How to know that , Is it Maintenance Cleanup Task is working properly from maintenance Plan or not through T-SQL?
If you want to be able to view the exact backup history for longer time than those four weeks that you keep the backup files on disk, make sure that you set the cleanup task not to delete the backup history from the msdb database, you can then run the backup report from SSMS (right click database, select reports, standard reports, backup and restore events) or run the following to view the backup history: 

First create this function - I'll just have it in msdb - which is not a good idea.

create FUNCTION dbo.fc_FileExists(@path varchar(8000))
RETURNS BIT
AS
BEGIN
     DECLARE @result INT
     EXEC master.dbo.xp_fileexist @path, @result OUTPUT
     RETURN cast(@result as bit)
END;
GO


Then you can run the following

SELECT     
 sd.name AS database_name, 
 ISNULL(bs.backup_finish_date, 0) AS backup_date,
 ISNULL(CASE bs.[type] 
  WHEN 'D' THEN 'Full' 
  WHEN 'I' THEN 'Differential'
  WHEN 'L' THEN 'Transaction Log' 
  END, 'NEVER') as type ,
 bm.physical_device_name, 
 dbo.fc_FileExists(bm.physical_device_name) as DoesFileExists,
 bs.backup_size / 1024 / 1024 AS backup_size, 
 DATEDIFF(mi, bs.backup_start_date, bs.backup_finish_date) AS duration
 FROM master.dbo.sysdatabases AS sd 
 LEFT OUTER JOIN msdb..backupset AS bs ON sd.name = bs.database_name 
 LEFT OUTER JOIN msdb..backupmediafamily AS bm ON bm.media_set_id =     bs.media_set_id
  GROUP BY 
   sd.name, 
   bm.physical_device_name, 
   bs.backup_size / 1024 / 1024, 
   DATEDIFF(mi, bs.backup_start_date, bs.backup_finish_date),
   bs.backup_finish_date,
   bs.[type],
   dbo.fc_FileExists(filename)
 order by bs.backup_finish_date desc