Варианты ИДЗ
.pdfМинистерство образования Российской федерации Тольяттинский Государственный Университет
Кафедра «Высшая математика и математическое моделирование»
ВЫСШАЯ МАТЕМАТИКА часть 1
Варианты ИДЗ для студентов, обучающихся
по технологии 30/70
Содержание |
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Модуль №1. Линейная алгебра........................................................................................................................................ |
5 |
Вариант 1...................................................................................................................................................................... |
5 |
Вариант 2...................................................................................................................................................................... |
6 |
Вариант 3...................................................................................................................................................................... |
7 |
Вариант 4...................................................................................................................................................................... |
8 |
Вариант 5.................................................................................................................................................................... |
10 |
Вариант 6.................................................................................................................................................................... |
11 |
Вариант 7.................................................................................................................................................................... |
12 |
Вариант 8.................................................................................................................................................................... |
14 |
Вариант 9.................................................................................................................................................................... |
15 |
Вариант 10.................................................................................................................................................................. |
16 |
Вариант 11.................................................................................................................................................................. |
18 |
Вариант 12.................................................................................................................................................................. |
19 |
Вариант 13.................................................................................................................................................................. |
20 |
Вариант 14.................................................................................................................................................................. |
22 |
Вариант 15.................................................................................................................................................................. |
23 |
Вариант 16.................................................................................................................................................................. |
24 |
Вариант 17.................................................................................................................................................................. |
26 |
Вариант 18.................................................................................................................................................................. |
27 |
Вариант 19.................................................................................................................................................................. |
28 |
Вариант 20.................................................................................................................................................................. |
30 |
Вариант 21.................................................................................................................................................................. |
31 |
Вариант 22.................................................................................................................................................................. |
32 |
Вариант 23.................................................................................................................................................................. |
34 |
Вариант 24.................................................................................................................................................................. |
35 |
Вариант 25.................................................................................................................................................................. |
36 |
Модуль №2. Векторная алгебра..................................................................................................................................... |
39 |
Вариант 1.................................................................................................................................................................... |
39 |
Вариант 2.................................................................................................................................................................... |
39 |
Вариант 3.................................................................................................................................................................... |
40 |
Вариант 4.................................................................................................................................................................... |
41 |
Вариант 5.................................................................................................................................................................... |
42 |
Вариант 6.................................................................................................................................................................... |
43 |
Вариант 7.................................................................................................................................................................... |
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Вариант 8.................................................................................................................................................................... |
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Вариант 9.................................................................................................................................................................... |
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Вариант 10.................................................................................................................................................................. |
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Вариант 11.................................................................................................................................................................. |
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Вариант 12.................................................................................................................................................................. |
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Вариант 13.................................................................................................................................................................. |
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Вариант 14.................................................................................................................................................................. |
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Вариант 15.................................................................................................................................................................. |
51 |
Вариант 16.................................................................................................................................................................. |
52 |
Вариант 17.................................................................................................................................................................. |
52 |
Вариант 18.................................................................................................................................................................. |
53 |
Вариант 19.................................................................................................................................................................. |
54 |
Вариант 20.................................................................................................................................................................. |
