حلول الأسئلة

السؤال

في كل معادلة مما يأتي أوجد قيمتي y،x الحقيقيتين اللتين تجعلانها صحيحة:

الحل

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction end cell row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction cross times fraction numerator 2 minus i square root of 3 over denominator 2 minus i square root of 3 end fraction end cell row blank cell fraction numerator 4 minus 2 i square root of 3 minus 2 i square root of 3 plus 3 i squared over denominator 4 minus 2 i square root of 3 plus 2 i square root of 3 minus 3 i squared end fraction end cell row blank cell fraction numerator 4 minus 4 i square root of 3 plus 3 i squared over denominator 4 minus 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 4 minus 4 i square root of 3 minus 3 over denominator 4 plus 3 end fraction end cell row blank cell fraction numerator 1 minus 4 i square root of 3 over denominator 7 end fraction end cell end table

شاهد حلول جميع الاسئلة

تدرب وحل المسائل

تدرب وحب المسائل

أوجد ناتج كل مما يأتي:

18)

11 i equals square root of negative 121 end root

19)

13 i equals square root of negative 169 end root

20)

10 i equals square root of negative 100 end root

21)

9 i equals square root of negative 81 end root

22)

negative 42 i equals left parenthesis negative 3 i right parenthesis left parenthesis negative 7 i right parenthesis left parenthesis 2 i right parenthesis

23)

negative 144 i equals 4 i cross times 36 i squared equals 4 i left parenthesis negative 6 i right parenthesis squared

24)

negative bold italic i equals bold italic i to the power of 11

25)

i equals i to the power of 25

26)

square root of negative 10 end root. square root of negative 24 end root

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell i square root of 10.2 i square root of 6 end cell row blank cell 2 i squared square root of 60 equals 4 i squared square root of 15 end cell end table

27)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 4 i left parenthesis 1 half bold i right parenthesis squared left parenthesis negative 2 i right parenthesis squared end cell row blank cell 4 i left parenthesis 1 fourth i squared right parenthesis left parenthesis 4 i squared right parenthesis end cell row blank cell i cubed cross times 4 i squared end cell row blank cell 4 i to the power of 5 end cell end table

حل كل معادلة مما يأتي:

28)

table attributes columnspacing 1em end attributes row cell 4 x squared plus 4 equals 0 end cell row cell 4 x squared equals negative 4 end cell row cell x squared equals negative 1 end cell row cell x equals plus-or-minus i end cell end table

29)

table attributes columnspacing 1em end attributes row cell 3 x squared plus 48 equals 0 end cell row cell 3 x squared equals negative 48 end cell row cell x squared equals negative 16 end cell row cell x equals plus-or-minus 4 i end cell end table

30)

table attributes columnalign right columnspacing 1em end attributes row cell 2 x squared plus 10 equals 0 end cell row cell 2 x squared equals negative 10 end cell row cell x squared equals negative 5 end cell row cell x equals plus-or-minus i square root of 5 end cell end table

31)

table attributes columnspacing 1em end attributes row cell 6 x squared plus 108 equals 0 end cell row cell bold 6 x squared equals negative bold 108 end cell row cell x squared equals negative 18 end cell end table

x equals plus-or-minus 3 i square root of 2

في كل معادلة مما يأتي أوجد قيمتي y،x الحقيقيتين اللتين تجعلانها صحيحة:

32)

x plus 1 plus 2 y i equals 3 minus 6 i

table attributes columnalign right left right left columnspacing 1em end attributes row cell 2 y end cell cell equals negative 6 end cell cell space space space space x plus 1 end cell cell equals 3 end cell row y cell equals negative 3 end cell x cell equals 2 end cell end table

33)

2 x plus 7 plus left parenthesis 3 minus y right parenthesis i equals negative 4 plus 6 i

table attributes columnalign right right columnspacing 1em end attributes row cell 3 minus y equals 6 end cell cell space space space space 2 x plus 7 equals negative 4 end cell row cell y equals negative 3 end cell cell 2 x equals negative 11 end cell row x cell equals negative 11 over 2 end cell end table

