Q14 of 18 Page 13

Show that the statement p: "If x is a real number such that x3 + 4x = 0, then x is 0" is true by

(i) Direct method.

(ii) Method of contradiction.

(iii) Method of contra positive.


(i) Direct method: x3 + 4x = 0 or x(x2 + 4) = 0
Now x2 + 4 ≠ 0 and x ∈ R and hence x = 0.

(ii) Method of Contradiction: Let x ≠ 0 and let x = p, p ∈ R is a root of x3 + 4x = 0. Therefore p3 + 4p = 0
(or) p(p2 + 4) = 0 as p = 0 Thus p2 + 4 = 0 which is not possible. Therefore, our supposition is wrong. Hence p = 0 or x = 0
 
(iii) Contra positive method: p is not true

Let x = 0 is not true Let x = p ≠ 0

Therefore p3 + 4p = 0

P being the root of x2 + 4 = 0

(or) p(p2 + 4) = 0

Now p = 0, also (p2 + 4) = 0

Which implies p(p2 + 4) = 0

Now p = 0, also p2 + 4 = 0

p(p2 + 4) ≠ 0 if p is not true.

Hence x = 0 is the root of x3 + 4x = 0

More from this chapter

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12
Write each of the following statements in the form of "if-then"

(i) You get a job implies that your credentials are good.

(ii) The Banana tree will bloom if it stays warm for a month.

(iii) A quadrilateral is a parallelogram if its diagonals bisect each other.

(iv) To get an A+ in the class, it is necessary that you do all the exercises of the book.

13
Given statements in (a) and (b).
Identify the statements given below as contra positive or converse of each other.
(a)
(i) If you live in Delhi, then you have winter clothes.
(ii) If you do not have winter clothes, then you do not live in Delhi.

(b)
 (i) If a quadrilateral is a parallelogram, then its diagonals bisect each other.
(ii) If the diagonals of a quadrilateral do not bisect each other, then the
quadrilateral is not a parallelogram.

15
Show that the statement "For any real number a and b, a2 = b2 implies that a = b" is not true by giving a counter example.

16
Show that the following statement is true by the method of contra positive.