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

1 Answer
Jul 16, 2017

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