56 |
Вариант 21.................................................................................................................................................................. |
57 |
2
Вариант 22.................................................................................................................................................................. |
58 |
Вариант 23.................................................................................................................................................................. |
59 |
Вариант 24.................................................................................................................................................................. |
60 |
Вариант 25.................................................................................................................................................................. |
61 |
Модуль №3. Аналитическая геометрия........................................................................................................................ |
63 |
Вариант 1.................................................................................................................................................................... |
63 |
Вариант 2.................................................................................................................................................................... |
65 |
Вариант 3.................................................................................................................................................................... |
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Вариант 4.................................................................................................................................................................... |
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Вариант 5.................................................................................................................................................................... |
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Вариант 6.................................................................................................................................................................... |
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Вариант 7.................................................................................................................................................................... |
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Вариант 8.................................................................................................................................................................... |
78 |
Вариант 9.................................................................................................................................................................... |
81 |
Вариант 10.................................................................................................................................................................. |
83 |
Вариант 11.................................................................................................................................................................. |
86 |
Вариант 12.................................................................................................................................................................. |
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Вариант 13.................................................................................................................................................................. |
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Вариант 14.................................................................................................................................................................. |
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Вариант 15.................................................................................................................................................................. |
96 |
Вариант 16.................................................................................................................................................................. |
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Вариант 17................................................................................................................................................................ |
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Вариант 18................................................................................................................................................................ |
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Вариант 19................................................................................................................................................................ |
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Вариант 20................................................................................................................................................................ |
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Вариант 21................................................................................................................................................................ |
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Вариант 22................................................................................................................................................................ |
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Вариант 23................................................................................................................................................................ |
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Вариант 24................................................................................................................................................................ |
118 |
Вариант 25................................................................................................................................................................ |
120 |
Модуль №4. Введение в анализ................................................................................................................................... |
123 |
Вариант 1.................................................................................................................................................................. |
123 |
Вариант 2.................................................................................................................................................................. |
124 |
Вариант 3.................................................................................................................................................................. |
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Вариант 4.................................................................................................................................................................. |
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Вариант 5.................................................................................................................................................................. |
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Вариант 6.................................................................................................................................................................. |
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Вариант 7.................................................................................................................................................................. |
134 |
Вариант 8.................................................................................................................................................................. |
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Вариант 9.................................................................................................................................................................. |
137 |
Вариант 10................................................................................................................................................................ |
139 |
Вариант 11................................................................................................................................................................ |
141 |
Вариант 12................................................................................................................................................................ |
143 |
Вариант 13................................................................................................................................................................ |
145 |
