(A) 7

(B) 3

(C) 4

(D) 9

Correct Answer: 7

Explanation

Given equation is

$$ x^4 - ax^2 + 9 = 0 $$

Let \(x^2 = t\). Then the equation becomes

$$ t^2 - at + 9 = 0 $$

For all roots of the original equation to be real and distinct, the quadratic in \(t\) must have two distinct positive real roots.

First, its discriminant must be positive:

$$ a^2 - 36 > 0 \Rightarrow a > 6 $$

Also, for both roots of \(t\) to be positive:

$$ \text{Sum of roots} = a > 0,\quad \text{Product of roots} = 9 > 0 $$

These conditions are already satisfied for positive \(a\).

Hence the smallest positive integer value of \(a\) greater than 6 is

$$ a = 7 $$

Therefore, the correct answer is 7.

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