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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – PHYSICS FIRST SEMESTER – NOVEMBER 2012 MT 1100 - MATHEMATICS FOR PHYSICS Date : 03/11/2012 Time : 1:00 - 4:00 Dept. No. Max. : 100 Marks SECTION A ANSWER ALL THE QUESTIONS: (10x2 =20) 1) Find the nth derivative of y sin(ax b) . 2) Write down the formula for subtangent and subnormal. 1 1 1 ... e 1 2! 4! 6! 3) Prove that . e 1 1 1 1 ... 1! 3! 5! 3 −1 2 4) Find the rank of the matrix (−6 2 −4). −3 1 −2 1 −𝑎𝑡 5) Show that [𝑒 ] = 𝑠+𝑎 . 6) State the formula for Laplace transformation of a periodic function. 7) Write down the expansion for sin 𝑛𝜃. 8) If 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1, Show that cos ℎ2 𝑥 − 𝑠𝑖𝑛ℎ2 𝑥 = 1. 9) What is the chance that a leap year selected at random will contain 53 Sundays? 10) Define Binomial distribution. SECTION B ANSWER ANY FIVE QUESTONS: 11) Find the angle of intersection of cardioids r a(1 cos ) and r b(1 cos ) . 12) Find the minimum and maximum value of the function 2 x3 3x 2 36 x 10 . 24 34 44 ... . 2! 3! 4! 14) Show that the system of equations xyz 6 x 2y 3z 14 x 4y 7z 30 13) Find the sum to infinity series 1 are consistent and solve them. et , 0 < t < 4 15) Find the L(f(t)) if f (t ) 0 , t > 4 (5x8 =40) s 16) Find a) L1 b) L(cos4 t ) . 2 ( s 2) 17) Prove that cos8θ = 1- 32sin2 θ + 160sin4 θ-256sin6 θ+128 sin8 θ. 18) Find the moment generating function for the Poisson distribution and hence find its mean and variance. SECTION C ANSWER ANY TWO QUESTIONS: 2 19) a) If y x 1 x m (2x20 = 40) then Prove that (1 x 2 ) yn 2 (2n 1) xyn 1 (n 2 m 2 ) yn 0 . b) Find 3 . (10+10) ( x 1)(2 x 1) 3 1 1 20) If 𝐴 = [−1 5 −1] then 1 −1 3 a) Find the characteristic value and characteristic vector of the matrix. b) Verify Cayley Hamilton Theorem and find A-1. (10+10) 5 3 21) a) Express cos θ sin θ in terms of sines of multiples of θ. b) Separate into real and imaginary parts of tan-1(α+iβ). (10+10) dx dy 22) a) Solve 2 x 3 y 0, y 2 x 0 with x(0) 8, y(0)=3 using Laplace transform. dt dt b) An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that: (i) two of the ball drawn is white; (ii) one is of each colour, (iii) none is red. (14+6) the nth derivative of ***********