34)

5 plus y plus left parenthesis 3 x minus 7 right parenthesis i equals 9 minus 3 i

table attributes columnalign right right columnspacing 1em end attributes row cell 3 x minus 7 equals negative 3 end cell cell space space 5 plus y equals 9 end cell row cell 3 x equals 4 end cell cell y equals 4 end cell row cell x equals 4 over 3 end cell blank end table

35)

left parenthesis 2 x minus 4 y right parenthesis i plus x plus 5 y equals 15 plus 58 i

2 x minus 4 y equals 58 space of 1em x plus 5 y equals 15

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 2 x minus 4 y end cell cell equals 58 end cell row cell 2 x plus 10 y end cell cell equals 30 end cell row cell negative 14 y end cell cell equals 28 end cell row y cell equals negative 2 end cell row cell x plus 5 left parenthesis negative 2 right parenthesis end cell cell equals 15 end cell row cell x minus 10 end cell cell equals 15 end cell row x cell equals 25 end cell end table

36)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis negative 3 plus i right parenthesis plus left parenthesis negative 4 minus i right parenthesis end cell row blank cell negative 3 plus i minus 4 minus i end cell row blank cell negative 7 end cell end table

37)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 11 minus 8 i right parenthesis minus left parenthesis 2 minus 8 i right parenthesis end cell row blank cell 11 minus 8 i minus 2 plus 8 i end cell end table

38)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 1 plus 2 i right parenthesis left parenthesis 1 minus 2 i right parenthesis end cell row blank cell 1 minus 2 i plus 2 i minus 4 i squared end cell row blank cell 1 minus 4 i squared space of 1em i squared equals negative 1 end cell row blank cell 1 minus 4 left parenthesis negative 1 right parenthesis end cell row blank cell 1 plus 4 end cell end table

39)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 3 plus 5 i right parenthesis left parenthesis 5 minus 3 i right parenthesis end cell row blank cell 15 minus 9 i plus 25 i minus 15 i squared end cell row blank cell 15 plus 16 i minus 15 i squared space of 1em i squared equals negative 1 end cell row blank cell 15 plus 16 i plus 15 end cell row blank cell 30 plus 16 i end cell end table

40)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 4 minus i right parenthesis left parenthesis 6 minus 6 i right parenthesis end cell row blank cell 24 minus 6 i minus 24 i plus 6 i squared end cell row blank cell 24 minus 30 i plus 6 i squared end cell row blank cell 24 minus 30 i minus 6 space of 1em i squared equals negative 1 end cell row blank cell 18 minus 30 i end cell end table

41)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 i over denominator 1 plus i end fraction end cell row blank cell fraction numerator 2 i over denominator 1 plus i end fraction cross times fraction numerator 1 minus i over denominator 1 minus i end fraction end cell row blank cell fraction numerator 2 i minus 2 i squared over denominator 1 minus i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 2 i plus 2 over denominator 1 plus 1 end fraction equals fraction numerator 2 i plus 2 over denominator 2 end fraction end cell row blank cell 1 plus i end cell end table

42)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 over denominator 2 plus 4 i end fraction end cell row blank cell fraction numerator 5 over denominator 2 plus 4 i end fraction cross times fraction numerator 2 minus 4 i over denominator 2 minus 4 i end fraction end cell row blank cell fraction numerator 10 minus 20 i over denominator 4 minus 16 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 10 minus 20 i over denominator 4 plus 16 end fraction end cell row blank cell fraction numerator 10 minus 20 i over denominator 20 end fraction end cell row blank cell 10 over 20 minus fraction numerator 20 i over denominator 20 end fraction end cell row blank cell 1 half minus i end cell end table

