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Unit I: Number System — Test 1

Concepts and applications · 8 questions · 19 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

Which expression is a rational number?

  1. √2
  2. π
  3. −7/4
  4. √3
Model answer and reasoning

(c) −7/4

  1. A rational number is p/q for integers p and q with q ≠ 0.
  2. −7/4 has integer numerator and non-zero integer denominator.

Question 2 · 1 mark

Which decimal expansion must be rational?

  1. 0.101001000100001… with increasing gaps
  2. A non-terminating, non-repeating decimal
  3. 0.272727…
  4. Every infinite decimal
Model answer and reasoning

(c) 0.272727…

  1. The block 27 repeats indefinitely.
  2. If x = 0.272727…, then 100x − x = 27, so x = 27/99 = 3/11.

Question 3 · 1 mark

Which statement is correct?

  1. Every square root is irrational
  2. √9 is irrational
  3. Every irrational number has a non-terminating, non-repeating decimal
  4. An irrational number is not a real number
Model answer and reasoning

(c) Every irrational number has a non-terminating, non-repeating decimal

  1. Rational decimals terminate or eventually repeat.
  2. Irrational real numbers have decimal expansions that neither terminate nor eventually repeat.

Question 4 · 3 marks

Describe how to mark −3/4 and 5/4 on a number line. State which point is closer to 0.

Model answer and reasoning

−3/4 is three quarter-steps left of 0; 5/4 is five quarter-steps right. −3/4 is closer.

  1. Divide each unit interval into four equal parts.
  2. Count three parts to the left for −3/4 and five to the right for 5/4.
  3. Distances from zero are 3/4 and 5/4; compare their magnitudes.

Question 5 · 3 marks

Find four rational numbers strictly between 1/3 and 2/3. Explain why there cannot be a smallest rational number greater than 1/3.

Model answer and reasoning

For example 6/15, 7/15, 8/15, 9/15; no smallest rational above 1/3 exists.

  1. Write the endpoints as 5/15 and 10/15.
  2. Choose the four fractions with numerators 6, 7, 8 and 9.
  3. For any rational r > 1/3, the rational (r + 1/3)/2 lies strictly between them.

Question 6 · 3 marks

Convert 0.125 and 0.1666… to fractions. Explain how long division detects an eventually repeating decimal.

Model answer and reasoning

0.125 = 1/8; 0.1666… = 1/6.

  1. 125/1000 reduces to 1/8.
  2. For x = 0.1666…, 10x = 1.666… and 100x = 16.666…; subtract to obtain 90x = 15.
  3. In long division record each remainder. A zero remainder terminates; a repeated remainder repeats the subsequent digits.

Question 7 · 4 marks

Prove that √2 is irrational using a fraction in lowest terms.

Model answer and reasoning

√2 is irrational.

  1. Assume √2 = p/q, where p and q are coprime integers and q ≠ 0.
  2. Then p² = 2q², so p² is even. An odd integer has odd square, hence p is even.
  3. Put p = 2k. Then q² = 2k², so q is also even.
  4. Both p and q are divisible by 2, contradicting lowest terms. The assumption is false.

Question 8 · 3 marks

Give ordered construction steps for the first three hypotenuses of a square root spiral. What lengths do they represent?

Model answer and reasoning

√2, √3 and √4 = 2 units.

  1. Begin with a unit segment from O. At its end construct a perpendicular unit segment; join its end to O to get √2.
  2. At the new endpoint construct a unit segment perpendicular to the latest radius; the next radius is √3.
  3. Repeat once more. By Pythagoras the new squared radius is 3 + 1 = 4.