1. (5) Show that, if −π/2<0<π/2, then L - log(1 + tan tan x) dx = 0 log sec 0. 2. (6) Suppose that f is continuously differentiable on [0,1] and that f(x)dx=0. Prove that | 2 [[" f (x)}² dx = [ " \f'(x) dx = [ " \f(x) | dr. dr 3. (7) How many n× n invertible matrices A are there for which all the entries of both A and A-1 are either 0 or 1? 4. (7) Let a be a nonzero real and u and v be real 3-vectors. Solve the equation for the vector x. 2ax + (v xx) + u = 0 5. (6) Let f(x) be a polynomial with real coefficients, evenly many of which are nonzero, which is palindromic. This means that the coefficients read the same in either direction, i.e. ak = an-k if f(x) = 0 akxk or, alternatively, f(x) = x^f(1/x), where n is the degree of the polynomial. Prove that f(x) has at least one root of absolute value 1. 6. (9) Let G be a subgroup of index 2 contained in Sn, the group all permutations of n elements. Prove that G = An, the alternating group of all even permutations. 7. (11) Let f(x) be a nonconstant polynomial that takes only integer values when x is an integer, and let P be the set of all primes that divide f(m) for at least one integer m. Prove that P is an infinite set.

1. (5) Show that, if −π/2<0<π/2, then L - log(1 + tan tan x) dx = 0 log sec 0. 2. (6) Suppose that f is continuously differentiable on [0,1] and that f(x)dx=0. Prove that | 2 [[" f (x)}² dx = [ " \f'(x) dx = [ " \f(x) | dr. dr 3. (7) How many n× n invertible matrices A are there for which all the entries of both A and A-1 are either 0 or 1? 4. (7) Let a be a nonzero real and u and v be real 3-vectors. Solve the equation for the vector x. 2ax + (v xx) + u = 0 5. (6) Let f(x) be a polynomial with real coefficients, evenly many of which are nonzero, which is palindromic. This means that the coefficients read the same in either direction, i.e. ak = an-k if f(x) = 0 akxk or, alternatively, f(x) = x^f(1/x), where n is the degree of the polynomial. Prove that f(x) has at least one root of absolute value 1. 6. (9) Let G be a subgroup of index 2 contained in Sn, the group all permutations of n elements. Prove that G = An, the alternating group of all even permutations. 7. (11) Let f(x) be a nonconstant polynomial that takes only integer values when x is an integer, and let P be the set of all primes that divide f(m) for at least one integer m. Prove that P is an infinite set.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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