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Question

The range of k for which both the roots of the quadratic equation (k+1)x23kx+4k=0 are greater than 1, is

A
(,1)
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B
[167,1]
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C
[167,1)
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D
(167,1)
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Solution

The correct option is C [167,1)
Given the quadratic equation:
(k+1)x23kx+4k=0, k+10
x23kxk+1+4kk+1=0
Let f(x)=x23kxk+1+4kk+1 and α,β are the roots of f(x)=0
Given: α,β>1


Conditions:
(i) D09k2(k+1)24×4kk+109k24(k+1)(4k)07k216k0k(7k+16)0k[167,0] (1)

(ii) b2a>13k2(k+1)>1
k22(k+1)>0k<1 or k>2 (2)

(iii) f(1)>02k+1k+1>0
k<1 or k>12 (3)

From (1),(2) and (3), we get


or k[167,1)

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