Question 3 Show that the following statement is true by the method of contrapositive.
p: If x is an integer and x^2 is even, then x is also even.
Class X1 - Maths -Mathematical Reasoning Page 342
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if x is not even . so, x is an odd number
i.e., x = 2n + 1 , where n is Natural number .
∴ x² = (2n + 1)²
x² = 4n² + 4n + 1
= 4(n² + n) + 1
= 4n(n + 1) + 1
we know, 4x(x + 1) + 1 is odd number , if x is odd.
here it is clear that if P is not true then q is also not true .
hence proved that if x is an integer and x² is even then x is also even number .
i.e., x = 2n + 1 , where n is Natural number .
∴ x² = (2n + 1)²
x² = 4n² + 4n + 1
= 4(n² + n) + 1
= 4n(n + 1) + 1
we know, 4x(x + 1) + 1 is odd number , if x is odd.
here it is clear that if P is not true then q is also not true .
hence proved that if x is an integer and x² is even then x is also even number .
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