def creerPolydegre2(nb_racines=2, rac_radical=True, rac_quotient=False): if nb_racines == 2: redo = True while redo: a = randrange(1, 4) * (-1) ** randrange(2) alpha = randrange(1, 10) * (-1) ** randrange(2) beta = randrange(1, 10) gamma = [1, randrange(1, 6)][rac_radical] if rac_quotient: den = randrange(2, 6) while pgcd(alpha, den) != 1 or pgcd(beta, den) != 1: den = randrange(2, 6) alpha = Fraction(alpha, den) beta = Fraction(beta, den) b = -2 * alpha * a c = a * (alpha ** 2 - gamma * beta ** 2) if abs(c) <= 10 and c != 0 and not factoriser(repr(Polynome([[a, 2], [b, 1], [c, 0]]))): redo = False if c.denominator != 1: c = 'Fraction(%s, %s)' % (c.numerator, c.denominator) else: c = c.numerator if b.denominator != 1: b = 'Fraction(%s, %s)' % (b.numerator, b.denominator) else: b = b.numerator return Polynome([[a, 2], [b, 1], [c, 0]]) elif nb_racines == 1: a, b = valeur_alea(-9, 9), valeur_alea(-9, 9) return Polynome([[a ** 2, 2], [2 * a * b, 1], [b ** 2, 0]]) else: pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)] while pol[1][0] ** 2 - 4 * pol[0][0] * pol[2][0] >= 0: pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)] return Polynome(pol)
def factorisation(): """Génère un exercice de factorisation utilisant les identités remarquables ou la distributivité """ l = [randrange(1, 11) for dummy in range(21)] diff = [True, False, False] shuffle(diff) exo = [id_rem1, id_rem2] lexo = [exo[randrange(2)](l[0], l[1])] lexo.append(id_rem3(l[2], l[3])) lexo.append(id_rem3bis(l[4], l[5], l[6])) lexo.append(facteur_commun1(l[7:13], diff=diff.pop())) shuffle(lexo) exo = [facteur_commun2, facteur_commun3] shuffle(exo) lexo.append(exo[0](l[13:17], diff=diff.pop())) lexo.append(exo[1](l[17:21], diff=diff.pop())) exo = [ "\\exercice", u"Factoriser chacune des expressions littérales suivantes :" ] exo.append("\\begin{multicols}{2}") cor = [ "\\exercice*", u"Factoriser chacune des expressions littérales suivantes :" ] cor.append("\\begin{multicols}{2}") for i in range(len(lexo)): p = [lexo[i]] while True: fact = factoriser(p[-1]) if fact: p.append(fact) else: break p = Priorites3.texify( [Priorites3.splitting(p[j]) for j in range(len(p))]) cor.append('\\\\\n'.join( ['$%s=%s$' % (chr(i + 65), p[j]) for j in range(len(p) - 1)])) cor.append('\\\\') cor.append('\\fbox{$%s=%s$}\\\\\n' % (chr(i + 65), p[-1])) exo.append('\\\\\n'.join([ '$%s=%s$' % (chr(i + 65), Priorites3.texify([Priorites3.splitting(lexo[i])])[0]) for i in range(len(lexo)) ])) exo.append("\\end{multicols}") cor.append("\\end{multicols}") return exo, cor
def factorisation(): """Génère un exercice de factorisation utilisant les identités remarquables ou la distributivité """ l = [randrange(1, 11) for dummy in range(21)] diff = [True, False, False] shuffle(diff) exo = [id_rem1, id_rem2] lexo = [exo[randrange(2)](l[0], l[1])] lexo.append(id_rem3(l[2], l[3])) lexo.append(id_rem3bis(l[4], l[5], l[6])) lexo.append(facteur_commun1(l[7:13], diff=diff.pop())) shuffle(lexo) exo = [facteur_commun2, facteur_commun3] shuffle(exo) lexo.append(exo[0](l[13:17], diff=diff.pop())) lexo.append(exo[1](l[17:21], diff=diff.pop())) exo = ["\\exercice", u"Factoriser chacune des expressions littérales suivantes :"] exo.append("\\begin{multicols}{2}") cor = ["\\exercice*", u"Factoriser chacune des expressions littérales suivantes :"] cor.append("\\begin{multicols}{2}") for i in range(len(lexo)): p = [lexo[i]] while True: fact = factoriser(p[-1]) if fact: p.append(fact) else: break p = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))]) cor.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), p[j]) for j in range(len(p) - 1)])) cor.append('\\\\') cor.append('\\fbox{$%s=%s$}\\\\\n' % (chr(i + 65), p[-1])) exo.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), Priorites3.texify([Priorites3.splitting(lexo[i])])[0]) for i in range(len(lexo))])) exo.append("\\end{multicols}") cor.append("\\end{multicols}") return exo, cor
