Question

Let `A= { x | x^2 + (m-1)x-2(m+1)=0, x in R}` `B= { x | ((m-1)x^2)+ m x +1=0, x in R}` Number of values of `m` such that `A uu B` has exactly 3 distinct elements, is? A) 4 B) 5 C) 6 D) 7

Answer 1

Consider the set `A`:

`A = { x in RR | x^2 + (m-1)x-2(m+1)=0 }`

We know that `x in RR => Delta_A ge 0`,and so:

`Delta_A = (m-1)^2-4(1)(-2(m+1))`
`\ \ \ \ \ = m^2-2m+1 + 8m+8`
`\ \ \ \ \ = (m-3)^2`

`Delta_A = 0 => m=3 => 1` solution
`Delta_A gt 0 => m!=3 => 2` solutions

And for set `B`, we have:

`B = { x in RR | ((m-1)x^2)+ mx + 1=0 }`

Similarly, We know that `x in RR => Delta_B ge 0`,and so:

`Delta_B = m^2-4(m-1)(1)`
`\ \ \ \ \ = m^2-4m+4`
`\ \ \ \ \ = (m-2)^2`

`Delta_B = 0 => m=2 => 1` solution
`Delta_B gt 0 => m!=2 => 2` solutions

Now we want `A uu B` to have `3` distinct elements, this requires

• One element from A, two elements from B:
  `=> Delta_A=0, Delta_B gt 0`
  `=> (m =3) nn (m!=2) => m= 3`

• One element from B, two elements from A # => Delta_B=0, Delta_A gt 0#
  `=> (m =2) nn (m!=3) => m= 2`

Therefore there are `2` values of `m` that satisfy the specified criteria