Download MATH 121B Final Exam: Solutions and Problems in Mathematics - Prof. V. Serganova and more Exams Mathematics in PDF only on Docsity! SAMPLE FINAL EXAM MATH 121B 1. A particle slides without a friction around a vertical cylinder under the force of gravity. Set up the Lagrange equation. 2. Let a and b be two critical points of the Legendre’s polynomial Pl (x) on interval [−1, 1]. Prove that if l > 0 then ∫ b a Pl (x) dx = 0. 3. Using power series show that the equation ( 1 − x2 ) y′′ − xy′ + m2y = 0 has a polynomial solution for every integer m. 4. (a) Find characteristic frequencies and normal modes of a square membrane with side a. (b) Find all simple combinations of normal modes which have nodal lines at the diagonal y = x. (c) Use (b) to find characteristic frequencies of a membrane in the form of a right triangle with angle 45◦. 5. A string has initial displacement y0 = x (l − x). Find the displacement as a function of x and t. 6. Find a steady state temperature in a semi-infinite covering the region 0 ≤ y ≤ 1, x ≥ 0, if the temperature along the x and y-axis is zero, and the top edge is kept at u (x, 1) = {100x, x<1 0, x>1 . 7. Find the Green function for the Laplace equation in the region x > 0, y > 0 in the plane with boundary condition u (x, 0) = u (0, y) = 0. 8. Five letters are put randomly into five envelopes (each envelope contains exactly one letter). What is the probability that at least one letter gets into the correct envelope? 9. A 300-page book has, on the average, one misprint every 5 pages. On about how many pages would you expect to find (a) no misprints, (b) one misprint, (c) two misprints? 10. 5 identical balls are randomly put in 10 boxes. What is the probability that all 5 balls got in the first five boxes for (a) Maxwell-Boltzmann statistics, 1 2 SAMPLE FINAL EXAM MATH 121B (b) Fermi-Dirac statistics, (c) Bose-Einstein statistics? In which case probability is the biggest, the smallest? 11. A multiple-choice test has 100 questions. Each question has four choices for the answer, and only one choice is correct. Suppose that you do not know the subject and choose all your answers randomly. Using Chebyshev inequality estimate the probability that you score for the exam is more than 50%.