def tex_answer(self): exo = [r'\exercice*'] exo.append(_(u'Déterminer les racines des polynômes :\\par')) noms = [r'P\,(x) &= ', r'Q\,(x) &= ', r'R\,(x) &= '] r = '' question = [[], [], []] for i in range(3): p = [] m = Polynome(list(self.exercice[i])).ordonne() if factoriser('%r' % Polynome(m)): p = [factoriser('%r' % Polynome(m))] while factoriser(p[-1]): p.append(factoriser(p[-1])) if p and eval(Priorites3.splitting(p[-1])[0]).degre() > 0: tmp = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))]) question[i].append('{$\\! \\begin{aligned}') question[i].append(noms[i] + str(Polynome(m, 'x')) + r'\\') question[i].append('\\\\\n'.join(['&=%s' % (tmp[j]) for j in range(len(tmp))])) question[i].append(r'\end{aligned}$}\par') lp = Priorites3.splitting(p[-1]) racines = [] for e in lp: if e[:9] == 'Polynome(': e = eval(e) if len(e) == 2: racines.append(str(Fraction(-e[1][0], e[0][0]).simplifie())) else: racines.append('0') if len(racines) > 1: question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), '$}\\underline{ et }\\fbox{$'.join(racines))) elif len(racines) == 1: question[i].append(_(u'\\underline{L\'unique racine de $%s$ est }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), racines[0])) elif len(m) == 2 and m[0][1] == 2 and m[1][1] == 0 and m[0][0] * m[1][0] > 0: question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'$\par') question[i][-1] = question[i][-1].replace('&', '') if m[1][0] > 0: question[i].append('$' + noms[i][:7] + ' \\geqslant %r$' % m[1][0]) else: question[i].append('$' + noms[i][:7] + ' \\leqslant %r$' % m[1][0]) question[i].append(_(u'car un carré est toujours positif.\\par\n\\underline{$%s$ n\'a donc pas de racine.}') % (noms[i].rstrip(r' &= '))) else: question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'\quad$') question[i][-1] = question[i][-1].replace('&', '') question[i].append(_(u'On calcule le discriminant de $%s$ avec $a=%s$, $b=%s$ et $c=%s$ :\\par\\medskip') % (noms[i].rstrip(r' &= '), m[0][0], m[1][0], m[2][0])) question[i].append(r'\begin{tabularx}{\linewidth}[t]{XXX}') question[i].append(r'{$\! \begin{aligned}') if m[1][0]>0: sol = [[str(m[1][0]), '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]] sol.extend(Priorites3.priorites('%s**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0]))) else: sol = [['(', str(m[1][0]), ')', '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]] sol.extend(Priorites3.priorites('(%s)**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0]))) solTeX = Priorites3.texify(sol) for s in solTeX: question[i].append(u'\\Delta &= %s\\\\' % s) question[i].append(r'\end{aligned}$}') question[i].append(r'&') question[i].append(r'{$\! \begin{aligned}') delta = sol[-1][0] print(sol) sol = [['Fraction(SquareRoot([[%s, None], [-1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]] sol.extend(Priorites3.priorites(sol[0][0])) sol = Priorites3.texify(sol) for s in sol: question[i].append(u'x_1 &= %s\\\\' % s) racines = [sol[-1]] question[i].append(r'\end{aligned}$}') question[i].append(r'&') question[i].append(r'{$\! \begin{aligned}') sol = [['Fraction(SquareRoot([[%s, None], [1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]] sol.extend(Priorites3.priorites(sol[0][0])) sol = Priorites3.texify(sol) for s in sol: question[i].append(u'x_2 &= %s\\\\' % s) racines.append(sol[-1]) question[i].append(r'\end{aligned}$}') question[i].append(r'\end{tabularx}\par') question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), _('$}\\underline{ et }\\fbox{$').join(racines))) if i == 1: question.append(question[1]) if len(question) == 4: question.pop(1) if question[0][0][-6:] == r'\quad$': question[1].insert(0, r'\par\medskip\begin{tabularx}{\linewidth}[t]{XX}') question[2].insert(0, r'&') question[2].append(r'\end{tabularx}\par\medskip') else: question[0].insert(0, r'\begin{tabularx}{\linewidth}[t]{XX}') question[1].insert(0, r'&') question[1].append(r'\end{tabularx}\par\medskip') for i in range(3): exo.extend(question[i]) return exo
