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Mathematics

Restate the following statements with appropriate conditions, so that they become true statements.

(i) All prime numbers are odd.

(ii) Two times a real number is always even.

(iii) For any x, 3x + 1 > 4.

(iv) For any x, x3 ≥ 0.

(v) In every triangle, a median is also an angle bisector.

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Answer

(i) We know that,

A natural number that has exactly two factors, i.e., 1 and the number itself, is a prime number. Example - 2, 3, 5, 7,…….

Hence, all prime numbers greater than 2 are odd.

(ii) Let real number be 1.5

On multiplying by 2, we get :

1.5 × 2 = 3, which is an odd number.

∴ The statement, two times a real number is always even, is incorrect.

But if we multiply, any natural number by 2, we always get an even number.

Hence, two times a natural number is always even.

(iii) Given,

⇒ 3x + 1 > 4

⇒ 3x > 4 - 1

⇒ 3x > 3

⇒ x > 33\dfrac{3}{3}

⇒ x > 1.

Hence, for x > 1, 3x + 1 > 4.

(iv) For any x ≥ 0, x3 ≥ 0.

(v) We know that,

A median is also the angle bisector in equilateral triangle.

Hence, in equilateral triangle, a median is also an angle bisector.

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