应用数学与信息技术专业入学考试大纲.pdf

应用数学与信息技术专业入学考试大纲专业名称：应用数学与信息技术 Прикладная математика и технологии Applied mathematics and Technology 考试课程名称：应用数学 Математика и программирование Mathematics and programming 考试内容： На экзамене предлагаются задания по дисциплинам: Математический анализ, линейная алгебра, дискретная математика, обыкновенные дифференциальные уравнения, уравнения в частных производных, теория вероятностей, численные методы, теория оптимизации программирование. The tasks are taken from the following topics: calculus, linear algebra, discrete mathematics (boolean functions, graph theory), ODE and PDE, probability theory, optimization theory, programming. 考试形式及要求：Очная форма (письменный экзамен) Written exam (offline form) 院系咨询人及电话咨询人：姚老师咨询电话：0755-28323285 考试样题： ÔÀÊÓËÜÒÅÒ ÂÛ×ÈÑËÈÒÅËÜÍÎÉ ÌÀÒÅÌÀÒÈÊÈ È ÊÈÁÅÐÍÅÒÈÊÈ ÌÃÓ-ÏÏÈ ÏÐÈÌÅÐÛ ÂÑÒÓÏÈÒÅËÜÍÎÃÎ ÇÀÄÀÍÈß Â ÌÀÃÈÑÒÐÀÒÓÐÓ 1. Íàéäèòå îáëàñòü ñõîäèìîñòè ôóíêöèîíàëüíîãî ðÿäà +∞ X 1 3n − 1 n=1 1−x 1 + 2x n . 2. Äàíà Ñè-ôóíêöèÿ equation(). Åñëè îíà ñîäåðæèò îøèáêó, îáîñíóéòå ýòî â êà÷åñòâå îòâåòà. Åñëè âîçìîæíî ïðèâåñòè êîä ôóíêöèé f() è g(), ïðè êîòîðûõ ôóíêöèÿ equation() âñåãäà âîçâðàùàåò çíà÷åíèå TRUE, íàïèøèòå êîä ýòèõ ôóíêöèé. Èíà÷å îáîñíóéòå, ÷òî õîòÿ áû äëÿ îäíîé èç ôóíêöèé f() è g() ýòî ñäåëàòü íåëüçÿ. #include int f(void); int g(const int *, int); int equation(void) { int a[1024] = {0}; int i; if (scanf(%d, &i) == 1 && 10 <= i && i < 100) { a[i] = 2023; return a[g(&a[10], 90)] == 2023; } return f() == 1 - f(); } c ÂÌÊ ÌÃÓ-ÏÏÈ, 2023. 3. Áàçèñîì ëèíåéíîãî ïðîñòðàíñòâà L1 ÿâëÿþòñÿ âåêòîðû ḡ1 (2, 3, 2) è ḡ2 (1, 1, 3), à áàçèñîì ëèíåéíîãî ïðîñòðàíñòâà L2 âåêòîðû f¯1 (1, 1, 1) è f¯2 (2, 1, 4). Íàéäèòå ðàññòîÿíèå îò òî÷êè M (3, 4, −1) äî ïåðåñå÷åíèÿ ïðîñòðàíñòâ L1 è L2 . 4. Íàéäèòå âñå ðåøåíèÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿ y − y 0 = y 2 + xy 0 , óäîâëåòâîðÿþùèå íà÷àëüíîìó óñëîâèþ y(2) = 3. 5. Ïóñòü â ïðîñòîì íåîðèåíòèðîâàííîì ãðàôå 65 âåðøèí è 2021 ðåáðî. Ñêîëüêî â íåì ìîæåò áûòü êîìïîíåíò ñâÿçíîñòè? Îáîñíóéòå âàø îòâåò! 6. Ïëîòíîñòü âåðîÿòíîñòè ñëó÷àéíîé âåëè÷èíû X èìååò âèä ( A, åñëè |x + 2| 6 3, f (x) = 0, åñëè |x + 2| > 3. Íàéäèòå çíà÷åíèå êîíñòàíòû A, ìàòåìàòè÷åñêîå îæèäàíèå M [X] è äèñïåðñèþ D[X] ñëó÷àéíîé âåëè÷èíû X . 