def tex_answer(self): exo = [r'\exercice*'] exo.append(_(u'Déterminer les racines des polynômes :\\par')) noms = [r'P\,(x) &= ', r'Q\,(x) &= ', r'R\,(x) &= '] r = '' question = [[], [], []] for i in range(3): p = [] m = Polynome(list(self.exercice[i])).ordonne() if factoriser('%r' % Polynome(m)): p = [factoriser('%r' % Polynome(m))] while factoriser(p[-1]): p.append(factoriser(p[-1])) if p and eval(Priorites3.splitting(p[-1])[0]).degre() > 0: tmp = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))]) question[i].append('{$\\! \\begin{aligned}') question[i].append(noms[i] + str(Polynome(m, 'x')) + r'\\') question[i].append('\\\\\n'.join(['&=%s' % (tmp[j]) for j in range(len(tmp))])) question[i].append(r'\end{aligned}$}\par') lp = Priorites3.splitting(p[-1]) racines = [] for e in lp: if e[:9] == 'Polynome(': e = eval(e) if len(e) == 2: racines.append(str(Fraction(-e[1][0], e[0][0]).simplifie())) else: racines.append('0') if len(racines) > 1: question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), '$}\\underline{ et }\\fbox{$'.join(racines))) elif len(racines) == 1: question[i].append(_(u'\\underline{L\'unique racine de $%s$ est }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), racines[0])) elif len(m) == 2 and m[0][1] == 2 and m[1][1] == 0 and m[0][0] * m[1][0] > 0: question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'$\par') question[i][-1] = question[i][-1].replace('&', '') if m[1][0] > 0: question[i].append('$' + noms[i][:7] + ' \\ge %r$' % m[1][0]) else: question[i].append('$' + noms[i][:7] + ' \\le %r$' % m[1][0]) question[i].append(_(u'car un carré est toujours positif.\\par\n\\underline{$%s$ n\'a donc pas de racine.}') % (noms[i].rstrip(r' &= '))) else: question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'\quad$') question[i][-1] = question[i][-1].replace('&', '') question[i].append(_(u'On calcule le discriminant de $%s$ avec $a=%s$, $b=%s$ et $c=%s$ :\\par\\medskip') % (noms[i].rstrip(r' &= '), m[0][0], m[1][0], m[2][0])) question[i].append(r'\begin{tabularx}{\linewidth}[t]{XXX}') question[i].append(r'{$\! \begin{aligned}') sol = [[str(m[1][0]), '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]] sol.extend(Priorites3.priorites('%s**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0]))) solTeX = Priorites3.texify(sol) for s in solTeX: question[i].append(u'\\Delta &= %s\\\\' % s) question[i].append(r'\end{aligned}$}') question[i].append(r'&') question[i].append(r'{$\! \begin{aligned}') delta = sol[-1][0] sol = [['Fraction(SquareRoot([[%s, None], [-1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]] sol.extend(Priorites3.priorites(sol[0][0])) sol = Priorites3.texify(sol) for s in sol: question[i].append(u'x_1 &= %s\\\\' % s) racines = [sol[-1]] question[i].append(r'\end{aligned}$}') question[i].append(r'&') question[i].append(r'{$\! \begin{aligned}') sol = [['Fraction(SquareRoot([[%s, None], [1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]] sol.extend(Priorites3.priorites(sol[0][0])) sol = Priorites3.texify(sol) for s in sol: question[i].append(u'x_2 &= %s\\\\' % s) racines.append(sol[-1]) question[i].append(r'\end{aligned}$}') question[i].append(r'\end{tabularx}\par') question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), _('$}\\underline{ et }\\fbox{$').join(racines))) if i == 1: question.append(question[1]) if len(question) == 4: question.pop(1) if question[0][0][-6:] == r'\quad$': question[1].insert(0, r'\par\medskip\begin{tabularx}{\linewidth}[t]{XX}') question[2].insert(0, r'&') question[2].append(r'\end{tabularx}\par\medskip') else: question[0].insert(0, r'\begin{tabularx}{\linewidth}[t]{XX}') question[1].insert(0, r'&') question[1].append(r'\end{tabularx}\par\medskip') for i in range(3): exo.extend(question[i]) return exo