Examine the following relationships among members of a family of $6$ persons $A, B, C, D, E,$ and $F$.
$1.$ The number of males equals that of females.
$2.$ $A$ and $E$ are sons of $F$.
$3.$ $D$ is the mother of two,one boy and one girl.
$4.$ $B$ is the son of $A$.
$5.$ There is only one married couple in the family at present.
Which one of the following inferences can be drawn from the above?

$(A)$ There are five friends.
$(B)$ They are standing in a row facing South.
$(C)$ Jayesh is to the immediate right of Alok.
$(D)$ Pramod is between Bhagat and Subodh.
$(E)$ Subodh is between Jayesh and Pramod.
Who is at the extreme left end?

In a group of $5$ persons $A, B, C, D,$ and $E,$ there is a professor,a businessman,and an artist. $A$ and $D$ are ladies who are unmarried and do not work. Of the married couple in the group,$E$ is the husband. $B$ is the brother of $A$ and is neither a businessman nor an artist. $E$'s wife is an artist. Which of the following groups include all the men?

Eight friends $A, B, C, D, E, F, G$ and $H$ are sitting in a circle facing the centre. $B$ is sitting between $G$ and $D$. $H$ is third to the left of $B$ and second to the right of $A$. $C$ is sitting between $A$ and $G$ and $B$ and $E$ are not sitting opposite to each other.
Which of the following statements is not correct?

$(i)$ Five friends $P, Q, R, S$ and $T$ travelled to five different cities of Chennai,Calcutta,Delhi,Bangalore and Hyderabad by five different modes of transport of 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.
Which of the following combinations of place and mode is not correct?

How many days of gap are there between Science and Philosophy?