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Question

Let m,n be positive integers and the quadratic equation 4x2+mx+n=0 has two distinct real roots p and q (pq). Also, the quadratic equations x2px+2q=0 and x2qx+2p=0 have a common root say α. If p and q are rational, then uncommon root of the equation x2px+2q=0 and x2qx+2p=0 is equal to

A
q
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B
q4
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C
q2
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D
q2
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Solution

The correct option is A q
x2px+2q=0
x2qx+2p=0
Subtracting the above equations, we get
x(qp)+2(qp)=0
(qp)[x+2]=0 ...(i)
Hence,
x=2 is the common root.
Substituting, we get
4+2p+2q=0
2(p+q)=4
p+q=2
Hence the equation, becomes
x2+(q+2)x+2q=0
(x+q)(x+2)=0
Hence, uncommon root is q.

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