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Some Standard Expansions - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

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

  • 21 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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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)

Study other Related Concepts

Some Standard Expansions Current Topic

Binomial Theorem - Formula, Expansion, Problems and Applications
Some Standard Expansions
General and Middle Terms
Greatest Term (numerically)
An Important Theorem
Problems on Divisibility
Finding last digits
Important Result (Comparison)
Multinomial Theorem
Series Involving Binomial Coefficients

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