43)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 plus i over denominator 3 i end fraction end cell row blank cell fraction numerator 5 plus i over denominator 3 i end fraction cross times i over i end cell row blank cell fraction numerator 5 i plus i squared over denominator 3 i squared end fraction end cell row blank cell fraction numerator 5 i over denominator 3 i squared end fraction plus fraction numerator i squared over denominator 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell negative 5 over 3 i plus 1 third end cell end table

44)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 4 minus 6 i right parenthesis plus left parenthesis 4 plus 6 i right parenthesis end cell row blank cell 4 minus 6 i plus 4 plus 6 i end cell row blank cell 4 plus 4 end cell end table

45)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 8 minus 5 i right parenthesis minus left parenthesis 7 plus i right parenthesis end cell row blank cell 8 minus 5 i minus 7 minus i end cell row blank cell 1 minus 6 i end cell end table

46)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis negative 6 minus i right parenthesis left parenthesis 3 minus 3 i right parenthesis end cell row blank cell negative 18 plus 18 i minus 3 i plus 3 i squared end cell row blank cell negative 18 plus 15 i plus 3 i squared space of 1em i squared equals negative 1 end cell row blank cell negative 18 plus 15 i minus 3 end cell row blank cell negative 21 plus 15 i end cell end table

47)

table attributes columnalign left right columnspacing 1em end attributes row cell left parenthesis 1 plus i right parenthesis left parenthesis 2 plus 3 i right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell left parenthesis 2 plus 3 i plus 2 i plus 3 i squared right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell cell i squared equals negative 1 end cell row cell left parenthesis 2 plus 5 i minus 3 right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell left parenthesis negative 1 plus 5 i right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell negative 4 plus 3 i plus 20 i minus 15 i squared end cell cell i squared equals negative 1 end cell row cell negative 4 plus 23 i plus 15 end cell blank row cell 11 plus 23 i end cell blank end table

48)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 minus i square root of 2 over denominator 4 plus i square root of 2 end fraction end cell row blank cell fraction numerator 4 minus i square root of 2 over denominator 4 plus i square root of 2 end fraction cross times fraction numerator 4 minus i square root of 2 over denominator 4 minus i square root of 2 end fraction end cell row blank cell fraction numerator 16 minus 4 i square root of 2 minus 4 i square root of 2 plus 2 i squared over denominator 16 minus 4 i square root of 2 plus 4 i square root of 2 minus 2 i squared end fraction end cell row blank cell fraction numerator 16 minus 8 i square root of 2 plus 2 i squared over denominator 16 minus 2 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 16 minus 8 i square root of 2 minus 2 over denominator 16 plus 2 end fraction end cell row blank cell fraction numerator 14 minus 8 i square root of 2 over denominator 18 end fraction end cell end table

49)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction end cell row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction cross times fraction numerator 2 minus i square root of 3 over denominator 2 minus i square root of 3 end fraction end cell row blank cell fraction numerator 4 minus 2 i square root of 3 minus 2 i square root of 3 plus 3 i squared over denominator 4 minus 2 i square root of 3 plus 2 i square root of 3 minus 3 i squared end fraction end cell row blank cell fraction numerator 4 minus 4 i square root of 3 plus 3 i squared over denominator 4 minus 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 4 minus 4 i square root of 3 minus 3 over denominator 4 plus 3 end fraction end cell row blank cell fraction numerator 1 minus 4 i square root of 3 over denominator 7 end fraction end cell end table

50) كهرباء: تبلغ المعاوقة في أحد أجزاء دائرة كهربائية 8i+ 7 أوم وفي الجزء الآخر منها 4i - 13 أوم اجمع هذين العددين المركبين لإيجاد المعاوقة الكلية في الدائرة الكهربائية.

table attributes columnalign right columnspacing 1em end attributes row cell left parenthesis 13 minus 4 i right parenthesis plus left parenthesis 7 plus 8 i right parenthesis equals end cell row cell 20 plus 4 i equals 13 minus 4 i plus 7 plus 8 i equals end cell end table

