Explanation
The cluster manager allocates resources to Spark applications and maintains the executor processes in client mode.
Correct. In cluster mode, the cluster manager is located on a node other than the client machine. From there it starts and ends executor processes on the cluster nodes as required by the Spark application running on the Spark driver.
The cluster manager allocates resources to Spark applications and maintains the executor processes in remote mode.
Wrong, there is no "remote" execution mode in Spark. Available execution modes are local, client, and cluster.
The cluster manager allocates resources to the DataFrame manager
Wrong, there is no "DataFrame manager" in Spark.
The cluster manager schedules tasks on the cluster in client mode.
No, in client mode, the Spark driver schedules tasks on the cluster - not the cluster manager.
The cluster manager schedules tasks on the cluster in local mode.
Wrong: In local mode, there is no "cluster". The Spark application is running on a single machine, not on a cluster of machines.