METU Department of Mathematics Math 303 History of Math Concepts Fall 2010

Midterm 2

December 9 2010, 17:30 – 19:00

Name and the student number:

Part II

Q1(7pts). What can be the number of normals to an ellipse, and how does it depend on the position of a point ? Sketch the evolute of an ellipse and explain its meaning. Why such a curve is also called astroid ?

Q2(7pts). Formulate the Pappus centroid theorems (for calculation the surface and volume). Apply it in the case of torus.

Q3(7pts). What is the Chinese Remainder Theorem ? Give an example of such problem and solve it.

Q4(6pts). Apply Heron’s formula to find the area of a triangle with the sides of length 9,10, and 11. Find the radius of the circle inscribed in this triangle.

Q5(6pts). Apply Brahmaputra’s formula to find the area of a cyclic quadrilateral with the sides of length 5,6,7,8. (Give a sketch.)

Q6(7pts). What is the discriminant of a cubic equation and what does it describe ? Show how it can be found. Who is the author of this approach ?