كهرباء: استعمل الصيغة I · C = V حيث V فرق الجهد وC شدة التيار وI المعاوقة في حل السؤالين 52، 51:

51) إذا كانت شدة التيار في دائرة كهربائية 6i + 3 أمبير والمعاوقة i - 5 أوم فكم يكون فرق الجهد؟

table attributes columnalign right columnspacing 1em end attributes row cell left parenthesis 3 plus 6 i right parenthesis left parenthesis 5 minus i right parenthesis end cell row cell 15 minus 3 i plus 30 i minus 6 i squared equals end cell row cell 321 plus 27 i equals end cell end table فولت

52) إذا كان فرق الجهد في دائرة كهربائية 12i - 20 فولت والمعاوقة 4i - 6 أوم فكم تكون شدة التيار؟

table attributes columnalign right columnspacing 1em end attributes row cell fraction numerator left parenthesis 20 minus 12 i right parenthesis over denominator left parenthesis 6 minus 4 i right parenthesis end fraction end cell row cell fraction numerator 20 minus 12 i over denominator 6 minus 4 i end fraction times fraction numerator 6 plus 4 i over denominator 6 plus 4 i end fraction equals end cell row cell fraction numerator 120 plus 80 i minus 72 i minus 48 i squared over denominator 36 minus 16 i squared end fraction equals end cell row cell fraction numerator 168 plus 8 i over denominator 52 end fraction equals end cell row cell 42 over 13 plus 2 over 13 i equals end cell end table

53) أوجد ناتج جمع:

table attributes columnalign right right right columnspacing 1em end attributes row cell i x squared minus 4 x minus 5 i x plus 7 end cell not stretchy leftwards arrow cell i x squared minus left parenthesis 4 plus 5 i right parenthesis x plus 7 end cell row cell stack 3 x squared plus 2 x plus 6 i x minus 8 i with _ below end cell not stretchy leftwards arrow cell fraction numerator 3 x squared plus left parenthesis 2 plus 6 i right parenthesis x minus 8 i over denominator blank end fraction end cell row blank cell left parenthesis 3 plus i right parenthesis x squared minus left parenthesis 2 plus i right parenthesis x minus 8 i plus 7 equals end cell blank end table

54) بسط العبارة:

table attributes columnalign right columnspacing 1em end attributes row cell left square bracket left parenthesis 2 plus i right parenthesis x squared minus i x plus 5 plus i right square bracket minus left square bracket left parenthesis negative 3 plus 4 i right parenthesis x squared plus left parenthesis 5 minus 5 i right parenthesis x minus 6 right square bracket end cell row cell 2 x squared plus i x squared minus i x plus 5 plus i plus 3 x squared minus 4 i x squared minus 5 x plus 5 i x plus 6 equals end cell row cell left parenthesis 2 plus i plus 3 minus 4 i right parenthesis x squared plus left parenthesis negative i minus 5 plus 5 i right parenthesis x plus i plus 11 equals end cell row cell left parenthesis 5 minus 3 i right parenthesis x squared plus left parenthesis negative 5 plus 4 i right parenthesis x plus i plus 11 equals end cell end table

55) تمثيلات متعددة: ستكتشف في هذه المسألة جمع الأعداد المركبة في المستوى المركب فالمستوى المركب يشبه إلى حد بعيد المستوى الحقيقي وفيه تكون الأعداد الحقيقية على المحور الأفقي والأعداد التخيلية البحتة على المحور الرأسي.

a) بيايناً:

التمثيل البياني

b) بيايناً:

التمثيل البياني

c) بيايناً:

التمثيل البياني

d) تحليلياً:

1 minus i النقطة C تمثل ناتج جمع العددين المركبين الممثلين بالنقطة A ،B

مشاركة الدرس

السؤال

في كل معادلة مما يأتي أوجد قيمتي y،x الحقيقيتين اللتين تجعلانها صحيحة:

