Verified Solution • 2025–26 Session moderate

QJEEM20Ma147c: JEE Main 2020 Mathematics - Class 11 JEEM20Ma147c Solution: If and be the coefficients of x 4 and x 2 respectively in the…

If α\alpha and β\beta be the coefficients of x4 and x2 respectively in the expansion of
(x+x2−1)6+(x−x2−1)6{\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6} , then

Multiple Choice Options

  • A α+β=60\alpha + \beta = 60
  • B α−β=60\alpha - \beta = 60
  • C α+β=−30\alpha + \beta = -30
  • D α−β=−132\alpha - \beta = -132

Step-by-Step Verified Solution

(x+a)n + (x – a)n = 2(T1 + T3 + T5 +.....)

(x+x2−1)6+(x−x2−1)6{\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}

= 2[T1 + T3 + T5 + T7 ]

= 2[6C0 x6 + 6C2 x4(x2 – 1) + 6C4 x2(x2 –1)2 + 6C6 x0(x2–1)3]

= 2[x6+ 15(x6 – x4) + 15x2 (x4 + 1 –2x2) + (x6 – 3x4 +3x2 –1)]

= 2[x6(2 + 15 + 15 + 1) + x4(–15 – 30 –3) + x2(15 + 3)]

Coefficient of x4 = α\alpha = -96

And coefficient of x2 = β\beta = 36

∴\therefore α−β=−96−36=−132\alpha - \beta = - 96 - 36 = -132

Related Learning Resources & Exam Preparation

JEE Main PYQ Question Bank
Browse year-wise authentic previous years' questions with solutions
Authentic CBT Mock Tests
Simulate official exam environment with live timer and percentile scoring