Alright, let's break down the problem step-by-step.
First, we are given three conditions:
1. [tex]\( ac = 1 \)[/tex]
2. [tex]\( \frac{b+c}{d} \)[/tex] is undefined
3. [tex]\( abc = d \)[/tex]
Let's analyze each condition:
### Condition 1: [tex]\( ac = 1 \)[/tex]
This implies that the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex] is equal to 1. This means neither [tex]\(a\)[/tex] nor [tex]\(c\)[/tex] can be zero, because any number multiplied by zero equals zero, not one.
### Condition 2: [tex]\( \frac{b+c}{d} \)[/tex] is undefined
The expression is undefined when division by zero occurs. Hence, for this expression to be undefined, [tex]\(d\)[/tex] must be zero.
### Condition 3: [tex]\( abc = d \)[/tex]
We know from condition 2 that [tex]\(d = 0\)[/tex], so substituting [tex]\(d\)[/tex] in condition 3 gives us:
[tex]\[ abc = 0 \][/tex]
Since neither [tex]\(a\)[/tex] nor [tex]\(c\)[/tex] can be zero (from condition 1), the only variable that can be zero to satisfy this equation is [tex]\(b\)[/tex].
Thus, the conclusion is that [tex]\(b\)[/tex] must be zero.
Therefore, among the given options, the correct statement must be:
D. [tex]\( b = 0 \)[/tex]