الحل

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction end cell row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction cross times fraction numerator 2 minus i square root of 3 over denominator 2 minus i square root of 3 end fraction end cell row blank cell fraction numerator 4 minus 2 i square root of 3 minus 2 i square root of 3 plus 3 i squared over denominator 4 minus 2 i square root of 3 plus 2 i square root of 3 minus 3 i squared end fraction end cell row blank cell fraction numerator 4 minus 4 i square root of 3 plus 3 i squared over denominator 4 minus 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 4 minus 4 i square root of 3 minus 3 over denominator 4 plus 3 end fraction end cell row blank cell fraction numerator 1 minus 4 i square root of 3 over denominator 7 end fraction end cell end table

تدرب وحل المسائل

تدرب وحب المسائل

أوجد ناتج كل مما يأتي:

18)

11 i equals square root of negative 121 end root

19)

13 i equals square root of negative 169 end root

20)

10 i equals square root of negative 100 end root

21)

9 i equals square root of negative 81 end root

22)

negative 42 i equals left parenthesis negative 3 i right parenthesis left parenthesis negative 7 i right parenthesis left parenthesis 2 i right parenthesis

23)

negative 144 i equals 4 i cross times 36 i squared equals 4 i left parenthesis negative 6 i right parenthesis squared

24)

negative bold italic i equals bold italic i to the power of 11

25)

i equals i to the power of 25

26)

square root of negative 10 end root. square root of negative 24 end root

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell i square root of 10.2 i square root of 6 end cell row blank cell 2 i squared square root of 60 equals 4 i squared square root of 15 end cell end table

27)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell 4 i left parenthesis 1 half bold i right parenthesis squared left parenthesis negative 2 i right parenthesis squared end cell row blank cell 4 i left parenthesis 1 fourth i squared right parenthesis left parenthesis 4 i squared right parenthesis end cell row blank cell i cubed cross times 4 i squared end cell row blank cell 4 i to the power of 5 end cell end table

حل كل معادلة مما يأتي:

28)

table attributes columnspacing 1em end attributes row cell 4 x squared plus 4 equals 0 end cell row cell 4 x squared equals negative 4 end cell row cell x squared equals negative 1 end cell row cell x equals plus-or-minus i end cell end table

29)

table attributes columnspacing 1em end attributes row cell 3 x squared plus 48 equals 0 end cell row cell 3 x squared equals negative 48 end cell row cell x squared equals negative 16 end cell row cell x equals plus-or-minus 4 i end cell end table

30)

table attributes columnalign right columnspacing 1em end attributes row cell 2 x squared plus 10 equals 0 end cell row cell 2 x squared equals negative 10 end cell row cell x squared equals negative 5 end cell row cell x equals plus-or-minus i square root of 5 end cell end table

31)

table attributes columnspacing 1em end attributes row cell 6 x squared plus 108 equals 0 end cell row cell bold 6 x squared equals negative bold 108 end cell row cell x squared equals negative 18 end cell end table

x equals plus-or-minus 3 i square root of 2

في كل معادلة مما يأتي أوجد قيمتي y،x الحقيقيتين اللتين تجعلانها صحيحة:

32)

x plus 1 plus 2 y i equals 3 minus 6 i

table attributes columnalign right left right left columnspacing 1em end attributes row cell 2 y end cell cell equals negative 6 end cell cell space space space space x plus 1 end cell cell equals 3 end cell row y cell equals negative 3 end cell x cell equals 2 end cell end table

33)

2 x plus 7 plus left parenthesis 3 minus y right parenthesis i equals negative 4 plus 6 i

table attributes columnalign right right columnspacing 1em end attributes row cell 3 minus y equals 6 end cell cell space space space space 2 x plus 7 equals negative 4 end cell row cell y equals negative 3 end cell cell 2 x equals negative 11 end cell row x cell equals negative 11 over 2 end cell end table

34)

