SSC CGL Tier-II practice question
If a = (√35 + √33) / (√35 − √33), find the value of (a³ + 1/a³) + 3(a + 1/a).
From a SSC CGL Tier-II Quantitative Aptitude practice set · Quantitative Aptitude
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Correct answer: (C) 314432
Rationalise a = (√35 + √33) / (√35 − √33) by multiplying numerator and denominator by (√35 + √33):
a = (√35 + √33)² / (35 − 33) = (35+33 + 2√1155) / 2 = 34 + √1155.
Then 1/a = 34 − √1155, since (34 + √1155)×(34 − √1155) = 34² − 1155 = 1.
So a + 1/a = 68 = 2×34.
Using a³ + 1/a³ = (a + 1/a)³ − 3(a + 1/a), the required expression becomes:
(a³ + 1/a³) + 3(a + 1/a) = (a + 1/a)³ = (68)³ = 314432.
हिंदी में प्रश्न
यदि a = (√35 + √33) / (√35 − √33) है, तो (a³ + 1/a³) + 3(a + 1/a) का मान ज्ञात कीजिए।
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सही उत्तर: (C) 314432
a = (√35 + √33) / (√35 − √33) को अंश व हर में (√35 + √33) से गुणा करके परिमेयीकरण करने पर:
a = (√35 + √33)² / (35 − 33) = (35+33 + 2√1155) / 2 = 34 + √1155।
तब 1/a = 34 − √1155, क्योंकि (34 + √1155)×(34 − √1155) = 34² − 1155 = 1।
अतः a + 1/a = 68 = 2×34।
सर्वसमिका a³ + 1/a³ = (a + 1/a)³ − 3(a + 1/a) का प्रयोग करने पर, अभीष्ट व्यंजक बन जाता है:
(a³ + 1/a³) + 3(a + 1/a) = (a + 1/a)³ = (68)³ = 314432।
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