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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 B.Sc. DEGREE EXAMINATION – PHYSICS THIRD SEMESTER – NOV 2006 AA 03 MT 3100 - ALLIED MATHEMATICS FOR PHYSICS (Also equivalent to MAT 100) Date & Time : 28-10-2006/9.00-12.00 Dept. No. Max. : 100 Marks Section A Answer ALL the questions (10 x 2 =20) 1) Evaluate lim x0 1 cos x . 2 x 2) Expand sin n and cos n . 3) Prove that cosh 2 x sinh 2 x cosh 2 x . 4) Find yn when y cosh 2 x . 5) Find L sin 3 2t 6) Find L 1 2s 3 ( s 1)5 7) Show that, in the curve r a , the polar subtangent varies as the square of the radius vector. 8) Find the coefficient of x n in the expansion of 1 . (1 2 x)(1 3x) 1 4 5 9) Find the rank of the matrix A 2 5 7 3 6 9 10) Mention a relation between binomial and Poisson distribution. Section B Answer any FIVE questions (5 x 8 = 40) 1 et 11) Find L t 1 12) Find L 1 s( s 1)( s 2) 13) If x iy tan( A iB) , prove that x 2 y 2 2 x cot 2 A 1 . 14) If y sin(m sin 1 x) , prove that (1 x 2 ) y2 xy1 m 2 y 0 . 15) Find the slope of the tangent with the initial line for the cardioid r a (1 cos ) at = . 6 1 1 2 16) Find the inverse of the matrix A 2 1 3 using Cayley Hamilton 3 2 3 theorem. 2.3 3.5 4.7 5.9 7 ... 2e . 3! 4! 5! 6! 2 18) Suppose on an average one house in 1000 in Telebakkam city needs television service during a year. If there are 2000 houses in that city, what is the probability that exactly 5 houses will need television service during the year. Section C 17) Show that Answer any TWO questions only (2 x 20 = 40) 19) (a) Using Laplace transform solve ( D 2 5 D 6) x e t when x (0) = 7.5, x’(0) = -18.5. (b) Prove that cosh5x = 16cosh5x – 20cosh3x + 5coshx. (12+8) 20) (a) Expand cos4 sin3 in terms of sines of multiples of angle. m (b) If y x 1 x 2 prove that (1 x2 ) yn2 (2n 1) xyn1 (n2 m2 ) yn 0 (8+12) 3 10 5 21) (a) Find the eigen values and eigen vectors of the matrix A 2 3 4 3 5 7 2 (b) Find the greatest term in the expansion of (1 x)31 3 when x . (15+5) 7 22) (a) Find the angle of intersection of the cardioid r a(1 cos ) and r b(1 cos ) . (b) Eight coins are tossed at a time, for 256 times. Number of heads observed at each throw is recorded and results are given below. No of heads at a throw 0 1 2 3 4 5 6 7 8 frequency 2 6 30 52 67 56 32 10 1 What are the theoretical values of mean and standard deviation? Calculate also the mean and standard deviation of the observed frequencies. (10+10) ______________