निम्नलिखित संख्याओं पर विचार करें:

I. tan 22.5°

II. cot 22.5°

III. tan 22.5° - cot 22.5°

उपरोक्त में से कितनी अपरिमेय संख्याएँ हैं?

This question was previously asked in
NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. कोई नहीं
  2. केवल एक
  3. केवल दो
  4. तीनों

Answer (Detailed Solution Below)

Option 3 : केवल दो
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NDA 01/2025: English Subject Test
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Detailed Solution

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व्याख्या:

(I) मान लीजिए tan 22.5° = tan 45/2 = t

⇒ tan45° = \(tan( 2 \times\frac{45}{2})\)

= \(\frac{2tan\frac{45}{2}}{1 -tan^2\frac{45}{2}}\)

⇒ 1 = \(\frac{2t}{1-t^2}\) ⇒ t2 + 2t -1 = 0

⇒ t = \(\frac{-2 ± √4+4}{2} = -1±√2\)

⇒ tan \(\frac{45}{2} = -1 ± √2 = -1 +√2\) (प्रथम चतुर्थांश में tan धनात्मक है)

यह एक अपरिमेय संख्या है।

(II) cot22.5° = \(\frac{1}{tan 22.5} = \frac{1}{-1 +√2} = 1+√2\)

यह भी एक अपरिमेय संख्या है

(III) tan22.5° - cot22.5° = -1+ √2 -(1 +√ 2)

= -2

यह एक अपरिमेय संख्या नहीं है

∴ विकल्प (ग) सही है।

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