Math IOQM

  THE MONOMIAL LEMMA So f(x) = 1 + x^n = 1001 = 1 + 10^n => n = 3 => f(20) = 1 + 20^3

 Q1. Solution: It gets 2 solutions 2/3 and 3/2. => (2/3)^x = 2/3,3/2 => x = -1,+1 Answer. Q2. Answer. Q3. So this is an A.P. And X_n is an H.P. After this: Q4. DS means diagonal sum here. Key Insight: It's difficult to solve a degree 4 equation in 'x'. But it's easier to solve a degree 2 equation in 'a'. Make a quadratic in 'a': You will get 2 different quadratic equations: Finally combine both solutions to get a >= 3/4 Key insight: This is essentially 2 quadratic equations hidden in one. We have to construct 2 separate equations. One with x >=0 => |x| = x. other with x <= 0. |x| = -x. Now for them to have 3 distinct real solutions, there are 3 cases: Case 1: First equation has D = 0, second has D > 0 Case 2: First equation has D > 0, second has D = 0 Case 3: Both have D > 0 but one of the equations has only one valid root, i.e. in x >= 0 case, one root turns out to be negative. Eventually you will get a = 1,3 as solutions. ...