Binomial Theorem 1 Question 7
8. The sum of the coefficients of all even degree terms is $x$ in the expansion of
(2019 Main, 8 April I)
$\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6},(x>1)$ is equal to
(a) 29
(b) 32
(c) 26
(d) 24
Show Answer
Answer:
Correct Answer: 8. $(a=2, n=4)$
Solution:
Key Idea Use formula :
$(a+b)^{n}+(a-b)^{n}=$
$$ 2\left[{ }^{n} C _0 a^{n}+{ }^{n} C _2 a^{n-2} b^{2}+{ }^{n} C _4 a^{n-4} b^{4}+\ldots \ldots\right] $$
Given expression is $\left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6}$
$$ \begin{aligned} & =2\left[{ }^{6} C _0 x^{6}+{ }^{6} C _2 x^{4}\left(\sqrt{x^{3}-1}\right)^{2}\right. \\ & \left.+{ }^{6} C _4 x^{2}\left(\sqrt{x^{3}-1}\right)^{4}+{ }^{6} C _6\left(\sqrt{x^{3}-1}\right)^{6}\right] \\ & {\because(a+b)^{n}+(a-b)^{n}\right. \\ & \left.=2\left[{ }^{n} C _0 a^{n}+{ }^{n} C _2 a^{n-2} b^{2}+{ }^{n} C _4 a^{n-4} b^{4}+\ldots\right] } \end{aligned} $$
$$ =2\left[{ }^{6} C _0 x^{6}+{ }^{6} C _2 x^{4}\left(x^{3}-1\right)+{ }^{6} C _4 x^{2}\left(x^{3}-1\right)^{2}+{ }^{6} C _6\left(x^{3}-1\right)^{3}\right] $$
The sum of the terms with even power of $x$
$$ \begin{aligned} & =2\left[{ }^{6} C _0 x^{6}+{ }^{6} C _2\left(-x^{4}\right)+{ }^{6} C _4 x^{8}+{ }^{6} C _4 x^{2}+{ }^{6} C _6\left(-1-3 x^{6}\right)\right] \\ & =2\left[{ }^{6} C _0 x^{6}-{ }^{6} C _2 x^{4}+{ }^{6} C _4 x^{8}+{ }^{6} C _4 x^{2}-1-3 x^{6}\right] \end{aligned} $$
Now, the required sum of the coefficients of even powers of $x$ in
$$ \begin{aligned} & \left(x+\sqrt{x^{3}-1}\right)^{6}+\left(x-\sqrt{x^{3}-1}\right)^{6} \\ & \quad=2\left[{ }^{6} C _0-{ }^{6} C _2+{ }^{6} C _4+{ }^{6} C _4-1-3\right] \\ & \quad=2[1-15+15+15-1-3]=2(15-3)=24 \end{aligned} $$
