Let [tex]\( a, b, c, \)[/tex] and [tex]\( d \)[/tex] be real numbers. Given that [tex]\( ac = 1 \)[/tex], [tex]\(\frac{b+c}{d}\)[/tex] is undefined, and [tex]\( abc = d \)[/tex], which of the following must be true?

A. [tex]\( a = 0 \)[/tex] or [tex]\( c = 0 \)[/tex]

B. [tex]\( a = 1 \)[/tex] and [tex]\( c = 1 \)[/tex]

C. [tex]\( a = -c \)[/tex]

D. [tex]\( b = 0 \)[/tex]

E. [tex]\( b + c = 0 \)[/tex]



Answer :

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]