1.

If(1+ 2x+3x^2 ) ^(10)=a _ 0 +a _ 1 x +a _2 x ^ 2 + … +a _(20) x ^ (20), then(a _ 2)/(a_1)isequal to

Answer»

`10.5 `
`21`
`10 `
`5.5 `

Solution : ` ( 1+ 2x+ 3x^ 2 ) ^ (10)= a _ 0+ a _ 1 x +a _ 2 x ^ 2+… + a _(20)x ^(20) `
`rArr( 1 + x ( 2 + 3x ) )^(10)= a _0 + a _1 x + a _ 2 x ^ 2+ … + a _ (20)x^(20) `
` rArr ""^(10) c _0 + ""^(10) c _1x ( 2 + 3x )+ ""^(10) c_2 x ^ 2( 2 + 3x) ^2+""^(10) c_3 x ^ 3`
`( 2 + 3x ) ^3 +.... +""^(10)c_(10)x ^(10)(2+ 3x) ^2 `
`=a _ 0+ a _1 x +a _2 x ^ 2 + .... +a _(20)x ^(20 ) `
`thereforea _ 1= 2 xx 10`
` a _2 =""^(10) c _ 1xx3xx ""^(10 c _ 2xx 4 `
` = 3 xx 10+(10 xx 9 ) /(1 xx 2 )xx 4`
`= 30+180 `
`= 210 `
`therefore (a _ 2 ) /(a_1) = (210)/(2 xx 10 )= (21 ) /(2)= 10.5 `


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