Feat: E6 technique développement et réduction
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\usepackage{amsmath}
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\author{Benjamin Bertrand}
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\title{Information chiffrée 1 - Exercices}
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\date{Octobre 2021}
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\pagestyle{empty}
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\xsimsetup{
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solution/print = false
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}
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\begin{document}
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\begin{exercise}[subtitle={Réductions}]
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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\item $- 3x - 10 + 4x + 7$
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\item $3x + 2 - 9x - 7$
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\item $- 9x^{2} - 7 - 7x^{2} + 6 + 6x - 7$
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\item $- 7x - 1 - 7x + 8 + 4x + 3x$
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\item $3x + 15 + 9x + 18x + 17$
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\item $- 4x - 5 + 4x + 1$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{flalign*}
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A =& - 3x - 10 + 4x + 7\\ =& - 3x - 10 + 4x + 7\\ =& - 3x + 4x - 10 + 7\\ =& (- 3 + 4) \times x - 3\\ =& x - 3
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\end{flalign*}
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\begin{flalign*}
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B =& 3x + 2 - 9x - 7\\ =& 3x + 2 - 9x - 7\\ =& 3x - 9x + 2 - 7\\ =& (3 - 9) \times x - 5\\ =& - 6x - 5
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\end{flalign*}
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\begin{flalign*}
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C =& - 9x^{2} - 7 - 7x^{2} + 6 + 6x - 7\\ =& - 9x^{2} - 7x^{2} - 7 - 1 + 6x\\ =& (- 9 - 7) \times x^{2} + 6x - 7 - 1\\ =& - 16x^{2} + 6x - 8
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\end{flalign*}
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\begin{flalign*}
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D =& - 7x - 1 - 7x + 8 + 4x + 3x\\ =& - 7x - 1 + 8 - 7x + (4 + 3) \times x\\ =& (- 7 - 7) \times x + 7 + 7x\\ =& - 14x + 7 + 7x\\ =& - 14x + 7x + 7\\ =& (- 14 + 7) \times x + 7\\ =& - 7x + 7
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\end{flalign*}
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\begin{flalign*}
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E =& 3x + 15 + 9x + 18x + 17\\ =& 3x + 15 + (9 + 18) \times x + 17\\ =& 3x + 15 + 17 + 27x\\ =& (3 + 27) \times x + 32\\ =& 30x + 32
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\end{flalign*}
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\begin{flalign*}
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F =& - 4x - 5 + 4x + 1\\ =& - 4x - 5 + 4x + 1\\ =& - 4x + 4x - 5 + 1\\ =& (- 4 + 4) \times x - 4\\ =& 0x - 4\\ =& - 4
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\end{flalign*}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Simple développement}]
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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\item $10(x + 3)$
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\item $8(- 10x + 2)$
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\item $3(- 9x + 3)$
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\item $- 10x(- 6x + 3)$
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\item $2x(10x - 2) - 4$
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\item $- 3x - 7x(- 10x + 9)$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{flalign*}
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A =& 10(x + 3)\\ =& 10x + 10 \times 3\\ =& 10x + 30
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\end{flalign*}
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\begin{flalign*}
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B =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
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\end{flalign*}
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\begin{flalign*}
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C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
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\end{flalign*}
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\begin{flalign*}
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D =& - 10x(- 6x + 3)\\ =& - 10x \times - 6x - 10x \times 3\\ =& - 10(- 6) \times x^{1 + 1} + 3(- 10) \times x\\ =& 60x^{2} - 30x
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\end{flalign*}
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\begin{flalign*}
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E =& 2x(10x - 2) - 4\\ =& 2x \times 10x + 2x(- 2) - 4\\ =& 2 \times 10 \times x^{1 + 1} - 2 \times 2 \times x - 4\\ =& 20x^{2} - 4x - 4
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\end{flalign*}
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\begin{flalign*}
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F =& - 3x - 7x(- 10x + 9)\\ =& - 3x - 7x \times - 10x - 7x \times 9\\ =& - 3x - 7(- 10) \times x^{1 + 1} + 9(- 7) \times x\\ =& - 3x - 63x + 70x^{2}\\ =& (- 3 - 63) \times x + 70x^{2}\\ =& 70x^{2} - 66x
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\end{flalign*}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Double développement}]
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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\item $(x + 10)(x + 6)$
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\item $(8x - 6)(- 5x - 10)$
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\item $(- 6x + 9)(7x - 3)$
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\item $(5x + 2)(- 6x + 2)$
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\item $(- 8x + 8)^{2}$
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\item $(2x - 6)^{2}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{flalign*}
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A =& 10(x + 3)\\ =& 10x + 10 \times 3\\ =& 10x + 30
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\end{flalign*}
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\begin{flalign*}
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B =& 8(- 10x + 2)\\ =& 8 \times - 10x + 8 \times 2\\ =& 8(- 10) \times x + 16\\ =& - 80x + 16
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\end{flalign*}
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\begin{flalign*}
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C =& 3(- 9x + 3)\\ =& 3 \times - 9x + 3 \times 3\\ =& 3(- 9) \times x + 9\\ =& - 27x + 9
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\end{flalign*}
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\begin{flalign*}
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D =& - 10x(- 6x + 3)\\ =& - 10x \times - 6x - 10x \times 3\\ =& - 10(- 6) \times x^{1 + 1} + 3(- 10) \times x\\ =& 60x^{2} - 30x
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\end{flalign*}
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\begin{flalign*}
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E =& 2x(10x - 2) - 4\\ =& 2x \times 10x + 2x(- 2) - 4\\ =& 2 \times 10 \times x^{1 + 1} - 2 \times 2 \times x - 4\\ =& 20x^{2} - 4x - 4
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\end{flalign*}
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\begin{flalign*}
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F =& - 3x - 7x(- 10x + 9)\\ =& - 3x - 7x \times - 10x - 7x \times 9\\ =& - 3x - 7(- 10) \times x^{1 + 1} + 9(- 7) \times x\\ =& - 3x - 63x + 70x^{2}\\ =& (- 3 - 63) \times x + 70x^{2}\\ =& 70x^{2} - 66x
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\end{flalign*}
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\end{multicols}
