JEE MAIN - Mathematics (2021 - 20th July Morning Shift - No. 4)
If $$\alpha$$ and $$\beta$$ are the distinct roots of the equation $${x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0$$, then the value of $${\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1)$$ is equal to :
56 $$\times$$ 325
56 $$\times$$ 324
52 $$\times$$ 324
28 $$\times$$ 325
Explanation
As, $$({\alpha ^2} + \sqrt 3 ) = - {(3)^{1/4}}.\alpha $$
$$ \Rightarrow ({\alpha ^4} + 2\sqrt 3 {\alpha ^2} + 3) = \sqrt 3 {\alpha ^2}$$ (On squaring)
$$\therefore$$ $$({\alpha ^4} + 3) = ( - )\sqrt 3 {\alpha ^2}$$
$$ \Rightarrow {\alpha ^8} + 6{\alpha ^4} + 9 = 3{\alpha ^4}$$ (Again squaring)
$$\therefore$$ $${\alpha ^8} + 3{\alpha ^4} + 9 = 0$$
$$ \Rightarrow {\alpha ^8} = - 9 - 3{\alpha ^4}$$
(Multiply by $$\alpha$$4)
So, $${\alpha ^{12}} = - 9{\alpha ^4} - 3{\alpha ^8}$$
$$\therefore$$ $${\alpha ^{12}} = - 9{\alpha ^4} - 3( - 9 - 3{\alpha ^4})$$
$$ \Rightarrow {\alpha ^{12}} = - 9{\alpha ^4} + 27 + 9{\alpha ^4}$$
Hence, $${\alpha ^{12}} = {(27)^2}$$
$$ \Rightarrow {({\alpha ^{12}})^8} = {(27)^8}$$
$$ \Rightarrow {\alpha ^{96}} = {(3)^{24}}$$
Similarly $${\beta ^{96}} = {(3)^{24}}$$
$$\therefore$$ $${\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1) = {(3)^{24}} \times 52$$
$$\Rightarrow$$ Option (3) is correct.
$$ \Rightarrow ({\alpha ^4} + 2\sqrt 3 {\alpha ^2} + 3) = \sqrt 3 {\alpha ^2}$$ (On squaring)
$$\therefore$$ $$({\alpha ^4} + 3) = ( - )\sqrt 3 {\alpha ^2}$$
$$ \Rightarrow {\alpha ^8} + 6{\alpha ^4} + 9 = 3{\alpha ^4}$$ (Again squaring)
$$\therefore$$ $${\alpha ^8} + 3{\alpha ^4} + 9 = 0$$
$$ \Rightarrow {\alpha ^8} = - 9 - 3{\alpha ^4}$$
(Multiply by $$\alpha$$4)
So, $${\alpha ^{12}} = - 9{\alpha ^4} - 3{\alpha ^8}$$
$$\therefore$$ $${\alpha ^{12}} = - 9{\alpha ^4} - 3( - 9 - 3{\alpha ^4})$$
$$ \Rightarrow {\alpha ^{12}} = - 9{\alpha ^4} + 27 + 9{\alpha ^4}$$
Hence, $${\alpha ^{12}} = {(27)^2}$$
$$ \Rightarrow {({\alpha ^{12}})^8} = {(27)^8}$$
$$ \Rightarrow {\alpha ^{96}} = {(3)^{24}}$$
Similarly $${\beta ^{96}} = {(3)^{24}}$$
$$\therefore$$ $${\alpha ^{96}}({\alpha ^{12}} - 1) + {\beta ^{96}}({\beta ^{12}} - 1) = {(3)^{24}} \times 52$$
$$\Rightarrow$$ Option (3) is correct.
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