Puzzle 50 - Integration

Evaluate

Puzzle 49

A rectangular sheet of paper ABCD is with width AD=1 and length AB more than the width but not more than double of it. The paper is folded through vertex A so that the edge along AD falls onto AB. Without unfolding, the paper is folded again along a line through B so that CB now lies on AB. Now the result is a triangular piece of paper. There is a region on this triangular sheet that has four layers of paper. Find the area of that region in terms of the length x, of the rectangle.

Puzzle 48 - Roots of a cubic

From Rolle’s theorem, we can conclude that between every pair of distinct roots of a polynomial P(x), there lies a root of its derivative P’(x). If P’(x) keeps its sign unchanged between any two distinct real numbers, then it is possible that P(x) does not have a root between them and hence may keep the same sign between those real numbers. Further if P(x) has two equal roots, then it will have a common root with its derivative. Consider with being an integer between and both inclusive, then find the number of values of if has to have
(i) one real root and two complex roots, (ii) only two equal roots

Puzzle 47

Semi circles are drawn on each side of a square. An elastic band is stretched tightly around the shape formed (see the figure). Find the length of the elastic band in terms of the length x, of the side of the square.

Puzzle 46 - Squares

Square ABCD, of side length 26, is centered at O. With D as center, draw another square EFGH with side length 22, none of its sides parallel to those of ABCD, so that OE is kept perpendicular to DA. Find the area of the square ABCD that is not inside the square EFGH.

Puzzle 45 - Similar Triangles

Let A is the set of all triangles in a plane. Define a function as implies that triangle is similar to with the dilation constant or scale factor ‘a’. Find all the possible triangles and such that

a. Lengths of the sides of and are integers
b. The perimeter and area of are same.

Puzzle 44

W(a, b) is the centre of the circle. TE is a tangent to the circle at E, which lies on the y-axis. The circle touches the x-axis at D. The equation of TE is. Calculate the values of a and b.