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Ch 4Complex Numbers and Quadratic Equationsसम्मिश्र संख्याएँ और द्विघातीय समीकरणComplex Numbers aur Quadratic Equations

The number i, complex numbers a + ib, equality, addition, subtraction, multiplication, division, powers of i, square roots of negative numbers, modulus, conjugate and the Argand plane — with every NCERT exercise solved.संख्या i, सम्मिश्र संख्याएँ a + ib, समानता, योग, अंतर, गुणा, भाग, i की घातें, ऋणात्मक संख्याओं के वर्गमूल, मापांक, संयुग्मी और आर्गंड तल — NCERT के सभी प्रश्नों के हल के साथ।Number i, complex numbers a + ib, equality, jod, ghatana, guna, bhaag, i ki powers, negative numbers ke square root, modulus, conjugate aur Argand plane — NCERT ke saare questions ke hal ke saath.

Didn’t understand a paragraph? Tap it (or its अ button) → हिंदी / Hinglish / Simple English.कोई पैराग्राफ़ समझ नहीं आया? उस पर (या अ बटन पर) टैप करें → English / Hinglish।Koi paragraph samajh nahi aaya? Us par (ya अ button par) tap kijiye → हिंदी / English.

👆 Words with a open a diagram or graph — tap any label to read what it means.👆 वाले शब्द पर टैप करें — चित्र या ग्राफ़ खुलेगा (हर लेबल पर टैप = उसका अर्थ)।👆 wale word par tap kijiye — diagram ya graph khulega (har label par tap = uska matlab).
📘 NCERT chapter (English PDF)📗 NCERT हिंदी PDF∑ Formulas✎ NCERT solutions🎯 Quiz

Diagrams in this chapterइस अध्याय के चित्रIs chapter ke diagrams

Powers of i repeat in a cycle of 4i की घातें 4 के चक्र में दोहराती हैंPowers of i repeat in a cycle of 4Section 4.3.5Argand plane: complex numbers as pointsआर्गंड तल: सम्मिश्र संख्याएँ बिंदुओं के रूप मेंArgand plane: complex numbers as pointsFig 4.1Modulus and conjugate in the Argand planeआर्गंड तल में मापांक और संयुग्मीModulus and conjugate in the Argand planeFig 4.2–4.3
In this chapter (NCERT sections)इस अध्याय में (NCERT खंड)Is chapter mein (NCERT sections)
  1. 4.2Complex Numbersसम्मिश्र संख्याएँComplex Numbers
  2. 4.3Algebra of Complex Numbersसम्मिश्र संख्याओं का बीजगणितComplex numbers ka algebra
  3. 4.4The Modulus and the Conjugateमापांक और संयुग्मीModulus aur Conjugate
  4. 4.5Argand Planeआर्गंड तलArgand Plane
  5. ★Yaad rakho — key formulasयाद रखो — मुख्य सूत्रYaad rakho — key formulas
  6. ⚠Common mistakesआम गलतियाँCommon mistakes
  7. ✎NCERT exercises — solutionsNCERT प्रश्नावली — हलNCERT exercises — solutions
  8. 🎯Quizक्विज़Quiz

4.2Complex Numbersसम्मिश्र संख्याएँComplex Numbers

The equation x^2 + 1 = 0 has no real solution, because the square of a real number is never negative. So we bring in a new symbol i with i^2 = -1, i.e. i = \sqrt{-1}.समीकरण x^2 + 1 = 0 का कोई वास्तविक हल नहीं है, क्योंकि किसी वास्तविक संख्या का वर्ग ऋणात्मक नहीं होता। इसलिए एक नया प्रतीक i लेते हैं जिसमें i^2 = -1, अर्थात् i = \sqrt{-1}।Equation x^2 + 1 = 0 ka koi real hal nahi hai, kyunki kisi real number ka square negative nahi hota. Isliye ek naya symbol i lete hain jismein i^2 = -1, yaani i = \sqrt{-1}.

Complex numberA number z = a + ib with a, b real. a = \text{Re}\, z is the real part, b = \text{Im}\, z the imaginary part. Example: 2 + i3 has Re = 2, Im = 3; -\frac{1}{5} + i has Re = -\frac{1}{5}, Im = 1.सम्मिश्र संख्यासंख्या z = a + ib, जहाँ a, b वास्तविक हैं। a = \text{Re}\, z वास्तविक भाग, b = \text{Im}\, z काल्पनिक भाग है। उदाहरण: 2 + i3 में Re = 2, Im = 3।Complex numberNumber z = a + ib, jahan a, b real hain. a = \text{Re}\, z real part, b = \text{Im}\, z imaginary part hai. Example: 2 + i3 mein Re = 2, Im = 3.