5 plus y plus left parenthesis 3 x minus 7 right parenthesis i equals 9 minus 3 i

table attributes columnalign right right columnspacing 1em end attributes row cell 3 x minus 7 equals negative 3 end cell cell space space 5 plus y equals 9 end cell row cell 3 x equals 4 end cell cell y equals 4 end cell row cell x equals 4 over 3 end cell blank end table

35)

left parenthesis 2 x minus 4 y right parenthesis i plus x plus 5 y equals 15 plus 58 i

2 x minus 4 y equals 58 space of 1em x plus 5 y equals 15

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell 2 x minus 4 y end cell cell equals 58 end cell row cell 2 x plus 10 y end cell cell equals 30 end cell row cell negative 14 y end cell cell equals 28 end cell row y cell equals negative 2 end cell row cell x plus 5 left parenthesis negative 2 right parenthesis end cell cell equals 15 end cell row cell x minus 10 end cell cell equals 15 end cell row x cell equals 25 end cell end table

36)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis negative 3 plus i right parenthesis plus left parenthesis negative 4 minus i right parenthesis end cell row blank cell negative 3 plus i minus 4 minus i end cell row blank cell negative 7 end cell end table

37)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 11 minus 8 i right parenthesis minus left parenthesis 2 minus 8 i right parenthesis end cell row blank cell 11 minus 8 i minus 2 plus 8 i end cell end table

38)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 1 plus 2 i right parenthesis left parenthesis 1 minus 2 i right parenthesis end cell row blank cell 1 minus 2 i plus 2 i minus 4 i squared end cell row blank cell 1 minus 4 i squared space of 1em i squared equals negative 1 end cell row blank cell 1 minus 4 left parenthesis negative 1 right parenthesis end cell row blank cell 1 plus 4 end cell end table

39)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 3 plus 5 i right parenthesis left parenthesis 5 minus 3 i right parenthesis end cell row blank cell 15 minus 9 i plus 25 i minus 15 i squared end cell row blank cell 15 plus 16 i minus 15 i squared space of 1em i squared equals negative 1 end cell row blank cell 15 plus 16 i plus 15 end cell row blank cell 30 plus 16 i end cell end table

40)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 4 minus i right parenthesis left parenthesis 6 minus 6 i right parenthesis end cell row blank cell 24 minus 6 i minus 24 i plus 6 i squared end cell row blank cell 24 minus 30 i plus 6 i squared end cell row blank cell 24 minus 30 i minus 6 space of 1em i squared equals negative 1 end cell row blank cell 18 minus 30 i end cell end table

41)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 i over denominator 1 plus i end fraction end cell row blank cell fraction numerator 2 i over denominator 1 plus i end fraction cross times fraction numerator 1 minus i over denominator 1 minus i end fraction end cell row blank cell fraction numerator 2 i minus 2 i squared over denominator 1 minus i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 2 i plus 2 over denominator 1 plus 1 end fraction equals fraction numerator 2 i plus 2 over denominator 2 end fraction end cell row blank cell 1 plus i end cell end table

42)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 over denominator 2 plus 4 i end fraction end cell row blank cell fraction numerator 5 over denominator 2 plus 4 i end fraction cross times fraction numerator 2 minus 4 i over denominator 2 minus 4 i end fraction end cell row blank cell fraction numerator 10 minus 20 i over denominator 4 minus 16 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 10 minus 20 i over denominator 4 plus 16 end fraction end cell row blank cell fraction numerator 10 minus 20 i over denominator 20 end fraction end cell row blank cell 10 over 20 minus fraction numerator 20 i over denominator 20 end fraction end cell row blank cell 1 half minus i end cell end table

43)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 5 plus i over denominator 3 i end fraction end cell row blank cell fraction numerator 5 plus i over denominator 3 i end fraction cross times i over i end cell row blank cell fraction numerator 5 i plus i squared over denominator 3 i squared end fraction end cell row blank cell fraction numerator 5 i over denominator 3 i squared end fraction plus fraction numerator i squared over denominator 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell negative 5 over 3 i plus 1 third end cell end table

