मात्रात्मक योग्यता प्रश्न 1729
प्रश्न: यदि $ \sin \theta \cos \theta =1/2, $ तो $ {{\sin }^{6}}\theta +{{\cos }^{6}}\theta $ किसके बराबर है?
विकल्प:
A) 1
B) 2
C) 3
D) $ \frac{1}{4} $
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उत्तर:
सही उत्तर: D
हल:
- दिया गया है, $ \sin \theta .\cos \theta =\frac{1}{2} $ $ {{\sin }^{6}}\theta +{{\cos }^{6}}\theta ={{({{\sin }^{2}}\theta )}^{3}}+{{({{\cos }^{2}}\theta )}^{3}} $ $ =({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )({{\sin }^{4}}\theta +{{\cos }^{4}}\theta -{{\sin }^{2}}\theta {{\cos }^{2}}\theta ) $ $ [\because {{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1] $ $ ={{({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )}^{2}}-2{{\sin }^{2}}\theta {{\cos }^{2}}\theta -{{\sin }^{2}}\theta {{\cos }^{2}}\theta $ $ =(1-3{{\sin }^{2}}\theta {{\cos }^{2}}\theta ) $ $ [ \because \sin \theta \cdot \cos \theta =\frac{1}{2} ] $ $ =1-3\times \frac{1}{4}=1-\frac{3}{4}=\frac{1}{4} $
BETA
