# 1.(i)    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,

If $x$ is a real number such that $x^3 + 4x = 0$, then $x$ is 0 : (if p then q)

p: x is a real number such that $\dpi{100} x^3 + 4x = 0$.

q: x is 0.

In order to prove the statement “if p then q”

Direct Method:  By assuming that p is true, prove that q must be true.

So,

p is true:There exists a real number x such that $\dpi{100} x^3 + 4x = 0 \implies x(x^2 + 4) = 0$

$\dpi{100} \implies x = 0\ or\ (x^2 + 4)= 0$

$\dpi{100} \implies x = 0\ or\ x^2 = -4\ (not\ possible)$

Hence, x = 0

Therefore q is true.

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