# problem sheets to be answered as final exam will have similar questions

problem sheets to be answered as final exam will have similar questions. These would be very good to use for studying we have been told. ( need questions answered in a simplest way possible for me to understand also.) Problem Sheet for Week 1 Question 1: Suppose you have the set of natural numbers N together with the binary operation ? (subtraction). Is the set closed under the operation ? ?

Question 2: Suppose we have integers a,b,c,d (elements of Z) with the properties that a ? b and c ? d. Is it always true that ac ? bd ? Verify the inequality is true provided 0 ? a and 0 ? c

Question 3:Verify that (a2 ?b2) = (a?b)(a + b) (a3 ?b3) = (a?b)(a2 + ab + b2) where a,b ?R

Question 4: Find factorizations for the following values and expressions: 25 91 221 437 439 x2 + x?6 x2 ?5x + 6 x2 + 9

Question 5: Find the solutions to the following equations – assuming they exist. Try factorizing the expressions on the left of the equations x2 ?3x + 2 = 0 2×2 ?x?1 = 0

Question 6: Does the following cubic equation have a real-valued solution? Explain. (Do not try to calculate the solution exactly – just give reasons why you know there must be a solution) x3 ?11×2 + 3x?10

Question 7: Determine if the number 991 is a prime number. If the number is prime at what point can you stop looking for divisors (factors) of the number 991 ? What about in general ? 1

Question 8: Using Euclid’s Algorithm (EA) calculate the greatest common divisor of A = 21 and B = 171. Repeat for A = 221 and B = 247. Repeat for A = 133 and B = 69.

Question 9: A function f(N) is said to be ”order Nk ( written as O(Nk))provided lim?? f(N) Nk ? C for some positive value C and positive value k. Show that f(N) = N3 ?5N2 + 6N ?2 is O(N3) but not O(N2)

Question 10:The exponential function f(x) = ex dominates any ”power of x” function as x increases to in?nity. That is lim xk ex = 0 for every positive value k. Use the power series given below to verify this fact. ex = 1 + x + x2 2! + x3 3! + x4 4! + x5 5! + … its in pdf format…

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