Вариант 14................................................................................................................................................................ |
147 |
Вариант 15................................................................................................................................................................ |
149 |
Вариант 16................................................................................................................................................................ |
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Вариант 17................................................................................................................................................................ |
153 |
Вариант 18................................................................................................................................................................ |
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Вариант 19................................................................................................................................................................ |
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Вариант 20................................................................................................................................................................ |
158 |
3
Вариант 21................................................................................................................................................................ |
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Вариант 22................................................................................................................................................................ |
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Вариант 23................................................................................................................................................................ |
164 |
Вариант 24................................................................................................................................................................ |
166 |
Вариант 25................................................................................................................................................................ |
168 |
4
Модуль №1. Линейная алгебра
Вариант 1
1. Найти матрицу 3A + 2B − 4C , если |
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0 1 −1 |
−2 1 0 |
;C |
1 2 3 |
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A = |
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2 1 −4 |
−3 0 2 |
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4 5 6 |
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2. Найти значение матричного многочлена A2 + 2A + 3E , если |
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1 3 1 |
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0 1 0 |
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A = |
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3. Найти произведение матриц AB и BA , если |
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3 4 0 |
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−3 0 1 |
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A = |
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4. Даны два линейных преобразования. Средствами матричного исчисления найти преобразование, выражающее X1'' , X2'' , X3'' через X1 , X 2 , X3 :
X1'
X2'X3'
=5X1 − X2 + 3X3
=X1 − 2X2
=7 X2 − X3
X |
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= 2X ' |
+ X ' |
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X |
2'' = X2' − 5X3' |
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X |
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= 2X1 |
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5. Вычислить определители
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7 |
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6. Доказать совместность системы и решить ее двумя способами: методом Крамера и средствами матричного исчисления:
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2X1 − X2 − 6X3 = −1 |
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− 2X2 − 4X3 = 5 |
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X1 |
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− X2 + 2X3 = −8 |
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X1 |
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7. Найти ранг матриц |
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2 −1 3 −2 |
4 |
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3 |
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−1 |
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= 2 |
−1 −3 4 . |
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A = |
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4 −2 5 1 |
7 ; B |
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2 −1 1 8 |
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7 |
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−1 |
7 |
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8. Решить системы уравнений |
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2X1 + 5X2 + X3 + 3X4 = 0 |
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2X1 + X2 = 5 |
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+ 6X2 + 3X3 + 5X4 = 0 |
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+ 3X3 |
= 16 |
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4X1 |
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X1 |
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+ 14X2 + X3 + 7 X4 = 0 |
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− X3 |
= 10 |
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4X1 |
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5X2 |
Вариант 2 |
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Найти матрицу 5A − 6B + 2C , если |
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A = 2 7 8 2 |
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= 1 0 0 2 |
;C = 4 5 1 2 |
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3 1 1 3 |
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2 1 8 9 |
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2 1 3 2 |
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2. |
Найти значение матричного многочлена 2A2 + 3A + 6E , если |
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4 |
8 |
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2 3 |
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0 |
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A = |
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3. |
Найти произведение матриц AB и BA , если |
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6 |
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8 |
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1 2 |
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2 1 |
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A = |
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4. |
Даны два линейных преобразования. Средствами матричного исчисления найти |
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преобразование, выражающее X1'' , X2'' , X3'' |
через X1, X2 , X3 : |
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X |
1' = X 1 + 2 X 2 + 2 X 3 |
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2' |
= − 3 X 2 + X 3 |
X |
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= 2 X 1 + 3 X 3 |
X |
3 |
5. Вычислить определители
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2 |
8 |
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8 |
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4 |
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X 1''X 2''X 3''
=3 X 1' + X 2'
=X 1' − 2 X 2' − X 3' .
=3 X 1' + 2 X 3'
6. Доказать совместность системы и решить ее двумя способами: методом Крамера и средствами матричного исчисления:
2X1 − 2X2 + X3 = 6 |
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+ 6X2 + 3X3 = 3 . |
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X1 |
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+ 3X2 + X3 = 0 |
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2X1 |
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7. Найти ранг матриц |
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−1 |
2 |
3 −2 4 |
2 −1 −3 4 |
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A = |
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−2 |
4 |
5 1 |
7 |
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; B = 1 3 5 |
−1 . |
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−1 |
2 |
1 8 |
2 |
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7 7 9 1 |
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8. Решить системы уравнений
3X1 + 2X |
2 + 5X3 + 4X4 |
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2 + 6X3 + 8X4 |
2X1 + 3X |
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− 9X3 − 20X4 |
X1 − 6X2 |
= 0 |
3X1 − 2X2 = 2 |
= 0 |
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X1 + 4X3 = 10 . |
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= 0 |