Two complex numbers are equal when real parts are equal AND imaginary parts are equal: a + ib = c + id \iff a = c,\ b = d.दो सम्मिश्र संख्याएँ समान होती हैं जब वास्तविक भाग बराबर हों और काल्पनिक भाग भी बराबर हों: a + ib = c + id \iff a = c,\ b = d।Do complex numbers barabar hote hain jab real parts barabar hon AUR imaginary parts bhi barabar hon: a + ib = c + id \iff a = c,\ b = d.

Example 1If 4x + i(3x - y) = 3 + i(-6) where x, y are real, find x and y.उदाहरण 1यदि 4x + i(3x - y) = 3 + i(-6), जहाँ x, y वास्तविक हैं, तो x और y ज्ञात कीजिए।Example 1Agar 4x + i(3x - y) = 3 + i(-6), jahan x, y real hain, to x aur y nikaliye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. Equate real parts and imaginary parts.वास्तविक भागों और काल्पनिक भागों को बराबर कीजिए।Real parts aur imaginary parts ko barabar kijiye.
    4x = 3 \Rightarrow x = \frac{3}{4};\quad 3x - y = -6 \Rightarrow y = \frac{9}{4} + 6 = \frac{33}{4}

4.3Algebra of Complex Numbersसम्मिश्र संख्याओं का बीजगणितComplex numbers ka algebra

Let z_1 = a + ib and z_2 = c + id.मान लीजिए z_1 = a + ib और z_2 = c + id।Maan lijiye z_1 = a + ib aur z_2 = c + id.

z_1 + z_2 = (a + c) + i(b + d)
Addition: add real parts, add imaginary partsयोग: वास्तविक भाग जोड़िए, काल्पनिक भाग जोड़िएJod: real parts jodiye, imaginary parts jodiye
z_1 - z_2 = (a - c) + i(b - d)
Subtractionव्यवकलन (अंतर)Ghatana
z_1 z_2 = (ac - bd) + i(ad + bc)
Multiplication: expand and use i^2 = -1गुणा: गुणा करके i^2 = -1 रखिएGuna: multiply karke i^2 = -1 rakhiye

Addition and multiplication obey the closure, commutative and associative laws, and multiplication distributes over addition. The additive identity is 0 = 0 + i0, the additive inverse of a + ib is -a - ib, the multiplicative identity is 1 = 1 + i0.योग और गुणा संवृत, क्रमविनिमेय और साहचर्य नियमों का पालन करते हैं, और गुणा योग पर वितरित होता है। योज्य तत्समक 0 = 0 + i0, a + ib का योज्य प्रतिलोम -a - ib, और गुणनात्मक तत्समक 1 = 1 + i0 है।Jod aur guna closure, commutative aur associative laws follow karte hain, aur guna jod par distribute hota hai. Additive identity 0 = 0 + i0, a + ib ka additive inverse -a - ib, multiplicative identity 1 = 1 + i0 hai.

Multiplicative inverse (reciprocal)For z = a + ib \ne 0: \frac{1}{z} = \frac{a}{a^2 + b^2} + i\frac{-b}{a^2 + b^2}. Division: \frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2}.गुणनात्मक प्रतिलोम (व्युत्क्रम)z = a + ib \ne 0 के लिए: \frac{1}{z} = \frac{a}{a^2 + b^2} + i\frac{-b}{a^2 + b^2}। भाग: \frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2}।Multiplicative inverse (reciprocal)z = a + ib \ne 0 ke liye: \frac{1}{z} = \frac{a}{a^2 + b^2} + i\frac{-b}{a^2 + b^2}. Bhaag: \frac{z_1}{z_2} = z_1 \cdot \frac{1}{z_2}.

4.3.5Powers of ii की घातेंi ki powers

Powers of i repeat in a cycle of 4i की घातें 4 के चक्र में दोहराती हैंPowers of i repeat in a cycle of 4Section 4.3.5 · 5 tappable partsअनुभाग 4.3.5 · 5 टैप करने लायक हिस्सेSection 4.3.5 · 5 tappable partsOpen

i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1, then the pattern repeats. So i^{4k} = 1,\ i^{4k+1} = i,\ i^{4k+2} = -1,\ i^{4k+3} = -i. Divide the power by 4 and look only at the remainder. Also i^{-1} = \frac{1}{i} = -i.i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1, फिर यही क्रम दोहराता है। अतः i^{4k} = 1,\ i^{4k+1} = i,\ i^{4k+2} = -1,\ i^{4k+3} = -i। घात को 4 से भाग देकर केवल शेषफल देखिए। साथ ही i^{-1} = \frac{1}{i} = -i।i^1 = i,\ i^2 = -1,\ i^3 = -i,\ i^4 = 1, phir yahi pattern repeat hota hai. Isliye i^{4k} = 1,\ i^{4k+1} = i,\ i^{4k+2} = -1,\ i^{4k+3} = -i. Power ko 4 se divide karke sirf remainder dekhiye. Saath hi i^{-1} = \frac{1}{i} = -i.

