 3 marks

A student claims that a rational number plus an irrational number is rational. Disprove the claim in general.

Model answer and reasoning

The sum of a rational number and an irrational number is irrational.

  1. Let r be rational and s irrational.
  2. If r + s were rational, then (r + s) − r would be rational because rationals are closed under subtraction.
  3. But this difference equals s, a contradiction.