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Model Paper 9 — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSModel Paper 93 Marks
Question
Find the expansion of $\left(3 x^2-2 a x+3 a^2\right)^3$ using binomial theorem.
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Answer

We have
$\left.\left(3 x^2-2 a x+3 a^2\right)^3=\left[\left(3 x^2-2 a x\right)+3 a^2\right)\right]^3$
$\begin{array}{l}={ }^3 C_0\left(3 x^2-2 a x\right)^3+{ }^3 C_1\left(3 x^2-2 a x\right)^2\left(3 a^2\right)+{ }^3 C_2\left(3 x^2-2 a x\right)\left(3 a^2\right)^2+{ }^3 C_3\left(3 a^2\right)^3 \\ =\left(3 x^2-2 a x\right)^3+3 \times 3 a^2\left(3 x^2-2 a x\right)^2+3 \times 9 a^4\left(3 x^2-2 a x\right)+27 a^6 \\ =\left(27 x^6-8 a^3 x^3-54 a x^5+36 a^2 x^4\right)+9 a^2\left(9 x^4+4 a^2 x^2-12 a x^3\right)+27 a^4\left(3 x^2-2 a x\right)+27 a^6 \\ =27 x^6-8 a^3 x^3-54 a x^5+36 a^2 x^4+81 a^2 x^4+36 a^4 x^2-108 a^3 x^3+81 a^4 x^2-54 a^5 x+27 a^6 \\ =27 x^6-54 a x^5+117 a^2 x^4-116 a^3 x^3+117 a^4 x^2-54 a^5 x 27 a^6\end{array}$

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