Question

Question 1 Show that the statement

p: “If x is a real number such that x^3 + 4x = 0, then x is 0” is true by

(i) direct method

(ii) method of contradiction

(iii) method of contrapositive

Class X1 - Maths -Mathematical Reasoning Page 342

Answer 1

(i) direct method :
x³ + 4x = 0
x( x² + 4) = 0
x = 0 , x² + 4 ≠ 0 , x ∈ R

(ii) method of contradiction :
Let x ≠ 0 , then Let x = Q , where Q ∈ R
and Q is a root of x³ + 4x = 0 .
so, put x = Q in equation ,
Q³ + 4Q = 0
Q( Q² + 4) = 0
Q = 0 , because Q² + 4 ≠ 0
hence, Q = x = 0

(iii) method of contrapositive :
Let x = 0 is not true statement .
and x = Q ≠ 0 .
∴ Q³ + 4Q = 0 ,Q being the root of x² + 4 < 0
or, Q( Q² + 4) = 0,
Now, Q = 0, also Q² + 4 = 0

∵ Q( Q² + 4) ≠ 0 if Q is not true.
∴ x = 0,is the root of x³ + 4x = 0