(a) Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let $L$ be the set of all rail lines on the railway track and $R$ be the relation on $L$ defined by

$R=\left\{l_1, l_2\right): l_1$ is parallel to $\left.l_2\right\}$

On the basis of the above information, answer the following questions:

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation R is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the equation $y=3 x+2$, then find the set of rail lines in R related to it.

OR

(b) Let $S$ be the relation defined by $S=\left\{\left(l_1, l_2\right): l_1\right.$ is perpendicular to $l_2$ \} check whether the relation $S$ is symmetric and transitive.

37. A rectangular visiting card is to contain $24 \mathrm{sq} . \mathrm{cm}$. of printed matter. The margins at the top and bottom of the card are to be 1 cm and the margins on the left and right are to be $1 \frac{1}{2} \mathrm{~cm}$ as shown below:

On the basis of the above information, answer the following questions:

(i) Write the expression for the area of the visiting card in terms of $x$.

(ii) Obtain the dimensions of the card of minimum area.

A departmental store sends bills to charge its customers once a month. Past experience shows that $70 \%$ of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month.

Based on the above information, answer the following questions:

(i) Let $E_1$ and $E_2$ respectively, denote the event of customer paying or not paying the first month bill in time.

(ii) Let A denotes the event of customer paying second month's bill in time, then find $P\left(A \mid E_1\right)$ and $P\left(A \mid E_2\right)$.

(iii) Find the probability of customer paying second month's bill in time.

OR

(iii) Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.