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Q.

Consider f(x)=x2+ax+3 and g(x) = x + b and F(x)=limn→∞ f(x)+x2ng(x)1+x2n.If F(x) is continuous at x = 1, thenIf f'(x) is continuous at x= -1, thenIf F(x) is continuous at x = ± 1, then f (x) = g(x) has

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a

b=a+3

b

b = a-1

c

a=b-2

d

none of these

e

a+b=-2

f

a-b=3

g

a+b=5

h

none of these

i

imaginary roots

j

both the roots positive

k

both the roots negative

l

roots of opposite signs

answer is , , .

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Detailed Solution

F(x)=limn→∞ f(x)+x2ng(x)1+x2n=f(x),f(x)+g(x),0≤x2<12x2=1g(x),x2>1=g(x),f(−1)+g(−1),x<−1f(x),−11If F(x) is continuous ∀x∈R,F(x) must be made continuousat x = ± IFor continuity at x = -1,f(−1)=g(−1) or 1−a+3=b−1 or a+b=5....... (1) For continuity at x =1 ,f (1)= g(1)or 1+a+3=1+b or a−b=−3Solving equations (1) and (2), we get a = 1 and b=4.f(x)=g(x)⇒x2+x+3=x+4 or x2=1 or x=±1
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