The children ___ playing in the park.

• is
• are
• has
• were
Correct Option: B  [ are ]

Remark: "Children" is a plural noun, so it takes a plural verb.

Each of the students ________ responsible for keeping this room clean.

• are
• has
• is
• have
Correct Option: C  [ is ]

Remark:

Unless we ______ now we can't be on time.

• will start
• start
• starts
• will be starting
Correct Option: B  [ start ]

Remark:

The Prime Minister with all the members of his cabinet __________ arrived.

• is
• have
• has
• are
Correct Option: C  [ has ]

Remark:

My friend, philosopher and guide _________ my father.

• is
• are
• were
• has
Correct Option: A  [ is ]

Remark: "friend, philosopher and guide" refers to one person.

The Moon _______ around the Earth

• move
• moving
• moved
• moves
Correct Option: D  [ moves ]

Remark: Subject is "Moon" a Singular noun, go with singular verb.
When universal truths are mentioned, the verb used should be the simple present tense.

Our Principal and Chairman of the occasion ________ arrived.

• is
• has
• have
• are
Correct Option: B  [ has ]

Remark:

Neither John nor Alex __________ scored the century.

• have
• was
• has
• is
Correct Option: C  [ has ]

Remark: in case "Neither-Nor" the verb must agree with the second subject (the subject closer to the verb).

Each of the students ___ to submit their assignments.

• have
• has
• is
• are
Correct Option: B  [ has ]

Remark: "Each" is a singular subject, so it takes a singular verb.

The news ___ very surprising.

• is
• are
• have
• has
Correct Option: A  [ is ]

Remark: "News" is an uncountable noun and always takes a singular verb.