Which option correctly describes the nature of \(\frac{1}{3}+\sqrt{11}\)?

Irrational number

\(\sqrt{11}\) is irrational while \(\frac{1}{3}\) is rational. Assume for contradiction that \(\frac{1}{3}+\sqrt{11}\) is rational; then \(\sqrt{11}=\big(\frac{1}{3}+\sqrt{11}\big)-\frac{1}{3}\) would be a difference of two rationals and hence rational, contradicting that \(\sqrt{11}\) is irrational. Thus the sum is irrational. The closest distractor, “rational,” fails because a rational plus an irrational cannot be rational. Exam tip: when one term is irrational, directly use contradiction (subtract the rational term) to show the sum is irrational.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.