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Question

If a < b < c < d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are


A

Real and distinct

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B

Real and equal

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C

Imaginary

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D

None of these

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Solution

The correct option is A

Real and distinct


Given equation can be rewritten as

3x2 – (a + c + 2b + 2d)x + (ac + 2bd) = 0

Its discriminant D

=(a+c+2b+2d)2 – 4.3(ac + 2bd)

= {(a+2d)+(c+2b)}2 – 12(ac + 2bd)

= {(a+2d)(c+2b)}2 + 4(a + 2d)(c + 2b) – 12(ac + 2bd)

= {(a+2d)(c+2b)}2 + 8(c – b)(d – a)

Which is +ve, since a < b < c < d. Hence roots are real and distinct.


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