Check The DATABASE_STATUS

SQL> select STARTUP_TIME,STATUS,LOGINS,DATABASE_STATUS from v$instance where INSTANCE_NUMBER=1;

STARTUP_T STATUS       LOGINS     DATABASE_STATUS
--------- ------------ ---------- -----------------
23-JUN-14 OPEN         ALLOWED    ACTIVE

SQL>

This would show when the DB was started, status and the mode of the DB Started.

How to check with the Database is running in Dedicated or Shared mode?

SQL> select username,status,server from v$session where username='SYS';

USERNAME                       STATUS   SERVER
------------------------------ -------- ---------
SYS                            ACTIVE   DEDICATED

SQL>

The Server states would show the Mode of Database Activation.