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\end{solution}
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\vfill
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\printexercise{exercise}{1}
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\printexercise{exercise}{2}
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\printexercise{exercise}{3}
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\vfill
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\printexercise{exercise}{1}
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\printexercise{exercise}{2}
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\printexercise{exercise}{3}
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\newpage
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\printsolutionstype{exercise}
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\end{document}
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\usepackage{amsmath}
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\author{Benjamin Bertrand}
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\title{Information chiffrée 1 - Exercices}
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\date{Octobre 2021}
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\xsimsetup{
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solution/print = false
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}
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\begin{document}
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\begin{exercise}[subtitle={Réductions}]
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Développer puis réduire les expressions suivantes
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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%- set a = Expression.random("{a}x + {b} + {c}x + {d}")
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\item $\Var{a}$
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%- set b = Expression.random("{a}x + {b} + {c}x + {d}")
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\item $\Var{b}$
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%- set c = Expression.random("{a}x^2 + {b} + {c}x^2 + {d} + {d}x + {e}")
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\item $\Var{c}$
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%- set d = Expression.random("{a}x + {b} + {c}x + {d} + {e}x + {f}x")
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\item $\Var{d}$
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%- set e = Expression.random("{a}*x + {b} + {c}x + {d}x + {e}", min_max=(2, 20))
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\item $\Var{e}$
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%- set f = Expression.random("{a}x + {b} + {c}x + {d}", conditions=["a+c==0"])
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\item $\Var{f}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{flalign*}
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A =& \Var{a.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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B =& \Var{b.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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C =& \Var{c.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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D =& \Var{d.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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E =& \Var{e.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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F =& \Var{f.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Simple développement}]
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Développer puis réduire les expressions suivantes
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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%- set a = Expression.random("{a}*(x + {b})", rejected=[-1,0,1])
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\item $\Var{a}$
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%- set b = Expression.random("{a}*({c}x + {d})", rejected=[-1,0,1])
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\item $\Var{b}$
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%- set c = Expression.random("{a}*({c}x + {d})", rejected=[-1,0,1])
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\item $\Var{c}$
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%- set d = Expression.random("{c}*x*({a}x + {b})", rejected=[-1,0,1])
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\item $\Var{d}$
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%- set e = Expression.random("{a}*x*({b}x + {c}) + {d}", rejected=[-1,0,1])
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\item $\Var{e}$
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%- set f = Expression.random("{c}*x + {d}*x*({a}x + {b})", rejected=[-1,0,1])
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\item $\Var{f}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\begin{solution}
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\begin{multicols}{2}
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\begin{flalign*}
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A =& \Var{a.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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B =& \Var{b.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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C =& \Var{c.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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D =& \Var{d.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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E =& \Var{e.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\begin{flalign*}
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F =& \Var{f.simplify().explain() | join('\\\ =& ')}
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\end{flalign*}
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\end{multicols}
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\end{solution}
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\begin{exercise}[subtitle={Double développement}]
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Développer puis réduire les expressions suivantes
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\begin{multicols}{3}
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\begin{enumerate}[label={\Alph*=}]
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%- set a = Expression.random("(x + {a})*(x + {b})", rejected=[-1,0,1])
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\item $\Var{a}$
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%- set b = Expression.random("({a}x + {b})*({c}x + {d})", rejected=[-1,0,1])
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\item $\Var{b}$
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%- set c = Expression.random("({a}x + {b})*({c}x + {d})", rejected=[-1,0,1])
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\item $\Var{c}$
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%- set d = Expression.random("({c}*x + {d})*({a}x + {b})", rejected=[-1,0,1])
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\item $\Var{d}$
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%- set e = Expression.random("({b}x + {c})^2", rejected=[-1,0,1])
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\item $\Var{e}$
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%- set f = Expression.random("({a}x + {b})^2", rejected=[-1,0,1])
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\item $\Var{f}$
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\end{enumerate}
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\end{multicols}
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\end{exercise}
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\newpage
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\printsolutionstype{exercise}
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\end{document}
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