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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 (p≤q). Also, the quadratic equations x2−px+2q=0 and x2−qx+2p=0 have a common root say α. If p and q are rational, then uncommon root of the equation x2−px+2q=0 and x2−qx+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 −qx2−px+2q=0x2−qx+2p=0Subtracting the above equations, we getx(q−p)+2(q−p)=0(q−p)[x+2]=0 ...(i)Hence,x=−2 is the common root.Substituting, we get4+2p+2q=02(p+q)=−4p+q=−2 Hence the equation, becomesx2+(q+2)x+2q=0(x+q)(x+2)=0Hence, uncommon root is −q.

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