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If ac lt b^2 then the sum of the coeff...

If `ac lt b^2` then the sum of the coefficient in the expansion of `(a alpha^2 x^2+ 2b alpha x+c)^n` is `(a, b,c, alpha in R and n in N)`

A

(a)`+ "ve , if " a gt 0 `

B

(b)`+ "ve, if c " gt 0 `

C

(c)`- " ve, if a " lt 0 ` , n is odd

D

(d)`+ " ve , if " c lt 0 ` , n is even

Text Solution

Verified by Experts

The correct Answer is:
a,b,c,d
Sum of coefficients ` = a alpha ^(2) + 2b alpha + c)^(n)`
Let ` f(alpha) = a alpha^(2) + 2 b alpha + c `
Now , ` D = 4b^(2) - 4ac = 4 (b^(2) - ac) lt 0 `
` therefore f(alpha ) lt o or f (alpha ) gt 0, AA alpha in R `
If ` a gt 0 , " then" f (alpha ) gt 0 `
` rArr (a alpha)^(2) + 2 b alpha + c^(n) gt 0 `
If ` c gt 0 , i.e. f(0) gt 0 rArr f(alpha ) gt 0 `
` rArr (a alpha^(2) + 2 b alpha + c)^(n) lt 0 ` , if n is odd
If ` c lt 0 , " then f(0) " lt 0 rArr f(alpha) lt 0 `
` rArr (a alpha^(2) + 2b alpha + c)^(n) gt 0 ` , if n is even .
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