JEE MAIN - Mathematics (2020 - 8th January Evening Slot - No. 11)
If $$\alpha $$ and $$\beta $$ be the coefficients of x4 and x2
respectively in the expansion of
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$, then
$$\alpha + \beta = 60$$
$$\alpha - \beta = 60$$
$$\alpha + \beta = -30$$
$$\alpha - \beta = -132$$
Explanation
(x+a)n + (x – a)n
= 2(T1
+ T3
+ T5
+.....)
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$
= 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 = $$\alpha $$ = -96
And coefficient of x2 = $$\beta $$ = 36
$$ \therefore $$ $$\alpha - \beta = - 96 - 36 = -132$$
$${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$$
= 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 = $$\alpha $$ = -96
And coefficient of x2 = $$\beta $$ = 36
$$ \therefore $$ $$\alpha - \beta = - 96 - 36 = -132$$
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