Question
Download Solution PDFযদি (a + b - c) = 20 এবং a2 + b2 + c2 = 152 হয়, তাহলে a3 + b3 - c3 + 3abc এর মান নির্ণয় কর।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFপ্রদত্ত:
(a + b - c) = 20;
a2 + b2 + c2 = 152
সূত্র ব্যবহৃত:
a3 + b3 = (a + b)3 - 3ab × (a + b)
গণনা:
আমাদের এই সমীকরণে 3টি ভেরিয়েবল রয়েছে এবং 2টি সমীকরণ দেওয়া হয়েছে।
তারপর c = 0 দিন
⇒ (a + b) = 20
⇒ a2 + b2 = 152
এখন,
(a + b) = 20
উভয় পক্ষের স্কোয়ারিং
⇒ a2 + b2 + 2ab = 400
⇒ 152 + 2ab = 400
⇒ 2ab = 400 - 152 = 248
⇒ ab = 248/2 = 124
a3 + b3 = (a + b)3 - 3ab × (a + b)
⇒ (20)3 - 3 × 124 × 20
⇒ 8000 - 7440 = 560
∴ সঠিক উত্তর হল 560।
Alternate Method
ধারণা:
প্রয়োজনীয় রাশি খুঁজতে বীজগণিতীয় পরিচয় ব্যবহার করুন।
হিসাব
ঘনক্ষেত্রের যোগফলের জন্য পরিচয় ব্যবহার করুন:
আমরা জানি,
a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)
যদি c ⇒ - c, তাহলে
a3 + b3 - c3 + 3abc = (a + b - c){(a2 + b2 + c2 - ab - b(-c) - (-c)a)}
⇒ a3 + b3 - c3 + 3abc = (a + b - c)(a2 + b2 + c2 - ab + bc + ca)
⇒ a3 + b3 - c3 + 3abc = (a + b - c){(a2 + b2 + c2 - (ab - bc - ca)}
প্রদত্ত:
a + b - c = 20
a2 + b2 + c2 = 152
খুঁজুন
( ab + bc - ca):
⇒ (a + b - c)2 = a2 + b2 + c2 + 2ab - 2bc - 2ca
⇒ 202 = 152 + 2ab - 2bc - 2ca
⇒ 400 = 152 + 2(ab - bc - ca)
⇒ 248 = 2(ab - bc - ca)
⇒ ab - bc - ca = 124
অভিব্যক্তিতে বিকল্প:
a3 + b3 - c3 + 3abc
⇒(a + b - c){(a2 + b2 + c2 - (ab - bc - ca)}
= 20 × (152 - 124)
= 20 × 28
= 560
∴ a3 + b3 - c3 + 3abc এর মান হল 560।
Last updated on Jun 13, 2025
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