44)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 4 minus 6 i right parenthesis plus left parenthesis 4 plus 6 i right parenthesis end cell row blank cell 4 minus 6 i plus 4 plus 6 i end cell row blank cell 4 plus 4 end cell end table

45)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis 8 minus 5 i right parenthesis minus left parenthesis 7 plus i right parenthesis end cell row blank cell 8 minus 5 i minus 7 minus i end cell row blank cell 1 minus 6 i end cell end table

46)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell left parenthesis negative 6 minus i right parenthesis left parenthesis 3 minus 3 i right parenthesis end cell row blank cell negative 18 plus 18 i minus 3 i plus 3 i squared end cell row blank cell negative 18 plus 15 i plus 3 i squared space of 1em i squared equals negative 1 end cell row blank cell negative 18 plus 15 i minus 3 end cell row blank cell negative 21 plus 15 i end cell end table

47)

table attributes columnalign left right columnspacing 1em end attributes row cell left parenthesis 1 plus i right parenthesis left parenthesis 2 plus 3 i right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell left parenthesis 2 plus 3 i plus 2 i plus 3 i squared right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell cell i squared equals negative 1 end cell row cell left parenthesis 2 plus 5 i minus 3 right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell left parenthesis negative 1 plus 5 i right parenthesis left parenthesis 4 minus 3 i right parenthesis end cell blank row cell negative 4 plus 3 i plus 20 i minus 15 i squared end cell cell i squared equals negative 1 end cell row cell negative 4 plus 23 i plus 15 end cell blank row cell 11 plus 23 i end cell blank end table

48)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 4 minus i square root of 2 over denominator 4 plus i square root of 2 end fraction end cell row blank cell fraction numerator 4 minus i square root of 2 over denominator 4 plus i square root of 2 end fraction cross times fraction numerator 4 minus i square root of 2 over denominator 4 minus i square root of 2 end fraction end cell row blank cell fraction numerator 16 minus 4 i square root of 2 minus 4 i square root of 2 plus 2 i squared over denominator 16 minus 4 i square root of 2 plus 4 i square root of 2 minus 2 i squared end fraction end cell row blank cell fraction numerator 16 minus 8 i square root of 2 plus 2 i squared over denominator 16 minus 2 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 16 minus 8 i square root of 2 minus 2 over denominator 16 plus 2 end fraction end cell row blank cell fraction numerator 14 minus 8 i square root of 2 over denominator 18 end fraction end cell end table

49)

table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction end cell row blank cell fraction numerator 2 minus i square root of 3 over denominator 2 plus i square root of 3 end fraction cross times fraction numerator 2 minus i square root of 3 over denominator 2 minus i square root of 3 end fraction end cell row blank cell fraction numerator 4 minus 2 i square root of 3 minus 2 i square root of 3 plus 3 i squared over denominator 4 minus 2 i square root of 3 plus 2 i square root of 3 minus 3 i squared end fraction end cell row blank cell fraction numerator 4 minus 4 i square root of 3 plus 3 i squared over denominator 4 minus 3 i squared end fraction space of 1em i squared equals negative 1 end cell row blank cell fraction numerator 4 minus 4 i square root of 3 minus 3 over denominator 4 plus 3 end fraction end cell row blank cell fraction numerator 1 minus 4 i square root of 3 over denominator 7 end fraction end cell end table

50) كهرباء: تبلغ المعاوقة في أحد أجزاء دائرة كهربائية 8i+ 7 أوم وفي الجزء الآخر منها 4i - 13 أوم اجمع هذين العددين المركبين لإيجاد المعاوقة الكلية في الدائرة الكهربائية.

