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Prove that (2+sqrt(x))^(4)+(2-sqrt(x))^(...

Prove that `(2+sqrt(x))^(4)+(2-sqrt(x))^(4)= 2(16+24x+x^(2))`.

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` ( a + b ) ^ ( 4 ) + ( a - b ) ^ ( 4) = [.^ ( 4 ) C_(0) a^ ( 4 ) + .^ ( 4) C_(1) a^(3)b +.^(4)C_(2)a^(2)b ^ ( 2) + .^(4) C_(3)ab^(3) + .^(4)C_(4)b^(4)]`
`+ [.^(4)C_(0) a^(4) + .^ (4)C_(1) a^(3) b + .^(4) C_(2)a^(2)b^(2) - .^(4)C_(3)ab^(3) + .^(4) C _(4)b^(4)]`
`=2[.^(4) C _ (0) a^(4) + .^(4) C _(2) a^(2) b^(2) + .^(4) C _(4) b^(4) ]= 2 [a^(4) + 6a^(2) b^(2) + b^(4)].`
Putting `a=2 and b= sqrt(x),` we get
`(2+ sqrt(x) )^(4) + ( 2 - sqrt(x))^(4) = 2[2^(4) + 6 xx 2 ^(2) xx ( sqrt(x))^(2) + ( sqrt(x))^(4) ] = 2 ( 16+ 24x + x^(2)).`
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RS AGGARWAL-BINOMIAL THEOREM-EXERCISE 10A
  1. Using binomial theorem, evaluate each of the following: (i)(104)^(4...

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  2. Using binomial theorem, prove that (2^(3n)-7n-1) is divisible by 49, w...

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  3. Prove that (2+sqrt(x))^(4)+(2-sqrt(x))^(4)= 2(16+24x+x^(2)).

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  4. Find the 7th term in the expansion of ((4x)/5+5/(2x))^(8)

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  5. Find the 9th term in the expansion of (a/b-b/(2a)^(2))^(12).

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  6. Find the 16th term in the expansion of (sqrt(x)-sqrt(y))^(17)

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  7. Find the 13^(t h)term in the expansion of (9x-1/(3sqrt(x)))^(18),x!=0

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  8. If the coefficients of x^7 and x^8 in the expansion of [2 +x/3]^n a...

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  9. The ratio of the coefficient of x^(15) to the term independent of x in...

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  10. Prove that the ratio of the coefficient of x^10 in (1 - x^2)^10 & the ...

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  11. Find the term independent of x in the expansion of (1+x+2x^3)[(3x^2//2...

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  12. Find the coefficient of :\ x\ in the expansion of (1-3x+7x^2)(1-x)^(1...

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  13. Find the coefficient of (i) x^(5)" in the expansion of "(x+3)^(8) (...

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  14. Show that the term containing to does not exist in the expansion of (3...

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  15. Does the expansion of (2x^2-1/x)^(20) contain any term involving x^9?

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  16. Show that the expansion of (x^2+1/x)^1 does not contain any term invol...

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  17. Write the general term in the expansion of (x^(2)-y)^(6).

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  18. Find the 5th term from the end in the expansion of (x-1/x)^(12).

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  19. Find the 4th term from the end in the expansion of ((4x)/5-5/(2x))^9do...

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  20. If the 7th terms from the beginning and end in the expansion of ( root...

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