$(i)$ Five friends $P, Q, R, S$ and $T$ travelled to five different cities: Chennai,Calcutta,Delhi,Bangalore,and Hyderabad by five different modes of transport: Bus,Train,Aeroplane,Car,and Boat from Mumbai.
$(ii)$ The person who travelled to Delhi did not travel by boat.
$(iii)$ $R$ went to Bangalore by car and $Q$ went to Calcutta by aeroplane.
$(iv)$ $S$ travelled by boat,whereas $T$ travelled by train.
$(v)$ Mumbai is not connected by bus to Delhi and Chennai.
The person travelling to Delhi went by which of the following modes?

$(i)$ $P, Q, R, S, T$ and $U$ are six students pursuing their Master's degree in six different subjects $-$ English,History,Philosophy,Physics,Statistics and Mathematics.
$(ii)$ Two of them stay in a hostel,two stay as paying guests $(PG)$ and the remaining two stay at their home.
$(iii)$ $R$ does not stay as a $PG$ and studies Philosophy.
$(iv)$ The students studying Statistics and History do not stay as $PG$.
$(v)$ $T$ studies Mathematics and $S$ studies Physics.
$(vi)$ $U$ and $S$ stay in a hostel. $T$ stays as a $PG$ and $Q$ stays at home.
Which of the following pairs of students stay one each at a hostel and at home?

The total income of $A$ and $B$ is greater than the total income of $C$ and $D$. The total income of $A$ and $C$ is equal to the total income of $B$ and $D$. Furthermore,the income of $A$ is equal to half the total income of $B$ and $D$. Which of the following statements is $NOT$ true?

Five men $A, B, C, D$ and $E$ read a newspaper. The one who reads first gives it to $C.$ The one who reads last had taken from $A.$ $E$ was not the first or last to read. There were two readers between $B$ and $A.$ $B$ passed the newspaper to whom?

$B$ is twice as old as $A$ but twice younger than $F$. $C$ is half the age of $A$ but twice the age of $D$. Which two persons form the pair of oldest and youngest?

Shekhar is taller than Kunal. Atul is taller than Pawan but not as tall as Kunal. Prashant is taller than Shekhar. Who among them is the shortest?