Q13: IIT Level Solved IV


We follow standard procedure

1. Find D
2. Find the range in which D is greater or lesser than 0
3. If D is negative, implies no roots and also that the entire graph will be above the x axis. In this case we are not dealing with the original quadratic, please note, but we are dealing with another quadratic expression which comes out as the value of D. 
4. And that gives us the range of ‘a’ for which the original quadratic expression is greater than zero or always positive. 


This question has a mod value and thus we have to take two cases i.e. when the x+2 is positive and when x+2 is negative. 

In Case I, x+2>0 or x>-2
solving the quadratic, we see that Q>0 (as given) only if x<-sqrt 2 and when x> sqrt 2

We combine to get the result shown in Case I

Then in Case II, we have two conditions again
x+2<0 i.e. x<-2
x is any Real Number

Again we combine to get the result shown

Finally, we combine the results obtained in Case I and II to get the final answer.