If `(x+2)` is a common factor of `(px^2+qx+r)` and `(qx^2+px+r)` then a) ` p=q` or `p+q+r=0` b)`p=r` or `p+q+r=0` c) `q=r` or `p+q+r=0` d)`p=q=-1/2r`

Options

A. `|d| le |D|`
B. `|d| ge|D|`
C. `|d| = |D|`
D. None of these

Answer

1

Explanation

`|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|)`