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Given |px^(2) + qx + r| le |Px^(2) + Qx ...

Given `|px^(2) + qx + r| le |Px^(2) + Qx + r|AA x in R` and `d=q^(2) - 4pr gt 0` and `D =Q^(2) PR gt 0`
Which of the following must be ture ?

A

`|p| ge |P|`

B

`|p| le |P|`

C

`|p| = |P|`

D

All of these

Text Solution

Verified by Experts

The correct Answer is:
2
`|px^(2) + qx + r| le| Px^(2) + Qx + R| AA x in R" "(1)`
Form the graph we can see that this is possible only when both equations have same roots.
Thus, `alpha` and `beta` are roots of `Px^(2) + Qx + R = 0` and also of
`px^(2) + qx + r=0`
So, from (1),
`|p| |x-alpha| |x - beta| le |P| |x-alpha| |x-beta|`
`rArr |p| le |p|`
Also `|(4pr-q^(2))/(4p)| le |(4PR -Q^(2))/(4p)|`
`rArr |d| le |(p)/(P)| |D|`
`rArr |d| le |D| " "(because |p| le |P|)`
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CENGAGE-THEORY OF EQUATIONS-Exercise (Comprehension)
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