7. ×èñëà A, B , C òàêîâû, ÷òî âåëè÷èíà Z1 (x3 − (Ax2 + Bx + C))2 dx −1 ïðèíèìàåò ìèíèìàëüíîå âîçìîæíîå çíà÷åíèå. Íàéäèòå ýòî çíà÷åíèå. 8. Íàéäèòå ðåøåíèå çàäà÷è (ôóíêöèþ u(x, t)) ( utt (x, t) = a2 uxx (x, t) + x2 t, −∞ < x < +∞, 0 6 t < +∞; √ u(x, 0) = x, ut (x, 0) = sin x, −∞ < x < +∞. c ÂÌÊ ÌÃÓ-ÏÏÈ, 2023. Îòâåòû: x ∈ (−∞; −2] ∪ (0; +∞). √ 2. 3. x+1 . 4. y = x−1 1. 5. 1. 1 , M [X] = −2, D [X] = 3. 6 7. 8/175. (A = C = 0, B = 3/5). √ √ x − at + x + at sin x sin at x2t3 a2t5 8. u(x, t) = + + + . 2 a 6 60 6. A= c ÂÌÊ ÌÃÓ-ÏÏÈ, 2023. SMBU, Computational mathematics and cybernetics Faculty MASTER'S DEGREE ENTRANCE EXAM (SAMPLE) 1. Find the convergence area of the functional series +∞ X 1 3n − 1 n=1 1−x 1 + 2x n . Given the C-function equation(). If it contain some mistakes, prove it, and answer ¾WRONG FUNCTION¿. If it's possible to create functions f() è g() such, that function equation() always returns TRUE, write the code of these functions. Otherwise, prove that at least one of the functions f() or g() cannot be created. 2. #include int f(void); int g(const int *, int); int equation(void) { int a[1024] = {0}; int i; if (scanf(%d, &i) == 1 && 10 <= i && i < 100) { a[i] = 2023; return a[g(&a[10], 90)] == 2023; } return f() == 1 - f(); } The space L1 is a linear span of vectors ḡ1 (2, 3, 2) and ḡ2 (1, 1, 3), and the space L2 is a linear span of vectors f¯1 (1, 1, 1) è f¯2 (2, 1, 4). Find the distance between point M (3, 4, −1) and intersection of spaces L1 and L2 . 3. c CMC SMBU, 2023. 4. Find all solutions of dierential equation y − y 0 = y 2 + xy 0 , that satisfy condition y(2) = 3. 5. Simple unoriented graph G has 65 vertices and 2021 edge. Could the graph G be disconnected? Prove your answer! 6. The probability density of random variable X given by the formula ( A, f (x) = 0, if |x + 2| 6 3, if |x + 2| > 3. Find the value of A, expectation M [X] and standard deviation D[X] of random variable X . 7. Find the minimum value of function Z1 f (A, B, C) = (x3 − (Ax2 + Bx + C))2 dx. −1 8. Solve the PDE (nd u(x, t)) ( utt (x, t) = a2 uxx (x, t) + x2 t, −∞ < x < +∞, 0 6 t < +∞; √ u(x, 0) = x, ut (x, 0) = sin x, −∞ < x < +∞. Answers: 1. x ∈ (−∞; −2] ∪ (0; +∞) √ 3. 2. 4. y= x+1 . x−1 5. No. 1 A = , M [X] = −2, D [X] = 3. 7. 8/175. (A = C = 0, B = 3/5). 6 √ √ x − at + x + at sin x sin at x2 t3 a2 t5 8. u(x, t) = + + + . 2 a 6 60 6. c CMC SMBU, 2023.