Let's analyze the given information and mathematical conditions step-by-step:
1. Given conditions:
- [tex]\(ac = 1\)[/tex]
- [tex]\(\frac{b+c}{d}\)[/tex] is undefined
- [tex]\(abc = d\)[/tex]
2. Condition for [tex]\(\frac{b+c}{d}\)[/tex] to be undefined:
[tex]\(\frac{b+c}{d}\)[/tex] is undefined when the denominator [tex]\(d = 0\)[/tex].
3. First, from [tex]\(ac = 1\)[/tex]:
Since [tex]\(a\)[/tex] and [tex]\(c\)[/tex] are real numbers, neither [tex]\(a\)[/tex] nor [tex]\(c\)[/tex] can be zero because their product is 1. Therefore, we can eliminate option (A).
4. Given [tex]\(d = abc\)[/tex]:
Since we established that [tex]\(d = 0\)[/tex] to make [tex]\(\frac{b+c}{d}\)[/tex] undefined, substitute [tex]\(d\)[/tex] with [tex]\(abc\)[/tex]:
[tex]\[
abc = 0
\][/tex]
Since neither [tex]\(a\)[/tex] nor [tex]\(c\)[/tex] can be zero (from [tex]\(ac = 1\)[/tex]), it follows that [tex]\(b\)[/tex] must be 0.
This confirms that option (D) must be true:
[tex]\[
b = 0
\][/tex]
5. Checking other options:
- Option (B) [tex]\(a = 1\)[/tex] and [tex]\(c = 1\)[/tex]: If this were true, [tex]\(ac = 1 \cdot 1 = 1\)[/tex], which holds true, but this specific combination is not necessary for the given conditions to hold.
- Option (C) [tex]\(a = -c\)[/tex]: Substituting in [tex]\(ac = 1\)[/tex], we get [tex]\(a \cdot (-a) = -a^2 = 1\)[/tex], which implies [tex]\(a^2 = -1\)[/tex]. This is not possible for real number [tex]\(a\)[/tex].
- Option (E) [tex]\(b + c = 0\)[/tex]: Since [tex]\(b = 0\)[/tex], we get [tex]\(0 + c = 0\)[/tex], which implies [tex]\(c = 0\)[/tex]. This contradicts our given fact that [tex]\(c\)[/tex] cannot be zero. Therefore, this is not a valid option.
Given the conditions for the expressions to be defined and what we derived from the given equations, the condition that must be true is:
D. [tex]\(b = 0\)[/tex]
