Verified Solution • 2025–26 Session moderate

QJEEM20Ma6e90: JEE Main 2020 Mathematics - Class 11 JEEM20Ma6e90 Solution: If , 14 and are the first three terms of an A.P. for some ,…

If 32sin⁡2α−1{3^{2\sin 2\alpha - 1}} , 14 and 34−2sin⁡2α{3^{4 - 2\sin 2\alpha }} are the first three terms of an A.P. for some α\alpha , then the sixth terms of this A.P. is:

Multiple Choice Options

  • A 66
  • B 81
  • C 65
  • D 78

Step-by-Step Verified Solution

Given that

34−sin⁡2α+32sin⁡2α−1=28{3^{4 - \sin 2\alpha }} + {3^{2\sin 2\alpha - 1}} = 28

Let 32sin⁡2α{3^{2\sin 2\alpha }} = t

⇒\Rightarrow 81t+t3=28{{81} \over t} + {t \over 3} = 28

⇒\Rightarrow t = 81, 3

∴\therefore 32sin⁡2α{3^{2\sin 2\alpha }} = 31, 34

sin⁡2α=12\sin 2\alpha = {1 \over 2} , 2 (rejected)

First term a = 32sin⁡2α−1{3^{2\sin 2\alpha -1}} = 30

⇒\Rightarrow a = 1

Given Second term = 14

∴\therefore Common difference d = 13

T6=a+5d{T_6} = a + 5d

T6=1+5×13{T_6} = 1 + 5 \times 13

T6=66{T_6} = 66

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