After the rearrangement of letters, the new order of the letters is as follows:

New Order : A E I M N R T

Given word: R A I M E N T

Clearly, positions of three letters I, M and T will remain unchanged.

Given word : CONTEXTUAL

After making changes as per given in the question , the above word becomes as follows:

XTUALCONTE . Hence, 4th letter from the right end is O.

quicker method:

    Total no of letters in the given word = 10.

    4th letter from the right means (10 - 4 + 1 =) 7th letter from the left of the word obtained after rearrangement. Again, read the question , we come to know that, the positions of the second and seventh letters are interchanged. Hence the required letter is the second letter of the given word which is O.

The specified letters are R, C, H, A and I. The only meaningful word  that can be made is CHAIR. The third letter of this word is A.

PORE and ROPE

9th letter from the right means (26 - 9 + 1 =) 18th letter from the left. Now, the middle letter between 18th letter from the left  and 8th letter from the left = [(18+8)/2] = 13th leter from the left = M.

6th to the left of 17th letter to the right of 7th letter from the left = 6th to the left of (17+7 =) 24th letter from the left = (24-6 =) 18th letter from the left.

Thus, we have to count from the left and we have to find out 18th letter from the left that falls into the second section of English alphabet.

Here, both the first half and the second half of the English alphabet have been reversed and we have to count from the left end, then the required letter can be found out using the formula given below. Suppose , the first and the second reversed sections of the English alphabet are α and  βrespectively.Then,

  The required letter = 2α+β+1-(required position of letter from left side)

Therefore, The required letter = 26 + 13 + 1 - 18 = 22 = V

Second vowel from the right of reversed English alphabet is E and 7th letter to the left of E in the new series is L.

2nd letter to the left of 17th letter from the right means (2 + 17 = ) 19th letter from the right = (26 - 19 + 1 =) 8th letter from the left end , i.e H.  Thus, 8th letter falls into the second section of the reversed sequence. Now, the sequence FGHIJ is reversed and becomes JIHGF. Clearly, position of H remains unchanged. Hence the required letter is H.