Hi, This Q can be solved by opening the modulus sign with + and - signs, one at a time. In this case you get a max. of 4 roots, two for each + and - signs. For getting exactly 3 roots, following may be the cases, a) One of the equations (modulus with + or - sign) has both equal roots, and the other 1 has two distinct roots. b) Both + and - give you distinct roots, but 1 root of + and 1 root of - become same. Now when opened with '+' : a1= 1, b1=3 and c1=(1+a). Solving b1^2-4a1c1=0, we get a =5/4. Similarly for opening with '-' sign: a2=1, b2= -1, c2=(1-a). Solving this you get a=3/4. Now there can be the last possibility: Put both b1^2 - 4a1c1 > 0 and b2^2 - 4a2c2 > 0. You get: a< 5/4 and a > 3/4, which gives you the last value, that is 1. So you get your final answer as:a = 3/4, 1, 5/4. Please always specify, whether the Q has single or multiple correct option, as you are specified in the exam, it is incomplete otherwise, as a Q is left after we get a value and nothing is mentioned to proceed after that. Thank you

Dear Tapasranjan

I have given the range of a for which given equation has  real solution

3/4 ≤ a ≤ 5/4

for exactly 3 solution ,

1)  any equation has 1 root and the other equation has 2 root then 3 solution are possible .

2)  Or both the equation has 2 roots , but one root in both the equation is common then also exactly   3 solution is possible .

 1st case is possible at end point of the calculated range of a  ie  a=3/4  and  5/4 (because at this point discrimnent of one equation is zero  already derived in earlier post )

and for the second case

formed two equation x²  -3x+a+1 =0   for x≥a

                              x²  -x+1-a =0     for x<a 

let β is a common root

then it is satisfied by both the equation

 

   β²  -3β+a+1 =0

   β²  -β+1-a =0

solve for β  

β =a

put this value of in any one of the equation

a²  -a+1-a =0

(a-1)² =0

a =1

so for a=1   also exactly 3 solution exicts

<sup>Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed  solution very  quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

All the best.
 
Regards,
Askiitians Experts
Badiuddin</sup>