Class XI Chapter 14 – Mathematical Reasoning Maths ______________________________________________________________________________ Exercise 14.5 Question 1: Show that the statement p: â€œ If x is a real number such that x 3 + 4x = 0, then x is 0â€ is true by (i)direct method (ii)method of contradiction (iii)method of contrapositive Solution 1: p: â€œ If x is a real number such that x 3 + 4x = 0, then x is 0â€. Let q: x is a real number such that x 3 + 4x = 0 r: x is 0. (i)To show that statement p is true, we assume that q, is true and then show that r is true. Therefore, let statement q be true. âˆ´ x 3 + 4x = 0 x (x 2 + 4) = 0 â‡’ x = 0 or x 2 + 4 = 0 However, since x is real, it is 0. Thus, statement r is true. Therefore, the given statement is true. (ii)To show statement p to be true by contradiction, we assume that p is not true. Let x be a real number such that x 3 + 4x = 0 and let x is not 0. Therefore, x 3 + 4x = 0 x x 2 4 0 x x 2 0or x 4 0 2 0or x 4 However, x is real. Therefore, x = 0, which is a contradiction since we have assumed that x is not 0. Thus, the given statement p is true. (iii)To prove statement p to be true by contrapositive method, we assume that r is false and prove that q must be false. Here, r is false implies that it is required to consider the negation of statement r. This obtains the following statement. â ¼r. x is not 0. 2 I can be seen that x 4 will always be positive. X a ‰ 0 implies that the product of any positive real number with x is not zero. 2 Let us consider the product of x with x 4. Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 14 – Mathematical Reasoning Maths ______________________________________________________________________________ 2 â ´ x (x + 4) â ‰ 0 3 â ‡ x + 4x â ‰ 0 This shows that statement q is not true. Thus, it has been proved that â ¼r â ‡’ â ¼q Therefore, the given statement p is true. Question 2: 2 2 Show that the statement â€œ For any real numbers a and b, a b implies that a a= b â€ is not true by giving a counter-example. Solution 2: The given statement can be written in the form of â€œ if-then â€ as follows. 2 2 If a and b are real numbers such that a b , then a=b. 2 2 Let p: a and b are real numbers such that a b . q: a=b The given statement has to be proved false. For this purpose, it has to be proved that if p, then â ¼q To sow this, two real numbers, a and b, with Let a = 1 and b = -1 a b 2 2 2 2 1 1and 1 1 a b are required such that â ‰ b. 2 2 â 2 2 ’ a b However, a â ‰ b Thus, it can be concluded that the given statement is false. Question 3: Show that the following statement is true by the method of contrapositive. p: If x is an integer and Solution 3: 2 x is even, then x is also even. 2 p: If x is an integer and x is even, then x is also even. 2 Let q: x is an integer and x is even. r: x is even. To prove that p is true by contrapositive method, we assume that r is false, and prove that q is also false. Let x is not even. Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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