Prove that x⁴ + 3x³ + 6x² + 9x + 12 cannot be expressed as a product of two polynomials of degree 2 with integer coefficients.

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

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

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 = _________

Samrud had to multiply a number by 35. By mistake he multiplied by 53 and got a result 720 more. The new product is _________.

Vishva plays football every 4th day. He played on a Tuesday . He plays football on a Tuesday again in _________ days.

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 _______.

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 ________ .