Question
Download Solution PDFदो दी गई प्राकृतिक संख्याओं के घनों के बीच का अंतर 6272 है, जबकि दोनों दी गई संख्याओं के बीच धनात्मक अंतर 8 है। दी गई दोनों संख्याओं के घनों का योग क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFx3 - y3 = 6272
x - y = 8
प्रयुक्त सूत्र:-
( A3 + B3) = ( A + B)3 - 3AB( A + B)
( A3 - B3) = ( A - B)3 + 3AB( A - B)
गणना:-
सूत्र के अनुसार,
x3 - y3 = ( x - y )3 + 3xy(x - y)
⇒ ( x - y )3 + 3xy(x - y) = 6272
⇒ 83 + 3xy(8) = 6272
⇒24xy = 6272 - 512 = 5760
⇒ xy = 240 .....(1)
अब,
(x - y) = 8 [दिया गया है]
दोनों पक्षों का वर्ग करने पर,
(x - y)2 = 64
⇒ x2 + y2 - 2xy = 64
⇒ x2 + y2 = 64 + 2× 240 [ xy = 240]
⇒ x2 + y2 = 544 ....(2)
हम जानते हैं कि
(x + y)2 = x2 + y2 + 2xy
⇒ (x + y)2 = 544 + 2× 240
⇒ (x + y)2 = 1024
⇒ (x + y) = 32
प्रश्न के अनुसार,
x3 + y3 = ( x + y )3 - 3xy(x + y)
⇒ x3 + y3 = 323 - 3 × 32 × 240
⇒ 32768 - 23040 = 9728
∴ अभीष्ट उत्तर 9728 है।
Last updated on Jul 19, 2025
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