Chapter 5 Complex Numbers and Quadratic Equations (सम्मिश्र संख्याएँ और द्विघातीय समीकरण).

प्रश्नावली 5.1

प्रश्न 1 से 10 तक की सम्मिश्र संख्याओं में प्रत्येक को a + ib के रूप में व्यक्त कीजिए।

प्रश्न 1.
(5i) ( \frac { -3 }{ 5 } i)
हल:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 1

प्रश्न 2.
i9 + i19
हल:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 2

प्रश्न 3.
i-39
हल:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 3

प्रश्न 4.
3(7 + i7) + i (7 + i7)
हल:
3(7 + i7) + i (7 + i7)
= 21 + 21i + 7i + 7i²
= 21 + 28i + 7(-1) [∵ i² = -1]
= 21 – 7 + 28i
= 14 + 28j

प्रश्न 5.
(1 – i) – (-1 + i6)
हल:
(1 – i) – (-1 + i6)
= (1 – i) + (1 – 6i)
= 1 – i + 1 – 6i
= 2 – 7i

UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 6
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 7

प्रश्न 8.
(1 – i)4
हल:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 8

प्रश्न 9.
( \frac { 1 }{ 3 } + 3i)3
हल:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 9
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 9.1

प्रश्न 10.
(-2 –  \frac { 1 }{ 3 } i)3
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 10

प्रश्न 11 से 13 तक की सम्मिश्र संख्याओं में प्रत्येक का गुणात्मक प्रतिलोम ज्ञात कीजिए।
प्रश्न 11.
4 – 3i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 11

प्रश्न 12.
√5 + 3i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 12

प्रश्न 13.
-i.
हल:
-1 का गुणात्मक प्रतिलोम
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 13

प्रश्न 14.
निम्नलिखित व्यंजक को a + ib के रूप में व्यक्त कीजिए:
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1 14

प्रश्नावली 5.2

प्रश्न 1 से 2 तक सम्मिश्र संख्याओं में प्रत्येक का मापांक और कोणांक ज्ञात कीजिए:
प्रश्न 1.
z = -1 – i√3
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 1.1

प्रश्न 2.
-√3 + i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 2

प्रश्न 3 से 8 तक सम्मिश्र संख्याओं में प्रत्येक को ध्रुवीय रूप में रूपांतरित कीजिए:
प्रश्न 3.
1 – i
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 3
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 3.1

प्रश्न 4.
-1 + i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 4

प्रश्न 5.
-1 – i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 5
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 5.1

प्रश्न 6.
-3.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 6

प्रश्न 7.
√3 + i
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 7
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 7.1

प्रश्न 8.
i.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.2 8

प्रश्नावली 5.3

निम्नलिखित समीकरणों में से प्रत्येक को हल कीजिए:
प्रश्न 1.
x² + 3 = 0.
हल:
x² + 3 = 0 या x² = -3 या x = ± √-3 = ± √3 i.

प्रश्न 2.
2x² + x + 1 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 2
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 2.1

प्रश्न 3.
x² + 3x + 9 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 3

प्रश्न 4.
-x² + x – 2 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 4

प्रश्न 5.
x² + 3x + 5 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 5
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 5.1

प्रश्न 6.
x² – x + 2 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 6

प्रश्न 7.
√2 x² + x + √2 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 7

प्रश्न 8.
√3 x² – √2 x + 3√3 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 8

प्रश्न 9.
x² + x +  \frac { 1 }{ \surd 2 } = 0
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 9

प्रश्न 10.
x² +  \frac { x }{ \surd 2 } + 1 = 0
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.3 10

अध्याय 5 पर विविध प्रश्नावली

UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 1.1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 2
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 3
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 4
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 4.1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 5.1

प्रश्न 6 से 9 में दिए गए प्रत्येक समीकरण को हल कीजिए:

प्रश्न 6.
3x² – 4x +  \frac { 20 }{ 3 } = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 6
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 6.1

प्रत 7.
x² – 2x +  \frac { 3 }{ 2 } = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 7

प्रश्न 8.
21x² – 28x + 10 = 0.
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 8

UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 9
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 10
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 11
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 11.1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 12
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 12.1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 12.2
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 13
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 13.1
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 13.2
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 14
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 15
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 16
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 17
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 18
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 19
UP Board Solutions for Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations 20

Chapter 5 Complex Numbers and Quadratic Equations (सम्मिश्र संख्याएँ और द्विघातीय समीकरण).