Q21 of 21 Page 14

Prove the following statement by contradiction method.
p: The sum of an irrational number and a rational number is irrational.

Let p be false.


It means the sum of an irrational and a rational number is rational.


Let r be a rational number.


Let √m is irrational, and n is rational number.


⇒ √m + n = r


⇒ √m = r – n


Now √m is irrational; whereas (r – n) is rational.


This is contradiction.


Hence our assumption was wrong.


∴ p is true.


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19

Prove that by direct method for any integer “n”, n3 – n is always even.

 20

Check validity of the following statements.
(i) p: 125 is divisible by 5 and 7.
(ii) q: 131 is a multiple of 3 or 11.


 22

Show that the following statement is true by the method of contrapositive.

p: If x is an integer and x2 is even, then x is also even.

 23

Form the biconditional statement p⟷q, where
(i) p: The unit digits of an integer is zero.
q: It is divisible by 5.
(ii) p: A natural number is odd.
q: Natural number is not divisible by 2.
(iii) p: A triangle is an equilateral triangle.
q: All three sides of a triangle are equal.