4.3.6Square roots of a negative real numberऋणात्मक वास्तविक संख्या के वर्गमूलNegative real number ke square root

For a > 0: \sqrt{-a} = i\sqrt{a}, and both \pm i\sqrt{a} satisfy x^2 = -a. Careful: \sqrt{a}\sqrt{b} = \sqrt{ab} holds only if at least one of a, b is \ge 0. If both are negative it fails: \sqrt{-1}\sqrt{-1} = i^2 = -1, not \sqrt{1} = 1.a > 0 के लिए: \sqrt{-a} = i\sqrt{a}, और \pm i\sqrt{a} दोनों x^2 = -a को संतुष्ट करते हैं। सावधान: \sqrt{a}\sqrt{b} = \sqrt{ab} तभी सत्य है जब a, b में से कम से कम एक \ge 0 हो। दोनों ऋणात्मक हों तो गलत: \sqrt{-1}\sqrt{-1} = i^2 = -1, न कि \sqrt{1} = 1।a > 0 ke liye: \sqrt{-a} = i\sqrt{a}, aur \pm i\sqrt{a} dono x^2 = -a satisfy karte hain. Dhyaan: \sqrt{a}\sqrt{b} = \sqrt{ab} tabhi sahi hai jab a, b mein se kam se kam ek \ge 0 ho. Dono negative hon to galat: \sqrt{-1}\sqrt{-1} = i^2 = -1, na ki \sqrt{1} = 1.

4.3.7Identitiesसर्वसमिकाएँIdentities

All the usual algebraic identities hold for complex numbers: (z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2, (z_1 - z_2)^3 = z_1^3 - 3z_1^2z_2 + 3z_1z_2^2 - z_2^3, z_1^2 - z_2^2 = (z_1 + z_2)(z_1 - z_2).सभी सामान्य बीजीय सर्वसमिकाएँ सम्मिश्र संख्याओं पर लागू होती हैं: (z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2, (z_1 - z_2)^3 = z_1^3 - 3z_1^2z_2 + 3z_1z_2^2 - z_2^3, z_1^2 - z_2^2 = (z_1 + z_2)(z_1 - z_2)।Saari normal algebraic identities complex numbers par bhi lagti hain: (z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2, (z_1 - z_2)^3 = z_1^3 - 3z_1^2z_2 + 3z_1z_2^2 - z_2^3, z_1^2 - z_2^2 = (z_1 + z_2)(z_1 - z_2).

Example 3Express (5 - 3i)^3 in the form a + ib.उदाहरण 3(5 - 3i)^3 को a + ib के रूप में व्यक्त कीजिए।Example 3(5 - 3i)^3 ko a + ib ke roop mein likhiye.
Solution — step by step (2)हल — चरण दर चरण (2)Hal — step by step (2)
  1. (5 - 3i)^3 = 5^3 - 3 \cdot 5^2 \cdot 3i + 3 \cdot 5 \cdot (3i)^2 - (3i)^3
  2. = 125 - 225i - 135 + 27i = -10 - 198i
Example 4Express (-\sqrt{3} + \sqrt{-2})(2\sqrt{3} - i) in the form a + ib.उदाहरण 4(-\sqrt{3} + \sqrt{-2})(2\sqrt{3} - i) को a + ib के रूप में व्यक्त कीजिए।Example 4(-\sqrt{3} + \sqrt{-2})(2\sqrt{3} - i) ko a + ib ke roop mein likhiye.
Solution — step by step (2)हल — चरण दर चरण (2)Hal — step by step (2)
  1. \sqrt{-2} = \sqrt{2}\,i \Rightarrow (-\sqrt{3} + \sqrt{2}\,i)(2\sqrt{3} - i) = -6 + \sqrt{3}\,i + 2\sqrt{6}\,i - \sqrt{2}\,i^2
  2. = (-6 + \sqrt{2}) + \sqrt{3}(1 + 2\sqrt{2})\,i