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6X2 − 3X3 = 6 |
Вариант 3 |
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1. Найти матрицу 5A + 6B − 7C , если |
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2 |
3 |
1 2 |
1 2 |
9 11 |
0 |
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A = |
5 |
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;C = |
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1 |
2 |
. |
4 |
6 7 |
2 4 |
2 4 |
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2. Найти значение матричного многочлена 3А2 +4A+7E, если
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7 |
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0 |
0 |
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7 |
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0 |
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0 |
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A = |
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3 |
7 |
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3. Найти произведение матриц AB и BA , если
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7 |
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5 |
3 |
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7 |
9 |
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A = |
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11 . |
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8 |
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0 |
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4. Даны два линейных преобразования. Средствами матричного исчисления найти преобразование, выражающее X1'' , X2'' , X3'' через X1, X2 , X3 :
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X 1' |
= 5 X 1 − X 2 + 3 X |
3 |
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X 1'' |
= 2 X 1' + X 3' |
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X 2' |
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= X 1 − 2 X 2 |
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X 2'' |
= X 2' |
− 5 X 3' |
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= 7 X 2 − X 3 |
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= 2 X |
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X 3 |
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X 3 |
1 |
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5. Вычислить определители |
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3 |
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3 7 |
9 |
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6. Доказать совместность системы и решить ее двумя способами: методом Крамера и средствами матричного исчисления:
2 X |
1 − 3 X 2 + 3 X |
3 |
= 3 |
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1 + 9 X 2 − 2 X 3 |
= 4 |
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6 X |
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1 0 X 1 + 3 X 2 − 3 X 3 = 3 |
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7. Найти ранг матриц |
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3 −1 |
2 |
−2 4 |
5 |
1 |
−1 |
7 |
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2 |
−1 −3 4 |
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A = |
5 −2 |
4 |
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7 |
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1 |
3 |
5 |
−1 |
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2 |
8 |
2 |
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1 −1 |
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7 |
9 |
1 |
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8. Решить системы уравнений
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7 X1 − 3X2 + 7 X3 + 17 X4 = 0 |
5X1 − 2X2 |
= 3 |
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− 6X2 |
− X3 − 5X4 = 0 |
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+ 4X3 = 11 |
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8X1 |
7 X2 |
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− 2X2 |
+ 3X3 + 7 X4 = 0 |
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+ 5X3 |
= 7 |
4X1 |
2X1 |
Вариант 4 |
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1. |
Найти матрицу 2A + 6B − 5C , если |
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2 4 2 2 |
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0 1 2 3 |
6 6 7 8 |
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A = |
5 3 1 1 |
, B = |
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,C = |
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1 0 3 2 |
8 7 6 5 |
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2. |
Найти значение матричного многочлена 5A2 + 3A + 8E , если |
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3 |
8 |
1 |
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0 |
0 |
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1 |
8 3 |
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0 1 0 |
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A = |
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= |
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8 |
3 |
1 |
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0 |
0 |
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3. |
Найти произведение матриц AB и BA , если |
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2 1 7 |
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A = |
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4. |
Даны два линейных преобразования. Средствами матричного исчисления найти |
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преобразование, выражающее X1'' , X2'' , X3'' |
через X1, X2 , X3 : |
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= 4X |
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+ 3X |
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+ 5X |
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2' = 6X1 + 7 X2 + X3 |
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= 9X1 + X2 |
+ 8X3 |
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5. Вычислить определители
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X1''X2''X3''
=4X1' + 5X2' − 3X3'
=X1' − X2' − X3'
=7 X1' + 4X2'
6. Доказать совместность системы и решить ее двумя способами: методом Крамера и средствами матричного исчисления:
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X1 + X2 − 6X3 = 6 |
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− X2 − 6X3 = 2 |
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3X1 |
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+ 3X2 + 9X3 = 6 |
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2X1 |
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7. Найти ранг матриц |
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−2 −1 3 2 4 |
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−3 4 |
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A = |
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4 7 |
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8. Решить системы уравнений |
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7 X1 |
− 3X2 + 7 X3 + 17 X4 = 0 |
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2X1 + 2X2 = 4 |
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− 6X2 − X3 − 5X4 = 0 |
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8X1 |
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3X1 − 2X3 = 1 |
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− 2X2 + 3X3 + 7 X4 = 0 |
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+ 2X2 + 3X3 = 6 |
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4X1 |
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Вариант 5 |
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1. Найти матрицу 5A + 7B − 8C , если |
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A = |
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0 2 4 6 |
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−1 4 3 6 |
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5 6 5 6 |
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, B = |
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6 5 6 5 |
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1 3 5 7 |
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2 0 −5 8 |
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2. Найти значение матричного многочлена 2A2 + 3A − 7E , если |
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A = |
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4 2 1 |
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0 1 0 |
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3. Найти произведение матриц AB и BA , если |
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1 |
0 |
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A = |
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8 1 0 |
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3 2 4 |
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4. Даны два линейных преобразования. Средствами матричного исчисления найти преобразование, выражающее X1'' , X2'' , X3'' через X1, X2 , X3 :
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