A particle is initially at a vertex of cube In each step it moves along the edge of a cube from onevertex to anothervertex . Find the number of different ways it can reach the opposite vertex in five orfewer steps ,It stops moving after reaching the opposite vertex .

Two satellites move along a circular orbit in the same direction at a small distance from each other. A container has to be thrown from the first satellite on the second a one. When will the container reach the second satellite earlier, if it is thrown in the direction of motion of the first satellite or in the opposite direction? The velocity of the container is small in comparison to that of the satellite.

Let C_1, C_2, ,C_n be a sequence of concentric circle. The nth circle has the radius n and it has n openings. A points P starts travelling on the smallest circle C_1 and leaves it at an opening along the normal at the point of opening to reach the next circle C_2 . Then it moves on the second circle C_2 and leaves it likewise to reach the third circle C_3 and so on. Find the total number of different path in which the point can come out of nth circle.

Let C_1, C_2, ,C_n be a sequence of concentric circle. The nth circle has the radius n and it has n openings. A points P starts travelling on the smallest circle C_1 and leaves it at an opening along the normal at the point of opening to reach the next circle C_2 . Then it moves on the second circle C_2 and leaves it likewise to reach the third circle C_3 and so on. Find the total number of different path in which the point can come out of nth circle.

Let C_1, C_2, ,C_n be a sequence of concentric circle. The nth circle has the radius n and it has n openings. A points P starts travelling on the smallest circle C_1 and leaves it at an opening along the normal at the point of opening to reach the next circle C_2 . Then it moves on the second circle C_2 and leaves it likewise to reach the third circle C_3 and so on. Find the total number of different path in which the point can come out of nth circle.

A regular square pyramid is placed in a cube so that the base of the pyramid and that of the cube coincide. The vertex of the pyramid lies on the face of the cube opposite to the base, as shown. An edge of the cube is 7 inches. How many square inches (approximately) are in the posi- tive difference between the surface area of the cube and the surface area of the pyramid ?

The Indian cricket team with eleven players, the team manager, the physiotherapist and two umpires are to travel from the hotel where they are staying to the stadium where the test match is to be played. Four of them residing in the same town own cars, each a four seater which they will drive themselves. The bus which was to pick them up failed to arrive in time after leaving the opposite team at the stadium. In how many ways can they be seated in the cars ? In how many ways can they travel by these cars so as to reach in time, if the seating arrangement in each car is immaterial and all the cars reach the stadium by the same route.

Two ants begin on opposite corners of a cube. On each move, they can travel along an edge to an adjacent vertex. The probability they both return to their starting position after 4 moves is (m)/(n) , where "m" and "n" are relatively prime positive integers, then "m+n" is

A rectangular farm house has a 1km difference between its sides. Two farmers simultaneously leave one of the vertex of the rectangle for a point at the opposite vertex. One farmer crosses the farmhouse along its diagonal and other walks along the edge. The speed of each farmer is 4km//hr . If one of them arrives half an hour earlier then the other then the size of farmhouse is .