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Binomial Theorem question

2019 · 10 Jan · Shift 2 · Q29
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  5. /2019 · 10 Jan · Shift 2 · Q29

Binomial Theorem question

2019 · 10 Jan · Shift 2 · Q29

JEE MainMathematicsBinomial TheoremMCQ+4 / −1
The positive value of λ\lambdaλ for which the co-efficient of x2 in the expression x2 (x+λx2)10{\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}(x​+x2λ​)10 is 720, is -
  1. A
    4
  2. B
    222\sqrt 222​
  3. C
    3
  4. D
    5\sqrt 55​
View written solutionFree

Correct answer: A

  1. Given expression

We need the coefficient of x2x^2x2 in

x2(x+λx2)10.x^2\left(\sqrt{x}+\frac{\lambda}{x^2}\right)^{10}.x2(x​+x2λ​)10.
  1. General term in the binomial expansion

Write

x=x1/2,λx2=λx−2.\sqrt{x}=x^{1/2}, \qquad \frac{\lambda}{x^2}=\lambda x^{-2}.x​=x1/2,x2λ​=λx−2.

So the general term of

(x1/2+λx−2)10\left(x^{1/2}+\lambda x^{-2}\right)^{10}(x1/2+λx−2)10

is

Tr+1=(10r)(x1/2)10−r(λx−2)r,T_{r+1}=\binom{10}{r}(x^{1/2})^{10-r}(\lambda x^{-2})^r,Tr+1​=(r10​)(x1/2)10−r(λx−2)r,

where r=0,1,2,…,10r=0,1,2,\dots,10r=0,1,2,…,10.

Simplifying,

Tr+1=(10r)λrx10−r2−2r.T_{r+1}=\binom{10}{r}\lambda^r x^{\frac{10-r}{2}-2r}.Tr+1​=(r10​)λrx210−r​−2r.

Now multiply by the outside factor x2x^2x2:

x2Tr+1=(10r)λrx2+10−r2−2r.x^2 T_{r+1}=\binom{10}{r}\lambda^r x^{2+\frac{10-r}{2}-2r}.x2Tr+1​=(r10​)λrx2+210−r​−2r.
  1. Find the term containing x2x^2x2

We want the power of xxx to be 222:

2+10−r2−2r=2.2+\frac{10-r}{2}-2r=2.2+210−r​−2r=2.

Subtract 222 from both sides:

10−r2−2r=0.\frac{10-r}{2}-2r=0.210−r​−2r=0.

Multiply by 222:

10−r−4r=010-r-4r=010−r−4r=0 10−5r=010-5r=010−5r=0 r=2.r=2.r=2.
  1. Coefficient of x2x^2x2

For r=2r=2r=2,

coefficient=(102)λ2.\text{coefficient} = \binom{10}{2}\lambda^2.coefficient=(210​)λ2.

Since

(102)=45,\binom{10}{2}=45,(210​)=45,

we get

45λ2=720.45\lambda^2=720.45λ2=720.

So,

λ2=72045=16.\lambda^2=\frac{720}{45}=16.λ2=45720​=16.

Thus,

λ=4or−4.\lambda=4 \quad \text{or} \quad -4.λ=4or−4.

Since the question asks for the positive value,

λ=4.\lambda=4.λ=4.
  1. Check with options

Option A is 444, so the correct choice is A.

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