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Question of 45

Q.a) Which one of the following sentences is a STATEMENT?

(1)
i) 275 is a perfect square.
ii) Mathematics is a difficult subject.
iii) Answer this question.
iv) Today is a rainy day.
b) Verify by method of contradiction : '2\sqrt{2} is irrational'. (3)
Kerala DhseKerala DHSE Plus One Board 2018Subjective· 4mImportance★★★★★
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A mathematical statement is a declarative sentence that is definitely true or definitely false; only (i) qualifies. √2 is shown irrational by assuming the contrary and reaching a contradiction.

a) (i) '275 is a perfect square' has a definite truth value (it is false, but still a statement) → this is a STATEMENT. (ii) is an opinion, (iii) is a command, (iv) is not definite, so none of those are statements. Answer: (i).

b) Prove '√2 is irrational' by contradiction.

Assume the opposite: √2 is rational, so √2 = p/q where p, q are integers with no common factor (q ≠ 0).

Then 2 = p²/q² ⇒ p² = 2q², so p² is even ⇒ p is even. Write p = 2m. …

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