Random Puzzles 

Workings

Hint
The word begins with the letter C.

Answer
CAT.
LO...E   =  LOCATE
EDU...E  =  EDUCATE
VA...E   =  VACATE
S...TER  =  SCATTER
S...HE   =  SCATHE
...ER    =  CATER

Workings

Hint
There are no squares of size 5 x 5.

Answer
There are 125 squares without a rook.

Reasoning
61 squares of size 1 x 1
39 squares of size 2 x 2
20 squares of size 3 x 3
 5 squares of size 4 x 4

Giving a total of 61 + 39 + 20 + 5 = 125 squares.

Workings

Hint
Try removing some letters.

Answer
Carrot.

Reasoning
Each of the other words remains a real word if you remove the first and last letter:
s-event-y b-raw-l c-love-r p-rope-r c-arro-t * s-win-g e-merge-d c-raft-y t-rave-l s-acre-d s-ton-e p-lung-e

Workings

Hint
One is the study of fossils.

Answer
Palaeontology, the study of fossils and Limnology, the study of lakes and rivers. There are others you might have chosen.