Sin ²⁰¹⁴ x + Cos ²⁰¹⁴ x = 1, x in the range of [-5π, 5π], how many values can x take

Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P

How many isosceles triangles with integer sides are possible such that sum of two of the side is 12

Region R is defined as the region in the first quadrant satisfying the condition 3x + 4y < 12. Given that a point P with coordinates (r, s) lies within the region R, what is the probability that r > 2

A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made

x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4, find z