Mohan correctly remembers that his father's birthday is before $20^{th}$ January but after $16^{th}$ January,whereas his sister correctly remembers that their father's birthday is after $18^{th}$ January but before $23^{rd}$ January. On which date in January is their father's birthday?

Reaching the place of meeting on Tuesday $15 \text{ minutes}$ before $8:30 \text{ hours}$,Anuj found himself half an hour earlier than the man who was $40 \text{ minutes}$ late. What was the scheduled time of the meeting? (in $\text{hours}$)

$A$ clock gaining $2 \, \text{minutes}$ every hour was synchronized at midnight with a clock losing $1 \, \text{minute}$ every hour. How many minutes behind will the second clock be at $11:00 \, \text{AM}$ the following morning compared to the clock gaining $2 \, \text{minutes}$ every hour?

$A$ man leaves for his office from his house. He walks towards East. After moving a distance of $20 \,m,$ he turns towards South and walks $10 \,m.$ Then he walks $35 \,m$ towards the West and further $5 \,m$ towards the North. He then turns towards East and walks $15 \,m.$ What is the straight distance in metres between his initial and final positions?

After walking a distance of $50 \, m$ to the south of her house,Lakshmi turns left and walks another $20 \, m$. Then she turns north and walks $30 \, m$ and then starts walking to her house. In which direction is she walking now?

$A$ boy rode his bicycle northwards,then turned left and rode $1 \, km$,and again turned left and rode $2 \, km$. He found himself exactly $1 \, km$ west of his starting point. How far did he ride northwards initially? (in $km$)