system which admits a solution. 2 3 4 6 h 7 2. (1.2; 12) Find the general solutions of the system whose augmented matrix is 1 −7 0 6 5 0 0 1 −2 −3 −1 7 −4 2 7 1 −2 4 3. (1.3; 17) Let a1 = 4 ‚ a2 = −3 ‚ b = 1 . For what −2 7 h value(s) of h is b in the plane spanned by a1 and a2? 4. (1.4; 15) Let A = b1 2 −1 and b = . Show that the equation −6 3 b2 Ax = b does not have a solution for all possible b‚ and describe the set of all b for which Ax = b does have a solution.
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A1-R4: IT TOOLS AND BUSINESS SYSTEMS NOTE: 1. There are TWO PARTS in this Module/Paper. PART ONE contains FOUR questions and PART TWO contains FIVE questions. 2. PART ONE is to be answered in the TEAR-OFF ANSWER SHEET only‚ attached to the question paper‚ as per the instructions contained therein. PART ONE is NOT to be answered in the answer book. 3. Maximum time allotted for PART ONE is ONE HOUR. Answer book for PART TWO will be supplied at the table when the answer sheet for PART ONE is returned
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Mathematical Database MATHEMATICAL INDUCTION 1. Introduction Mathematics distinguishes itself from the other sciences in that it is built upon a set of axioms and definitions‚ on which all subsequent theorems rely. All theorems can be derived‚ or proved‚ using the axioms and definitions‚ or using previously established theorems. By contrast‚ the theories in most other sciences‚ such as the Newtonian laws of motion in physics‚ are often built upon experimental evidence and can never be
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________________________________ Signature _________________________ Date ---------------------------------------------------------------------------------------------------------------- FOR OFFICIAL USE Assignment Received By: ACNB A1 QCF SEP 2014 DUE Date: 1 Unit Outcomes Outcome Evidence for the criteria Feedback Assessor’s decision First attempt Explain the importance of the key essential elements required to the formation of a valid contract
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be independent or disjoint. Problem 1 Solution We prove this part by induction‚ for k = 2 we have P (A1 ∪ A2 ) = P (A1 ) + P (A2 ) − P (A1 ∩ A2 ) ≤ P (A1 ) + P (A2 ) (1) Now‚ assume that the statement is true for k = n n P (A1 ∪ A2 . . . ∪ An ) ≤ P (Ai ) (2) i=1 For k = n + 1‚ we can write P (A1 ∪ A2 . . . ∪ An+1 ) = P (An+1 ) + P (A1 ∪ A2 . . . ∪ An ) − P (An+1 ∩ (A1 ∪ A2 . . . ∪ An )) n+1 ≤ P (Ai ) i=1 (3) This proves the union bound. 2. (Independence) For
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About the Authors Titu Andreescu received his BA‚ MS‚ and PhD from the West University of Timisoara‚ Romania. The topic of his doctoral dissertation was “Research on Diophantine Analysis and Applications.” Professor Andreescu currently teaches at the University of Texas at Dallas. Titu is past chairman of the USA Mathematical Olympiad‚ served as director of the MAA American Mathematics Competitions (1998–2003)‚ coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993–2002)
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The Universal Declaration of Human Rights‚ n.d‚ http://www.un.org/en/documents/udhr/index.shtml#a1 (accessed November 12‚ 2012) Warchild.org [2] Un.org. The Universal Declaration of Human Rights‚ n.d‚ http://www.un.org/en/documents/udhr/index.shtml#a1 (accessed November 12‚ 2012) [3] Un.org [4] Un.org. The Universal Declaration of Human Rights‚ n.d‚ http://www.un.org/en/documents/udhr/index.shtml#a1 (accessed November 12‚ 2012) [5] Un.org
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BIO 219 Group 1 Section 66 October 19‚ 2012 The extraction of purified DNA from A. fischeri by restriction digestion using Sal I enzyme and pGEM for shotgun cloning Introduction: The ultimate goal of this experiment is to isolate the lux operon‚ a targeted piece of DNA that causes bioluminescence‚ from Aliivibrio fischeri and insert it into the DNA of Escherichia coli in order to make it glow. A. fischeri is a gram-negative bacteria which participates in a symbiotic relationship with
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Develop a workplace learning environment Name Sukhwinder Singh Id 1035 Activity 1 Q1. ... A1. Learning opportunities can be considered using following dimensions: Formal learning includes the hierarchically structured school that runs from primary school through to university and organised school like program created in business for technical and professional training. Informal learning describe a lifelong process whereby individuals acquire attitudes‚ value‚ skills and knowledge from daily
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and the pin configuration of 8255 are shown in fig. • The 8-bit data bus buffer is controlled by the read/write control logic. The read/write control logic manages all of the internal and external transfers of both data and control words. • RD‚ WR‚ A1‚ A0 and RESET are the inputs provided by the microprocessor to the READ/ WRITE control logic of 8255. The 8-bit‚ 3-state bidirectional buffer is used to interface the 8255 internal data bus with the external system data bus. PIO 8255 • This buffer
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