If a<b<c<d and a,b,c,d∈R then the equation 3(x−a)(x−c)+5(x−b)(x−d)=0 has
A
real and equal roots
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B
real and distinct roots
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C
one root lies in (a,b) and the other lies in (c,d)
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D
non-real roots
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Solution
The correct option is C one root lies in (a,b) and the other lies in (c,d) Let f(x)=3(x−a)(x−c)+5(x−b)(x−d)
Given: a<b<c<d f(a)=5(a−b)(a−d)>0f(b)=3(b−a)(b−c)<0f(c)=5(c−b)(c−d)<0f(d)=3(d−a)(d−c)>0 ∴f(a)>0,f(b)<0,f(c)<0,f(d)>0
So, one root lies in (a,b) and the other lies in (c,d)
Therefore, real and distinct roots.