JEE MAIN - Mathematics (2020 - 5th September Morning Slot - No. 9)
If $${3^{2\sin 2\alpha - 1}}$$, 14 and $${3^{4 - 2\sin 2\alpha }}$$ are the first three terms of an A.P. for some $$\alpha $$, then the sixth
terms of this A.P. is:
66
81
65
78
Explanation
Given that
$${3^{4 - \sin 2\alpha }} + {3^{2\sin 2\alpha - 1}} = 28$$
Let $${3^{2\sin 2\alpha }}$$ = t
$$ \Rightarrow $$ $${{81} \over t} + {t \over 3} = 28$$
$$ \Rightarrow $$t = 81, 3
$$ \therefore $$ $${3^{2\sin 2\alpha }}$$ = 31, 34
$$\sin 2\alpha = {1 \over 2}$$, 2 (rejected)
First term a = $${3^{2\sin 2\alpha -1}}$$ = 30
$$ \Rightarrow $$ a = 1
Given Second term = 14
$$ \therefore $$ Common difference d = 13
$${T_6} = a + 5d$$
$${T_6} = 1 + 5 \times 13$$
$${T_6} = 66$$
$${3^{4 - \sin 2\alpha }} + {3^{2\sin 2\alpha - 1}} = 28$$
Let $${3^{2\sin 2\alpha }}$$ = t
$$ \Rightarrow $$ $${{81} \over t} + {t \over 3} = 28$$
$$ \Rightarrow $$t = 81, 3
$$ \therefore $$ $${3^{2\sin 2\alpha }}$$ = 31, 34
$$\sin 2\alpha = {1 \over 2}$$, 2 (rejected)
First term a = $${3^{2\sin 2\alpha -1}}$$ = 30
$$ \Rightarrow $$ a = 1
Given Second term = 14
$$ \therefore $$ Common difference d = 13
$${T_6} = a + 5d$$
$${T_6} = 1 + 5 \times 13$$
$${T_6} = 66$$
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