Binomial Expansion (WS – 2)

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Question No: 6 [ Maximum Mark: 7 ] (Do not use GDC)

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Expand \(\left(x^2-\frac{1}{x}\right)^5 \)

Answer: \(\left(x^2-\frac{1}{x}\right)^5=x^{10}-5 x^7+10 x^4-10 x+\frac{5}{x^2}-\frac{1}{x^5} \)

\(\left(x^2-\frac{1}{x}\right)^5=1\left(x^2\right)^5+5\left(x^2\right)^4\left(-\frac{1}{x}\right)^1+10\left(x^2\right)^3\left(\frac{-1}{x}\right)^2+10\left(x^2\right)^2\left(-\frac{1}{x}\right)^3+5\left(x^2\right)^1\left(\frac{-1}{x}\right)^4+1\left(\frac{-1}{x}\right)^5 \)

\(\left(x^2-\frac{1}{x}\right)^5=x^{10}+5 x^8\left(\frac{-1}{x}\right)+10 x^6\left(\frac{1}{x^2}\right)+10 x^4\left(\frac{-1}{x^3}\right)+5 x^2\left(\frac{1}{x^4}\right)+\left(\frac{-1}{x^5}\right) \)

\(\left(x^2-\frac{1}{x}\right)^5=x^{10}-5 x^7+10 x^4-10 x+\frac{5}{x^2}-\frac{1}{x^5} \)

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Question No: 9 [ Maximum Mark: 7 ] (Do not use GDC)

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Use \({ }_n C_r \) to find the first three terms and the last two terms in the expansion of \( \left(3 x-\frac{2}{x}\right)^{20}\)

Answer : \(3486784401 x^{20}-46490458680 x^{18}+294439571640 x^{16}+\cdots \cdot-\frac{31457280}{x^{18}}+\frac{1048576}{x^{20}} \)

\( \left(3 x-\frac{2}{x}\right)^9=1(3 x)+20(3 x)^{19}\left(\frac{-2}{x}\right)^1+20 C_2(3 x)^{18}\left(\frac{-2}{x}\right)^2+\cdots \cdot+20(3 x)\left(\frac{-2}{x}\right)^{19}+1\left(\frac{-2}{x}\right)^{20}\)

\( =3^{20} x^{20}+20 \times 3^{19} x^{19} \times\left(\frac{-2}{x}\right)+190 \times 3^{18} x^{18} \times \frac{4}{x^2}+\cdots+20(30 x) \frac{(-2)^{19}}{x^{19}}+\frac{2^{20}}{x^{20}}\)

\( =3486784401 x^{20}-46490458680 x^{18}+294439571640 x^{16}+\cdots-\frac{31457280}{x^{18}}+\frac{1048576}{x^{20}}\)

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Question No: 10 [ Maximum Mark: 7 ] (Do not use GDC)

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Use \({ }_n C_r \) to find the first three and last two terms in the expansion of \( \left(2 x^2-\frac{3}{x}\right)^9\)

Answer :\(\left(2 x^2-\frac{3}{x}\right)^9=512 x^{18}-6912 x^{15}+41472 x^{12}+\cdots+\frac{118098}{x^6}-\frac{19683}{x^9} \)

\(\left(2 x^2-\frac{3}{x}\right)^9=9 c_0\left(2 x^2\right)^9+9 c_1\left(2 x^2\right)^8\left(\frac{-3}{x}\right)+9 c_2\left(2 x^2\right)^7\left(\frac{-3}{x}\right)^2+\cdots \cdots+9 c_8\left(2 x^2\right)^1\left(\frac{-3}{x}\right)^8+9 c_9\left(\frac{-3}{x}\right)^9 \)

\(=2^9 x^{18}+9 \times 2^8 x^{16} \times \frac{-3}{x}+36 \times 2^7 x^{14} \times \frac{9}{x^2}+\cdots \cdot+9 \times 2 x^2 \times 3^8+\frac{\left(-3^9\right)}{x^8}\)

\( =512 x^{18}-6912 x^{15}+41472 x^{12}+\cdots+\frac{118098}{x^6}-\frac{19683}{x^9}\)

Question No: 6 [ Maximum Mark: 7 ] (Do not use GDC)

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Question No: 9 [ Maximum Mark: 7 ] (Do not use GDC)

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Question No: 10 [ Maximum Mark: 7 ] (Do not use GDC)

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Binomial Expansion (WS – 2)

Question No: 6 [ Maximum Mark: 7 ] (Do not use GDC)

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Question No: 9 [ Maximum Mark: 7 ] (Do not use GDC)