Explanation
Pay attention here to which variables are quoted. fileSchema is a variable and thus should not be in quotes.
parquet is not a variable and therefore should be in quotes.
SparkSession.read (here referenced as spark.read) returns a DataFrameReader which all subsequent calls reference - the DataFrameReader is not callable, so you should not use parentheses here.
Finally, there is no open method in PySpark. The method name is load.
Static notebook | Dynamic notebook: See test 1

Explanation
Correct code block:
transactionsDf.join(itemsDf, transactionsDf.productId==itemsDf.itemId,
how='inner').filter(~isnull(col('value'))).count()
Expressions col("transactionsDf.productId") and col("itemsDf.itemId") are invalid. col() does not accept the name of a DataFrame, only column names.
Static notebook | Dynamic notebook: See test 2

Explanation
This question tests your knowledge about how to use the groupBy and agg pattern in Spark. Using the documentation, you can find out that there is no average() method in pyspark.sql.functions.
Static notebook | Dynamic notebook: See test 2

Explanation
Spark achieves great performance by storing data in the DAG format, whereas Hadoop can only use parquet files.
Wrong, there is no "DAG format". DAG stands for "directed acyclic graph". The DAG is a means of representing computational steps in Spark. However, it is true that Hadoop does not use a DAG.
The introduction of the DAG in Spark was a result of the limitation of Hadoop's map reduce framework in which data had to be written to and read from disk continuously.
Graph DAG in Apache Spark - DataFlair
Spark achieves great performance by storing data in the HDFS format, whereas Hadoop can only use parquet files.
No. Spark can certainly store data in HDFS (as well as other formats), but this is not a key performance advantage over Hadoop. Hadoop can use multiple file formats, not only parquet.
Spark achieves higher resiliency for queries since, different from Hadoop, it can be deployed on Kubernetes.
No, resiliency is not asked for in the question. The question is about performance improvements.
Both Hadoop and Spark can be deployed on Kubernetes.
Spark achieves performance gains for developers by extending Hadoop's DataFrames with a user-friendly API.
No. DataFrames are a concept in Spark, but not in Hadoop.

Explanation
transactionsDf.groupBy("productId").count()
Correct. This code block first groups DataFrame transactionsDf by column productId and then counts the rows in each group.
transactionsDf.groupBy("productId").select(count("value"))
Incorrect. You cannot call select on a GroupedData object (the output of a groupBy) statement.
transactionsDf.count("productId")
No. DataFrame.count() does not take any arguments.
transactionsDf.count("productId").distinct()
Wrong. Since DataFrame.count() does not take any arguments, this option cannot be right.
transactionsDf.groupBy("productId").agg(col("value").count())
False. A Column object, as returned by col("value"), does not have a count() method. You can see all available methods for Column object linked in the Spark documentation below.
More info: pyspark.sql.DataFrame.count - PySpark 3.1.2 documentation, pyspark.sql.Column - PySpark
3.1.2 documentation
Static notebook | Dynamic notebook: See test 3