JEE Main 2018 — Mathematics Algebra

$$\left[\frac{1}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8+\left[\frac{1}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8$$[5x3+1​−5x3−1​1​]8+[5x3+1​+5x3−1​1​]8 After rationalise the polynomial we get $$\begin{aligned} &{\left[\frac{1}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}} \times \frac{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}\right]^8} \\ &+\left[\frac{1}{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}} \times \frac{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}\right]^8 \end{aligned}$$​[5x3+1​−5x3−1​1​×5x3+1​+5x3−1​5x3+1​+5x3−1​​]8+[5x3+1​+5x3−1​1​×5x3+1​−5x3−1​5x3+1​−5x3−1​​]8​ $$\left[\frac{\sqrt{5 x^3+1}+\sqrt{5 x^3-1}}{\left(5 x^3+1\right)-\left(5 x^3-1\right)}\right]^8+\left[\frac{\sqrt{5 x^3+1}-\sqrt{5 x^3-1}}{\left(5 x^3+1\right)-\left(5 x^3-1\right)}\right]^8$$[(5x3+1)−(5x3−1)5x3+1​+5x3−1​​]8+[(5x3+1)−(5x3−1)5x3+1​−5x3−1​​]8 $$\begin{aligned} &=\frac{1}{2^8}\left[\left(\sqrt{5 x^3+1}+\sqrt{5 x^3-1}\right)^8+\left(\sqrt{5 x^3+1}-\sqrt{5 x^3-1}\right)^8\right] \\ &\left.=\frac{1}{2^8}\left[\begin{array}{l} \left.{ }^8 C_0\left(\sqrt{5 x^3+1}\right)^8+{ }^8 C_4\left(\sqrt{5 x^3+1}\right)^6\left(\sqrt{5 x^3-1}\right)^2\right)^4\left(\sqrt{5 x^3-1}\right)^4+ \\ { }^8 C_6\left(\sqrt { 5 x ^ { 3 } + 1 } { } ^ { 2 } \left(\sqrt{\left.5 x^3-\right)^1}{ }^6+{ }^8 C_8\left(\sqrt{5 x^3-1}{ }^8\right.\right.\right. \end{array}\right]\right) \\ &\frac{1}{2^8}\left[\begin{array}{l} { }^8 C_0\left(5 x^3+1\right)^4+{ }^8 C_2\left(5 x^3+1\right)^3\left(5 x^3-1\right)+8_{C_4} \\ \left(5 x^3+1\right)^2\left(5 x^3-1\right)^2+ \\ { }^8 C_6\left(5 x^3+1\right)\left(5 x^3-1\right)^3+8_{C_8}\left(5 x^3-1\right)^4 \end{array}\right] \\ & \end{aligned}$$​=281​[(5x3+1​+5x3−1​)8+(5x3+1​−5x3−1​)8]=281​​8C0​(5x3+1​)8+8C4​(5x3+1​)6(5x3−1​)2)4(5x3−1​)4+8C6​(5x3+1​2(5x3−)1​6+8C8​(5x3−1​8​​​281​​8C0​(5x3+1)4+8C2​(5x3+1)3(5x3−1)+8C4​​(5x3+1)2(5x3−1)2+8C6​(5x3+1)(5x3−1)3+8C8​​(5x3−1)4​​​ So, the degree of polynomial is 12 , Now, coefficient of $$\begin{aligned} &x^{12}=\left[{ }^8 \mathrm{C}_0 5^4+{ }^8 \mathrm{C}_2 5^4+{ }^8 \mathrm{C}_4 5^4+{ }^8 \mathrm{C}_6 5^4+{ }^8 \mathrm{C}_8 5^4\right] \\ &=5^4 \times \frac{2^8}{2}=5^4 \times 2^4 \times \frac{2^2}{2} \\ &=10^4 \times 2^3=8(10)^4 \end{aligned}$$​x12=[8C0​54+8C2​54+8C4​54+8C6​54+8C8​54]=54×228​=54×24×222​=104×23=8(10)4​