Question
Download Solution PDFयदि α और β समीकरण 4x2 + 2x - 1 = 0 के मूल हैं, तो निम्नलिखित में से कौन-सा सही है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा:
सामान्य द्विघात समीकरण
ax2 + bx + c = 0
- मूलों का गुणनफल (αβ) = c/a
- मूलों का योग (α + β) = -b/a
- इस सूत्र का उपयोग करके मूल प्राप्त करना,
- sin 3θ = 3 sinθ - 4 sin3θ
गणना:
दिया गया: 4x2 + 2x - 1 = 0
a = 4, b = 2, c = -1
⇒
⇒
जैसा कि हम जानते हैं कि
तो, जैसा कि हम जानते हैं α = sin 18° और β = - sin 54° -----(i)
अब, दिए गए सूत्र से, sin 3θ = 3 sinθ - 4 sin3θ
उपरोक्त त्रिकोणमितीय सूत्र में θ = 18° रखने पर, हम प्राप्त करते हैं
⇒ sin 54° = 3 sin18° - 4 sin318° ----(ii)
(i) और (ii) से, हम प्राप्त करते हैं
⇒ - β = 3α - 4α3
⇒ β = 4α3 - 3α
∴ सही संबंध β = 4α3 - 3α है।
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