MA211_ST1_202001_Solution.pdf - ADVANCED CALCULUS MA211...

This preview shows page 1 - 4 out of 10 pages.

JR ALL THE BEST Page 1 of 10 ____________________________________________________________________________________ Number of Questions: 5 Time allowed: 90 minutes Total marks: 60 NOTE: 1. All questions are compulsory 2. Show all relevant working Question 1 [3 + 4 + 4 = 10 marks] a). Show that if A and B are not both zero, then the graph of the polar equation sin cos r A θ B θ is a circle. Find its center and radius. b). In the late seventeenth century the Italian astronomer Giovanni Domenico Cassini (1625 1712) introduced the family of curves 2 2 2 2 4 2 2 4 0 0, 0 x y a b a x a b in his studies of the relative motions of the Earth and the Sun. These curves, which are called Cassini ovals. Show that if a b , then the polar equation of the Cassini oval is 2 2 2 cos 2 , r a θ which is a lemniscate. c). Find the area of the region inside the rose 2 cos 2 r a θ and outside the circle 2 r a . Solution a). We use the concept of polar coordinates here: 2 2 2 2 2 2 2 sin cos sin cos 2 2 4 r A θ B θ r Ar θ Br θ x y Ay Bx B A A B x y Therefore, it is a circle centered at , 2 2 B A with radius 2 2 1 2 r A B . b). The solution is as follows: ADVANCED CALCULUS MA211 SHORT TEST 1 Semester 1, 2020