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Number systems — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsNumber systems1 Mark
MCQ
Which one of the following statements is true?
  • A
    The sum of two irrational numbers is always an irrational number.
  • B
    The sum of two irrational numbers is always a rational number.
  • C
    The sum of two irrational numbers may be a rational number or an irrational number.
  • D
    The sum of two irrational numbers is always an integer.
✓

Answer

  1. The sum of two irrational numbers may be a rational number or an irrational number.
    Solution:
    If two irrational numbers i.e. $\sqrt{2},\sqrt{5},2+\sqrt{3},2-\sqrt{3}$ etc. are added it is not necessary that sum comes out to be an irrational number always, or a rational nnumber always, or a rational number always...
    Since $\sqrt{2}+\sqrt{5}=$ an irrational number
    $2+\sqrt{\not\text{3}}+2-\sqrt{\not\text{3}}=4=$ a rational number
    So we see that $\sqrt{2}$ and $\sqrt{5}$ are irrational numbers, and their sum is also irrational.
    But $2+\sqrt{3}$ and $2-\sqrt{3}$ are also irrational numbers, and their sum is rational number '4'.
    So sum of two irrational numbers can be either an irrational number or a rational number depending which numbers are being added.
    So options (a) and (b) are totally wrong, because they are not 'always' true.
    Option (c) is correcrt because sum can be either irrational or rational and option (c) is verifying this statement.
    Option (d) - again it is not always true, if we add two irrational numbers like $2+\sqrt{3}$ and $2-\sqrt{3}.$
    Sum is an integer = 4, but if we add $\sqrt{3}$ and $\sqrt{3},$ sum is $2\sqrt{3}$ which is not an integer but again an irrational number.
    So option (d) is also incorrect.
    Hence, correct option is (c).

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