Question

Consider the following statements:
(i) For every positive real number X, x-10 is positive.
(ii) Let n be a natural number. If n² is even, then n is even.
(iii) If a natural number is odd, then its square is also odd.
Then

• A. (i) False, (ii) True and (iii) True
• B. (i) False, (ii) False and (iii) True
• C. (i) True, (ii) False and (iii) True
• D. (i) True, (ii) True and (iii) True
• E. (i) False, (ii) True and (iii) False

Answer 1

The Correct Option is A

(i) "For every positive real number \( x \), \( x-10 \) is positive."
This statement is false. For \( x = 5 \), \( x - 10 = -5 \), which is not positive. Hence, this is false. 

(ii) "Let \( n \) be a natural number. If \( n^2 \) is even, then \( n \) is even."
This statement is true. If the square of a number is even, then the number itself must be even. For example, \( 2^2 = 4 \), which is even, and \( n = 2 \) is even. Hence, this is true.

(iii) "If a natural number is odd, then its square is also odd."
This statement is true. The square of any odd number is always odd. For example, \( 3^2 = 9 \), which is odd. Hence, this is true.

The correct option is (A) : (i) False, (ii) True and (iii) True

Answer 2

(i) For every positive real number X, x-10 is positive.

This statement is false. Consider x = 5, which is a positive real number. Then, x - 10 = 5 - 10 = -5, which is not positive.

(ii) Let n be a natural number. If n² is even, then n is even.

This statement is true. We can prove this by contrapositive. The contrapositive of the statement is: "If n is odd, then n² is odd." If n is odd, then n = 2k + 1 for some integer k. Then, n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. Since the contrapositive is true, the original statement is also true.

(iii) If a natural number is odd, then its square is also odd.

This statement is true. If a natural number is odd, it can be represented as 2k + 1, where k is an integer. Its square is (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is also odd.

Therefore, (i) False, (ii) True and (iii) True.