Let α , β be the roots of the equation x 2 - 6 x + 3 = 0 such that Im ( α ) > Im ( β ) . Let a , b be integers not divisible by 3 and n be a natural number such that α 99 β + α 98 = 3 n ( a + i b ) , i = - 1 . Then n + a + b is equal to ___________.

Step-by-step solution

Given, x 2 - 6 x + 3 = 0 has roots α & β Now, solving x 2 - 6 x + 3 = 0 we get, x = 6 ± i 6 2 = 3 1 2 ± 1 2 i Now, taking α = 3 e i π 4 , β = 3 e - i π 4 Now, solving α 99 β + α 98 = α 98 α β + 1 = α 98 ( α + β ) β = 3 49 e i π 4 98 6 3 e - i π 4 = 3 49 e i 99 π 4 × 2 = 3 49 cos 99 π 4 + i sin 99 π 4 × 2 = 3 49 cos 25 π - π 4 + i sin 25 π - π 4 × 2 = 3 49 ( - 1 + i ) = 3 n ( a + i b ) So, on comparing we get, n = 49 , a = - 1 , b = 1 ∴ n + a + b = 49 - 1 + 1 = 49