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  4. Some Standard Expansions - Practice Questions & MCQ

Some Standard Expansions - Practice Questions & MCQ

Updated on Sep 18, 2023 18:34 AM

Quick Facts

  • Some Standard Expansions (Part 2) is considered one of the most asked concept.

  • 36 Questions around this concept.

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QuestionEASY

The sum of coefficients of integral powers of  x  in  the  binomial  expansion  of

\left ( 1-2\sqrt{x} \right )^{50}\; is:

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The number of non-zero terms in the expansion of (1+3 \sqrt{2} x)^9+(1-3 \sqrt{2} x)^9

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The value of (\sqrt{2}+1)^6+(\sqrt{2}-1)^6 will be:

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Concepts Covered - 2

Some Standard Expansions (Part 1)

We know the binomial expansion,

\\\mathrm{(x+y)^{n}=^{n} C_{0} x^{n}+^{n} C_{1} x^{n-1} y+^{n} C_{2} x^{n-2} y^{2}+\cdots+^{n} C_{n} y^{n}}\\\text{i.e.}\;\;\;\;\;\;\;\;(x+y)^{n}=\sum _{r=0} ^{n}\;^nC_rx^{n-r}y^r

 

  1. Replace ‘y’ with ‘-y’ in the binomial expansion, we get

\\\mathrm{(x-y)^{n}=^{n} C_{0} x^{n}-^{n} C_{1} x^{n-1} y+^{n} C_{2} x^{n-2} y^{2}-\cdots+(-1)^r\;^nC_rx^{n-r}y^r+\cdots+(-1)^n\;^{n} C_{n} y^{n}}\\\text{or}\;\;\;\;\;\;\;\;\mathrm{(x-y)^{n}=\sum_{r=0}^{n}(-1)^r\;^nC_rx^{n-r}y^r}
 

  1. In the binomial expansion,  (x + y)n replace ‘x’ by 1 and ‘y’ by x

\\\mathrm{(1+x)^{n}=^{n} C_{0}\; x^{0}+^{n} C_{1} \;x^{1}+^{n} C_{2}\; x^{2} +\cdots+^nC_r\;x^r+\cdots+^{n} C_{n} \;x^{n}}\\\text{or}\;\;\;\;\;\;\;\;\mathrm{(1+x)^{n}=\sum_{r=0}^{n}\;{^n}C_r\;x^{r}}

  1. In the binomial expansion,  (x + y)n replace ‘x’ by ‘1’ and ‘y’ by ‘-x’

\\\mathrm{(1-x)^{n}=^{n} C_{0}\; x^{0}-^{n} C_{1} \;x^{1}+^{n} C_{2}\; x^{2} -\cdots+(-1)^r\;^nC_r\;x^r+\cdots+(-1)^n\;^{n} C_{n} \;x^{n}}\\\text{or}\;\;\;\;\;\;\;\;\mathrm{(1-x)^{n}=\sum_{r=0}^{n}(-1)^r\;^nC_r\;x^{r}}

Some Standard Expansions (Part 2)

         4. Addition: (x + y)n +  (x - y)n

\mathrm{(x+y)^{n}+(x-y)^{n}=2\left[^{n} C_{0}\; x^{n}\; y^{0}+^{n} C_{2} \;x^{n-2}\; y^{2}+^{n} C_{4}\; x^{n-4} \;y^{4}+\ldots .\right]}

            \begin{array}{l}{\text { If }^{\prime} n^{\prime} \text { is odd then number of terms is } \frac{n+1}{2}} \\ {\text { If } n^{\prime} \text { is even then number of terms is } \frac{n}{2}+1}\end{array}

         5. Subtraction: (x + y)n -  (x - y)n

\mathrm{(x+y)^{n}-(x-y)^{n}=2\left[^{n} C_{1}\; x^{n-1} \;y^{1}+^{n} C_{3} \;x^{n-3}\; y^{3}+^{n} C_{5} \;x^{n-5}\; y^{5}+\ldots \ldots\right]}

            \begin{array}{l}{\text { If }n \text { is odd, then the number of terms is } \frac{n+1}{2}} \\ {\text { If }n \text { is even, then the number of terms is } \frac{n}{2}}\end{array}

Study it with Videos

Some Standard Expansions (Part 1)
Some Standard Expansions (Part 2)

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