Answer :
Let's analyze each of the given expressions to determine whether they are real or non-real complex numbers.
### Option A: [tex]\( 2 - \frac{1}{\sqrt{11}} \)[/tex]
[tex]\[ A = 2 - \frac{1}{\sqrt{11}} \][/tex]
Here, both [tex]\(2\)[/tex] and [tex]\(\frac{1}{\sqrt{11}}\)[/tex] are real numbers. Subtracting one real number from another will also result in a real number. Thus, [tex]\( A \)[/tex] is a real number.
Conclusion: [tex]\( A \)[/tex] is real.
### Option B: [tex]\( \frac{9 + 3 \sqrt{5}}{2} \)[/tex]
[tex]\[ B = \frac{9 + 3 \sqrt{5}}{2} \][/tex]
Here, both [tex]\(9\)[/tex] and [tex]\(3 \sqrt{5}\)[/tex] are real numbers. The numerator is a sum of two real numbers, yielding another real number. Dividing a real number by 2, a real number, will result in a real number. Thus, [tex]\( B \)[/tex] is real.
Conclusion: [tex]\( B \)[/tex] is real.
### Option C: [tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]
[tex]\[ C = \frac{8}{3} + \sqrt{-\frac{7}{3}} \][/tex]
Here, [tex]\(\frac{8}{3}\)[/tex] is a real number. However, [tex]\(\sqrt{-\frac{7}{3}}\)[/tex] involves taking the square root of a negative number, which will result in an imaginary number. Specifically, we can rewrite it as:
[tex]\[ \sqrt{-\frac{7}{3}} = \sqrt{-1} \cdot \sqrt{\frac{7}{3}} = i \sqrt{\frac{7}{3}} \][/tex]
Since [tex]\(\sqrt{-\frac{7}{3}}\)[/tex] is an imaginary number, adding it to a real number [tex]\(\frac{8}{3}\)[/tex] will result in a non-real complex number. Thus, [tex]\( C \)[/tex] is a non-real complex number.
Conclusion: [tex]\( C \)[/tex] is non-real.
### Option D: [tex]\( 5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}} \)[/tex]
[tex]\[ D = 5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}} \][/tex]
Here, each term involves a real number. Both [tex]\( 5 \sqrt{\frac{1}{3}} \)[/tex] and [tex]\(\frac{9}{\sqrt{7}}\)[/tex] are real numbers because [tex]\(\sqrt{\frac{1}{3}}\)[/tex] and [tex]\(\sqrt{7}\)[/tex] are real. Subtracting one real number from another will also result in a real number. Thus, [tex]\( D \)[/tex] is real.
Conclusion: [tex]\( D \)[/tex] is real.
### Final Conclusion
Among the given options, only Option C ([tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]) is a non-real complex number.
Answer: C. [tex]\(\frac{8}{3} + \sqrt{-\frac{7}{3}}\)[/tex]
### Option A: [tex]\( 2 - \frac{1}{\sqrt{11}} \)[/tex]
[tex]\[ A = 2 - \frac{1}{\sqrt{11}} \][/tex]
Here, both [tex]\(2\)[/tex] and [tex]\(\frac{1}{\sqrt{11}}\)[/tex] are real numbers. Subtracting one real number from another will also result in a real number. Thus, [tex]\( A \)[/tex] is a real number.
Conclusion: [tex]\( A \)[/tex] is real.
### Option B: [tex]\( \frac{9 + 3 \sqrt{5}}{2} \)[/tex]
[tex]\[ B = \frac{9 + 3 \sqrt{5}}{2} \][/tex]
Here, both [tex]\(9\)[/tex] and [tex]\(3 \sqrt{5}\)[/tex] are real numbers. The numerator is a sum of two real numbers, yielding another real number. Dividing a real number by 2, a real number, will result in a real number. Thus, [tex]\( B \)[/tex] is real.
Conclusion: [tex]\( B \)[/tex] is real.
### Option C: [tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]
[tex]\[ C = \frac{8}{3} + \sqrt{-\frac{7}{3}} \][/tex]
Here, [tex]\(\frac{8}{3}\)[/tex] is a real number. However, [tex]\(\sqrt{-\frac{7}{3}}\)[/tex] involves taking the square root of a negative number, which will result in an imaginary number. Specifically, we can rewrite it as:
[tex]\[ \sqrt{-\frac{7}{3}} = \sqrt{-1} \cdot \sqrt{\frac{7}{3}} = i \sqrt{\frac{7}{3}} \][/tex]
Since [tex]\(\sqrt{-\frac{7}{3}}\)[/tex] is an imaginary number, adding it to a real number [tex]\(\frac{8}{3}\)[/tex] will result in a non-real complex number. Thus, [tex]\( C \)[/tex] is a non-real complex number.
Conclusion: [tex]\( C \)[/tex] is non-real.
### Option D: [tex]\( 5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}} \)[/tex]
[tex]\[ D = 5 \sqrt{\frac{1}{3}} - \frac{9}{\sqrt{7}} \][/tex]
Here, each term involves a real number. Both [tex]\( 5 \sqrt{\frac{1}{3}} \)[/tex] and [tex]\(\frac{9}{\sqrt{7}}\)[/tex] are real numbers because [tex]\(\sqrt{\frac{1}{3}}\)[/tex] and [tex]\(\sqrt{7}\)[/tex] are real. Subtracting one real number from another will also result in a real number. Thus, [tex]\( D \)[/tex] is real.
Conclusion: [tex]\( D \)[/tex] is real.
### Final Conclusion
Among the given options, only Option C ([tex]\( \frac{8}{3} + \sqrt{-\frac{7}{3}} \)[/tex]) is a non-real complex number.
Answer: C. [tex]\(\frac{8}{3} + \sqrt{-\frac{7}{3}}\)[/tex]