table attributes columnalign right columnspacing 1em end attributes row cell left parenthesis 13 minus 4 i right parenthesis plus left parenthesis 7 plus 8 i right parenthesis equals end cell row cell 20 plus 4 i equals 13 minus 4 i plus 7 plus 8 i equals end cell end table

كهرباء: استعمل الصيغة I · C = V حيث V فرق الجهد وC شدة التيار وI المعاوقة في حل السؤالين 52، 51:

51) إذا كانت شدة التيار في دائرة كهربائية 6i + 3 أمبير والمعاوقة i - 5 أوم فكم يكون فرق الجهد؟

table attributes columnalign right columnspacing 1em end attributes row cell left parenthesis 3 plus 6 i right parenthesis left parenthesis 5 minus i right parenthesis end cell row cell 15 minus 3 i plus 30 i minus 6 i squared equals end cell row cell 321 plus 27 i equals end cell end table فولت

52) إذا كان فرق الجهد في دائرة كهربائية 12i - 20 فولت والمعاوقة 4i - 6 أوم فكم تكون شدة التيار؟

table attributes columnalign right columnspacing 1em end attributes row cell fraction numerator left parenthesis 20 minus 12 i right parenthesis over denominator left parenthesis 6 minus 4 i right parenthesis end fraction end cell row cell fraction numerator 20 minus 12 i over denominator 6 minus 4 i end fraction times fraction numerator 6 plus 4 i over denominator 6 plus 4 i end fraction equals end cell row cell fraction numerator 120 plus 80 i minus 72 i minus 48 i squared over denominator 36 minus 16 i squared end fraction equals end cell row cell fraction numerator 168 plus 8 i over denominator 52 end fraction equals end cell row cell 42 over 13 plus 2 over 13 i equals end cell end table

53) أوجد ناتج جمع:

table attributes columnalign right right right columnspacing 1em end attributes row cell i x squared minus 4 x minus 5 i x plus 7 end cell not stretchy leftwards arrow cell i x squared minus left parenthesis 4 plus 5 i right parenthesis x plus 7 end cell row cell stack 3 x squared plus 2 x plus 6 i x minus 8 i with _ below end cell not stretchy leftwards arrow cell fraction numerator 3 x squared plus left parenthesis 2 plus 6 i right parenthesis x minus 8 i over denominator blank end fraction end cell row blank cell left parenthesis 3 plus i right parenthesis x squared minus left parenthesis 2 plus i right parenthesis x minus 8 i plus 7 equals end cell blank end table

54) بسط العبارة:

table attributes columnalign right columnspacing 1em end attributes row cell left square bracket left parenthesis 2 plus i right parenthesis x squared minus i x plus 5 plus i right square bracket minus left square bracket left parenthesis negative 3 plus 4 i right parenthesis x squared plus left parenthesis 5 minus 5 i right parenthesis x minus 6 right square bracket end cell row cell 2 x squared plus i x squared minus i x plus 5 plus i plus 3 x squared minus 4 i x squared minus 5 x plus 5 i x plus 6 equals end cell row cell left parenthesis 2 plus i plus 3 minus 4 i right parenthesis x squared plus left parenthesis negative i minus 5 plus 5 i right parenthesis x plus i plus 11 equals end cell row cell left parenthesis 5 minus 3 i right parenthesis x squared plus left parenthesis negative 5 plus 4 i right parenthesis x plus i plus 11 equals end cell end table

55) تمثيلات متعددة: ستكتشف في هذه المسألة جمع الأعداد المركبة في المستوى المركب فالمستوى المركب يشبه إلى حد بعيد المستوى الحقيقي وفيه تكون الأعداد الحقيقية على المحور الأفقي والأعداد التخيلية البحتة على المحور الرأسي.

a) بيايناً:

التمثيل البياني

b) بيايناً:

التمثيل البياني

c) بيايناً:

التمثيل البياني

d) تحليلياً:

1 minus i النقطة C تمثل ناتج جمع العددين المركبين الممثلين بالنقطة A ،B