Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
i) True. Consider the function $f(x) = x^3 - 4x - 16$. Evaluating at the endpoints of the interval: $f(3) = 27 - 12 - 16 = -1$ and $f(4) = 64 - 16 - 16 = 32$. Since $f(3) < 0$ ________ _______ ______ ____ _______ _________ _____ ____ __________.
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Find one root of the equation:
x3 - 2x 5 = 0
in the interval [3,2] using Birge-Vieta method. Perform only one iteration.
Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.
Show that (x, 5) is not a principal ideal in Z[x].
Find the first term of the series whose second and subsequent terms are ,0,3,8 − 0,1 using difference table.
Let R = Z20
Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:
i) The equation x3 4x 16 = 0 has a root in the interval [3, 4].
ii) The order of convergence of the secant method is 0.62.
iii) For the system of linear equations:
5x + y + 2z = 34
4y − 3z = 12
10x − 2y + z = −4
the matrix is diagonally dominant.
iv) The numerical method:
is relatively stable.
v) The method:
converges to 1.5 for any choice of initial approximation.
Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.