99 lines
3.4 KiB
Plaintext
99 lines
3.4 KiB
Plaintext
(*<*)
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theory MathExam
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imports "Isabelle_DOF.mathex"
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HOL.Real
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begin
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(*>*)
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(* open_monitor*[exam::MathExam] *)
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section*[header::Header,examSubject= "[algebra]",
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date="''02-05-2018''", timeAllowed="90::int"] \<open>Exam number 1\<close>
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text\<open>
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\begin{itemize}
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\item Use black ink or black ball-point pen.
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\item Draw diagrams in pencil.
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\item Answer all questions in the spaces provided.
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\end{itemize}
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\<close>
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text*[idir::Author, affiliation="''CentraleSupelec''",
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email="''idir.aitsadoune@centralesupelec.fr''"]
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\<open>Idir AIT SADOUNE\<close>
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figure*[figure::figure, spawn_columns=False,
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relative_width="80",
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src="''figures/Polynomialdeg5''"]
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\<open>A Polynome.\<close>
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subsubsection*[exo1 :: Exercise, content="[q1::Task,q2::Task]"]\<open>Exercise 1\<close>
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text\<open>
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Here are the first four lines of a number pattern.
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\begin{itemize}
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\item Line 1 : @{term "1*6 + 2*4 = 2*7"}
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\item Line 2 : @{term "2*7 + 2*5 = 3*8"}
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\item Line 3 : @{term "3*8 + 2*6 = 4*9"}
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\item Line 4 : @{term "4*9 + 2*7 = 5*10"}
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\end{itemize}
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\<close>
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declare [[show_sorts=false]]
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subsubsection*[exo2 :: Exercise, content="[q1::Task,q2::Task]"]\<open>Exercise 2\<close>
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text\<open>Find the roots of the polynome:
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@{term "(x^3) - 6 * x^2 + 5 * x + 12"}.
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Note the intermediate steps in the following fields and submit the solution.\<close>
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text\<open>
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\begin{Form}[action={http://your-web-server.com/path/receiveform.cgi}]
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\begin{tabular}{l}
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From @{term "(x^3) - 6 * x^2 + 5 * x + 12"} \\\\
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\TextField{have 1} \\\\
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\TextField{have 2} \\\\
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\TextField{have 3} \\\\
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\TextField{finally show} \\\\
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\CheckBox[width=1em]{Has the polynomial as many solutions as its degree ? } \\\\
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\Submit{Submit}\\
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\end{tabular}
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\end{Form}
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\<close>
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(* a bit brutal, as long as lemma* does not yet work *)
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(*<*)
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lemma check_polynome :
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fixes x::real
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shows "(x^3) - 6 * x^2 + 5 * x + 12 = (x-4) * (x+1) * (x - 3)"
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proof -
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have * : "(x-4) * (x+1) * (x - 3) = (x-4) * ((x+1) * (x-3))"
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by simp
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have ** : "... = (x-4) * (x^2 - 2*x - 3)"
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apply(auto simp: right_diff_distrib add.commute semiring_normalization_rules(1)[symmetric])
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by (simp add: semiring_normalization_rules(29))
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have *** : "... = x^3 - 6 * x^2 + 5 * x + 12"
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apply(auto simp: right_diff_distrib left_diff_distrib add.commute semiring_normalization_rules(1)[symmetric])
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by (simp add: numeral_3_eq_3 semiring_normalization_rules(29))
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show ?thesis
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by(simp only: * ** ***)
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qed
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(*>*)
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text*[a1::Answer_Formal_Step]\<open>First Step: Fill in term and justification\<close>
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text*[a2::Answer_Formal_Step]\<open>Next Step: Fill in term and justification\<close>
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text*[a3::Answer_Formal_Step]\<open>Next Step: Fill in term and justification\<close>
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text*[a4::Answer_Formal_Step]\<open>Next Step: Fill in term and justification\<close>
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text*[q1::Task, local_grade="oneStar", mark="1::int", type="formal"]
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\<open>Complete Line 10 : @{term "10*x + 2*y = 11*16"}\<close>
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subsubsection*[exo3 :: Exercise, content="[q1::Task,q2::Task]"]\<open>Exercise 3\<close>
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text*[q2::Task, local_grade="threeStars", mark="3::int", type="formal"]
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\<open>Prove that @{term "n*(n+5) + 2*(n+3) "} is always the product of two numbers
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with a difference of 5.
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\<close>
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(* this does not work on the level of the LaTeX output for known restrictions of the Toplevel. *)
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(* close_monitor*[exam :: MathExam] *)
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end
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