Solution:

16/81

Explanation:
It is a question of geometric progression but we can solve it step by step as well in case you are not aware of the formula.

At first bounce
The height at which the ball would have reached = 2/3 of the original height from which it was dropped.

At second bounce
The height at which the ball would have reached = 2/3 * 2/3 of the original height from which it was dropped.

At third bounce
The height at which the ball would have reached = 2/3 * 2/3 * 2/3 of the original height from which it was dropped.

At fourth bounce
The height at which the ball would have reached = 2/3 * 2/3 * 2/3 * 2/3 of the original height from which it was dropped.

Now, 2/3 * 2/3 * 2/3 * 2/3 = 16/81

Therefore, 16/81 of the original height from which the ball was dropped is the required fraction height.

Solution:

7 Races.

Solution:

The answer is 'R'

It represents the gear stick. The number 1-5 represents gears and the last one thus should be reverse i.e. R.

Solution:

Batsman hit six and gone retired hurt while the new batsman come to play and get runout for his wicket and the other batsman hit six and made both century and team win..

Solution:

Firstly: Policeman should take the thug across the river thames and escort the boat back with him. (Thug across, everyone else on this side)
Then, Policeman will take one cat along, leave it there, and on the way back, bring back the thug (since thug cannot be left with anyone without policeman's supervision).
Then Mrs. Mother will take the second cat on the other side. Drop it there and bring the boat back. For the 4th trip, Mrs. Mother And Mr. Father Will travel to the other side, and Mrs. Mother will stay there, and MR. Father will bring the boat back. (Now Mrs. Mother and 2 cats are on the other side.)
5th trip, the policeman takes the thug across. Both of them stay there and Mrs. Mother will come back. Then for the next trip, Mr. Father and Mrs. Mother will go on the boat, and Mrs. Mother will stay there, and Mr. Father will return back.
Then for the 7th trip, Mr. Father will travel with one dog. On the return, the policeman and the thug will return. Then in the next trip, the Policeman will take the second dog. (Now Mrs. Mother and her two cats; Mr. Father and his two dogs and the Policeman is here).
Now finally the policeman takes the boat on the other side and brings back the thug.
Tough one, right ?

Solution:

What is the correct time ?

Solution:

The clerk said that every day the password is CHANGED. Changed is the password.

Solution:

Upon reaching each room, he found out that the window facing the dead body was closed. He had to open the window himself. This tells that there was somebody else with him who closed the window. If he would have committed suicide, at least one window would have been open.

Solution:

One of the cop will move clockwise while the other will move anti-clockwise every day. In such a manner, they will catch the thief in minimum seven days.

Solution:

Playing chess

Game needs two players so Brother-8 is playing chess with Brother-2

Solution:

60 minutes

Explanation:
Roy will be able to consume all the five tablets in an hour.

Tablet 1 will be taken in 0 min.
Tablet 2 will be taken in 15 min.
Tablet 3 will be taken in 30 min.
Tablet 4 will be taken in 45 min.
Tablet 5 will be taken in 60 min.