Verbal Reasoning - Mental Ability Questions

First is a hatred for the second

The letters move four places forward each in the alphabet.

 The alphabets in the given words are coded as follows :

$$\begin{gathered} M I S T A K E N D \\ 9 7 6 5 4 1 2 8 3 \end{gathered}$$

D is coded as 3,

I as 7,
S as 6,
T as 5,
A as 4  and
N as 8.
So, the code for DISTANT is 3765485.

The given words are related in a way that, an optimist is a person whose outlook is cheerful. Similarly, a pessimist is a person whose outlook is gloomy.

The answer is not (b) because a pessimist does not have to be mean. Options (c) and (a) are incorrect because neither words describe the outlook of a pessimist.

Hence, Optimist is to cheerful as Pessimist is to Gloomy.

Second is the ultimate source for the first.

That is, Battery is the source of motorcycle. Similarly, Sun is the ultimate source of life.

Hence, Motorcycle : Battery :: Life : Sun.

Jagat is the brother of Priya and Priya is the daughter of Sachin. So. Jagat is the son of Sachin. Now, Rajan is the brother of Sachin. 

Thus, Rajan is the uncle of Jagat 

EXTRA
The two letters , as mentioned in the question, are E and A. We know that E comes after A in alphabetical order.