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Greatest Binomial Coefficients

If 1+2x+3x2...

Question

If ${(1 + 2 x + 3 x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{20} x^{20}$ , then $a_{1}$ equals

A

6

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B

8

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C

210

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D

None of these

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Solution

The correct option is C None of these
Given
$(1 + 2 x + 3 x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + - - - + a_{20} x_{20}$
Differentiating Both the sides w.r.t 'x'
$\Rightarrow 10 (1 + 2 x + 3 x^{2})^{9} (2 + 6 x) = a_{1} + 2 a_{2} x + - - - + a^{20} .20 x^{19}$
put x = 0
$\Rightarrow 10 \times 1 \times 2 = a_{1}$
$∴ a_{1} = 20$

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Similar questions

{(1 + 2 x + 3 x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . + a_{20} x^{20}

(1 + 2 x + 3 x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20},

a_{1}

(1 + 2 x + 3 x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}

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i f (1 + x)^{10} = a_{0} + a_{1} x + . . . . a_{10} x^{10}, t h e n (a_{0} - a_{2} + a_{4} - a_{6} + a_{8} - a_{10})^{2} + (a_{1} - a_{3} + a_{5} - a_{7} + a_{9})^{2}

{(1 + x + 2 x^{2})}^{20} = a_{0} + a_{1} x + a_{2} x^{2} + . . . . . + a_{10} x^{40}

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