Question 3Restate 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.

Question

Question 3Restate 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.

Accepted 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 1.5On 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}33​⇒ 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.

Proofs in Mathematics — Question 3

Question 3