Question

If $a<b<c<d$ and $a,b,c,d\in R$ then the equation
$3(x-a)(x-c)+5(x-b)(x-d)=0$ has

• A. real and equal roots
• B. real and distinct roots
• C. one root lies in $(a,b)$ and the other lies in $(c,d)$
• D. non-real roots

Answer 1

Answer: (B), (C)

The correct options are
B real and distinct roots
C one root lies in $(a,b)$ and the other lies in $(c,d)$

Let $f\left(x\right)=3(x-a)(x-c)+5(x-b)(x-d)$
Given: $a<b<c<d$
$f\left(a\right)=5(a-b)(a-d)>0f\left(b\right)=3(b-a)(b-c)<0f\left(c\right)=5(c-b)(c-d)<0f\left(d\right)=3(d-a)(d-c)>0$
$\therefore f\left(a\right)>0,f\left(b\right)<0,f\left(c\right)<0,f\left(d\right)>0$


So, one root lies in $(a,b)$ and the other lies in $(c,d)$
Therefore, real and distinct roots.