We know that, February has $28\;\text{days}/29\;\text{days (leap year)}.$

First, we assume, that particular year is not a leap year (means it has $28$ days).

- $1^{\text{st}}\;\text{February} \longrightarrow \text{Monday}$
- $8^{\text{th}}\;\text{February} \longrightarrow \text{Monday}$
- $15^{\text{th}}\;\text{February} \longrightarrow \text{Monday}$
- $22^{\text{th}}\;\text{February} \longrightarrow \text{Monday}$
- ${\color{Red}{28^{\text{th}}\;\text{February} \longrightarrow \text{Sunday}}}$

We can say that, if February has $28$ days, then $5$ Monday is not possible.

Now, we assume, that particular year is a leap year (means it has $29$ days).

- ${\color{Teal}{1^{\text{st}}\;\text{February} \longrightarrow \text{Monday}}}$
- ${\color{Purple}{2^{\text{nd}}\;\text{February} \longrightarrow \text{Tuesday}}}$
- ${\color{Blue}{7^{\text{th}}\;\text{February} \longrightarrow \text{Sunday}}}$

- ${\color{DarkOrange}{8^{\text{th}}\;\text{February} \longrightarrow \text{Monday}}}$
- ${\color{Blue}{14^{\text{th}}\;\text{February} \longrightarrow \text{Sunday}}}$

- $15^{\text{th}}\;\text{February} \longrightarrow \text{Monday}$
- ${\color{Blue}{21^{\text{th}}\;\text{February} \longrightarrow \text{Sunday}}}$

- ${\color{DarkOrange}{22^{\text{th}}\;\text{February} \longrightarrow \text{Monday}}}$
- ${\color{Blue}{28^{\text{th}}\;\text{February} \longrightarrow \text{Sunday}}}$

- ${\color{Green}{29^{\text{th}}\;\text{February} \longrightarrow \text{Monday}}}$

Now, we can check all the options.

- The $2^{\text{nd}}$ February of that year is a Tuesday ${\color{Lime}{-\text{True.}}}$
- There will be five Sundays in the month of February in that year ${\color{Red}{-\text{False.}}}$
- The $1^{\text{st}}$ February of that year is a Sunday ${\color{Red}{-\text{False.}}}$
- All Mondays of February in that year have even dates ${\color{Red}{-\text{False.}}}$

Correct Answer $:\text{A}$