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Question

# If f:R−{−1,k}→R−{α,β} is a bijective function defined by f(x)=(2x−1)(2x2−4px+p3)(x+1)(x2−p2x+p2) for p≥0, then which of the following statements is (are) CORRECT ?

A
If k(1,1), then α+β=2
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B
If k(1,1), then α+β=6
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C
If k(1,3), then α+β=4
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D
If k(1,3), then α+β=6
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Solution

## The correct options are A If k∈(−1,1), then α+β=2 D If k∈(1,3), then α+β=6f:R−{−1,k}→R−{α,β} f(x)=(2x−1)(2x2−4px+p3)(x+1)(x2−p2x+p2) For the domain to be R−{−1,k},k must be a repeated root of x2−p2x+p2=0 ∴p4−4p2=0⇒p2(p2−4)=0 ∴p=0,±2, but p≥0 ⇒p=0,2 For p=0,k=0 f(x)=(2x−1)(2x2)(x+1)(x2) ⇒Df=R−{−1,0}, Rf=R−{4,−2} ∴α+β=2 For p=2,k=2 f(x)=(2x−1)2(x−2)2(x+1)(x−2)2 ⇒Df=R−{−1,2}, Rf=R−{4,2} ∴α+β=6

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