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 contrapositive

Solution:

It is given that, p: “If x is a real number such that x³ + 4x = 0 , then x is 0”

Let us assume that q : "x is a real number such that, r : x³ + 4x = 0" and r: "x is 0".

(i) To show that p is true, we prove "if q then r".

Therefore, let statement q be true.

x³ + 4x = 0

x (x² + 4) = 0

x = 0 or x² + 4 = 0

x = 0 or x² = - 4

Since x is real, x² ≠ -4. Hence x = 0.

Thus, statement is true.

Therefore, the given statement is true.

(ii) To show statement p to be true by contradiction, let us assume that the statement p is not true. Then

Let x be a real number such that

x³ + 4x = 0 and x ≠ 0

Therefore,

x³ + 4x = 0

x (x² + 4) = 0

x = 0 or x² + 4 = 0

x = 0 or x² = - 4

Since x is real,x² ≠ -4. Hence x = 0.

Here, x = 0 which is a contradiction since we have assumed that x ≠ 0.

Thus, the given statement p is true.

(iii) To prove statement p to be true by contrapositive method, we prove "if not r then not q".

So we assume r to be false. Then

r : x ≠ 0.

Then, (x² + 4) will always be positive.

x ≠ 0 implies that the product of any positive real number with x is not zero.

Then x (x ² + 4) = x³ + 4x ≠ 0

This shows that statement q is not true.

Thus, we have proved that ~ r ⇒ ~ p

Therefore, the given statement p is true

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NCERT Solutions Class 11 Maths Chapter 14 Exercise 14.5 Question 1

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 contrapositive

Summary:

(i) The given statement is proved by direct method.. (ii) The given statement is proved by the method of contradiction. (iii) The given statement is proved by the method of contrapositive