Manchester Trucking Company $($MTC$)$ bought a truck on $1$ January $2017$ for $$10,500,000.$The equipment has a useful life of $5$ years with a residual value of $$800,000.$ The truck's original engine was replaced with a new more powerful and efficient one on the $1$ April $2020$ at a cost of $$2,700,000.$ At this date the remaining useful of the Truck is eight $(8)$ years.At the $1$ October $2023$ the truck is involved in an accident any is no longer fully operational.The recoverable of the truck is determined to be $$2,900,000.($a$)$ Determine if the truck is impaired and the amount, showing all workings$($b$)$ Show the journal entry to record the installation of the new truck engine on $1$April $2020.(6$ marks$)(2$ marks$)$