A cube has edge length x (an integer). three faces meeting at a corner are painted blue. The cube is then cut into smaller cubes of unit length. If exactly 343 of these cubes have no faces painted blue, then the value of x is ________ .

There are 4 coins in a row and all are showing heads to start with. The coins can be flipped with the following rules :

(a) The fourth coin (from the left) can be flipped any time

(b) An intermediate coin can be changed to tail only if its immediate neighbor on the right is heads and all other coins (if any) to its right are tails.

(c) Only one coin can be flipped in one step.

The minimum number of steps required to bring all coins to show tails is _______.

The four digit number 8ab9 is a perfect square. The value of a² + b² is

(A) 52 

(B) 62 

(C) 54 

(D) 68

If x²(y + z) = a², y²(z + x) = b², z²(x + y) = c², xyz = a b c prove that a² + b² + c² + 2abc = 1

The sum of the ages of a man and his wife is six times the sum of the ages of their children. Two years ago the sum of their ages was ten times the sum of the ages of their children. Six years hence the sum of their ages will be three times the sum of the ages of their children. How many children do they have?

The difference between the biggest and the smallest three digit number each of which has different digits is

(A) 864 

(B) 875 

(C) 885 

(D) 895

ABC is a triangle in which the angles are in the ratio 3 : 4 : 5 . PQR is a triangle in which the angles are in the ratio 5 : 6 : 7. The difference between the least angle of ABC and the least angle of PQR is aº. Then a = _________

The number of ways in which 26 identical chocolates be distributed between Amy, Bob, Cathy and Daniel so that each receives at least one chocolate and Amy receives more chocolates than Bob is _______.

When a two digit number divides 265, the remainder is 5. The number of such two digit numbers is ________

We have four sets S₁, S₂, S₃, S₄ each containing a number of parallel lines. The set S_i contains i + 1 parallel lines i = 1,2,3,4. A line in S_i is not parallel to lines in S_j when i ≠ j. In how many points do these lines intersect ?

(A) 54 

(B) 63 

(C) 71

(D) 95