Question

When expressed in a decimal form, which of the following numbers will be non - terminating as well as non- repeating?

• A. (\(\frac {π}{2}\))[(\(\frac{1}{π}\))+1]-\(\frac {π}{2}\)
• B. sin\(^2\) 1\(^∘\)+ sin\(^2\) 2\(^∘\)+.... + sin\(^2\) 89\(^∘\)
• C. \(\sqrt{2}\)(3\(\sqrt{2}\) - \(\frac{4}{(\sqrt{2})}\))+ \(\sqrt 3\)
• D. (\(\frac{\sqrt[3]{729}} {3}\)) + \(\frac{22}{7}\)
• E. (4-π) [1+(\(\frac{π}{4}\))+\((\frac{π}{4})^2\) +\((\frac{π}{4})^3\) +……..(infinite terms)]

Answer 1

The Correct Option is C

To determine which of the given options will be a non-terminating and non-repeating decimal, we need to evaluate each expression:

1. \(<\frac{π}{2}>[(<\frac{1}{π}>)+1]-<\frac{π}{2}\): This simplifies to \(\frac{π}{2} \times \frac{1+π}{π} - \frac{π}{2} = \frac{π}{2π} + \frac{π}{2} - \frac{π}{2} = \frac{1}{2}\), which is a terminating decimal system.
2. sin²1° + sin²2° + ... + sin²89°: The sum of sine squares from 1° to 89° does not provide a definite pattern, but it is known that the result is 45, a terminating decimal. 
3. \(\sqrt{2}(3\sqrt{2} - \frac{4}{\sqrt{2}})+\sqrt{3}\):
   • Simplify the expression:
     \(= \sqrt{2} \times (3\sqrt{2} - 2) + \sqrt{3}\)
     \(= (3 \cdot 2 - 2) + \sqrt{3}\)
     \(= 4 + \sqrt{3}\)
   • \(4 + \sqrt{3}\) includes \(\sqrt{3}\), an irrational number, making the decimal non-terminating and non-repeating.
4. (\(\frac{\sqrt[3]{729}}{3}\)) + \(\frac{22}{7}\):
   • Simplify the expression:
     \(= \frac{9}{3} + \frac{22}{7}\)
     \(= 3 + \frac{22}{7}\), where \(\frac{22}{7}\) is an approximation of \(\pi\), so the result is a non-terminating repeating decimal.
5. (4-π)[1+(\(\frac{π}{4}\))+(\(\frac{π}{4}\))²+(\(\frac{π}{4}\))³+... ] : This infinite series is a geometric series with a common ratio of \(\frac{π}{4}\). Evaluating this gives non-rational numbers, but \((4-π)\) times this series will yield non-terminating results since \(4-π\) is irrational. However, it simplifies to a terminating decimal due to the geometric convergence.

Conclusion: The correct option is \(\sqrt{2}(3\sqrt{2} - \frac{4}{\sqrt{2}})+\sqrt{3}\), which produces a non-terminating, non-repeating decimal.