4.4The Modulus and the Conjugateमापांक और संयुग्मीModulus aur Conjugate

Modulus and conjugateFor z = a + ib: modulus |z| = \sqrt{a^2 + b^2} (a non-negative real number) and conjugate \bar{z} = a - ib. Example: |3 + i| = \sqrt{10}, \overline{3 + i} = 3 - i.मापांक और संयुग्मीz = a + ib के लिए: मापांक |z| = \sqrt{a^2 + b^2} (एक अऋणात्मक वास्तविक संख्या) और संयुग्मी \bar{z} = a - ib। उदाहरण: |3 + i| = \sqrt{10}, \overline{3 + i} = 3 - i।Modulus aur conjugatez = a + ib ke liye: modulus |z| = \sqrt{a^2 + b^2} (non-negative real number) aur conjugate \bar{z} = a - ib. Example: |3 + i| = \sqrt{10}, \overline{3 + i} = 3 - i.
z\,\bar{z} = |z|^2
so z^{-1} = \frac{\bar{z}}{|z|^2}अतः z^{-1} = \frac{\bar{z}}{|z|^2}isliye z^{-1} = \frac{\bar{z}}{|z|^2}
|z_1 z_2| = |z_1|\,|z_2|
and \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}और \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}aur \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}
\overline{z_1 z_2} = \bar{z}_1\,\bar{z}_2
also \overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2 and \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}साथ ही \overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2 और \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}saath hi \overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2 aur \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}
Example 5Find the multiplicative inverse of 2 - 3i.उदाहरण 52 - 3i का गुणनात्मक प्रतिलोम ज्ञात कीजिए।Example 52 - 3i ka multiplicative inverse nikaliye.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. \bar{z} = 2 + 3i,\ |z|^2 = 4 + 9 = 13 \Rightarrow z^{-1} = \frac{2 + 3i}{13} = \frac{2}{13} + \frac{3}{13}i
Example 6Express in the form a + ib: (i) \frac{5 + \sqrt{2}\,i}{1 - \sqrt{2}\,i} (ii) i^{-35}.उदाहरण 6a + ib के रूप में व्यक्त कीजिए: (i) \frac{5 + \sqrt{2}\,i}{1 - \sqrt{2}\,i} (ii) i^{-35}।Example 6a + ib ke roop mein likhiye: (i) \frac{5 + \sqrt{2}\,i}{1 - \sqrt{2}\,i} (ii) i^{-35}.
Solution — step by step (2)हल — चरण दर चरण (2)Hal — step by step (2)
  1. (i) Multiply top and bottom by the conjugate of the denominator.(i) अंश और हर दोनों को हर के संयुग्मी से गुणा कीजिए।(i) Numerator aur denominator dono ko denominator ke conjugate se guna kijiye.
    \frac{(5 + \sqrt{2}i)(1 + \sqrt{2}i)}{1 + 2} = \frac{5 + 5\sqrt{2}i + \sqrt{2}i - 2}{3} = \frac{3 + 6\sqrt{2}i}{3} = 1 + 2\sqrt{2}\,i
  2. (ii)\ i^{-35} = \frac{1}{i^{35}} = \frac{1}{i^{32} \cdot i^3} = \frac{1}{-i} = \frac{i}{-i^2} = i = 0 + i

4.5Argand Planeआर्गंड तलArgand Plane

The complex number x + iy is shown as the point P(x, y) in a plane — the Argand plane (complex plane). The x-axis is the real axis, the y-axis the imaginary axis. Tap the points in the Argand plane picture.सम्मिश्र संख्या x + iy को तल में बिंदु P(x, y) से दिखाते हैं — यह आर्गंड तल (सम्मिश्र तल) है। x-अक्ष वास्तविक अक्ष और y-अक्ष काल्पनिक अक्ष है। आर्गंड तल चित्र में बिंदुओं को छुइए।Complex number x + iy ko plane mein point P(x, y) se dikhate hain — ye Argand plane (complex plane) hai. x-axis real axis aur y-axis imaginary axis hai. Argand plane picture mein points ko tap kijiye.

Argand plane: complex numbers as pointsआर्गंड तल: सम्मिश्र संख्याएँ बिंदुओं के रूप मेंArgand plane: complex numbers as pointsFig 4.1 · 8 tappable partsचित्र 4.1 · 8 टैप करने लायक हिस्सेFig 4.1 · 8 tappable partsOpen

|x + iy| = \sqrt{x^2 + y^2} is the distance of P from the origin. The conjugate x - iy is the point Q(x, -y), the mirror image of P in the real axis.|x + iy| = \sqrt{x^2 + y^2}, मूलबिंदु से P की दूरी है। संयुग्मी x - iy बिंदु Q(x, -y) है, जो वास्तविक अक्ष में P का दर्पण प्रतिबिंब है।|x + iy| = \sqrt{x^2 + y^2}, origin se P ki doori hai. Conjugate x - iy point Q(x, -y) hai, jo real axis mein P ka mirror image hai.

Modulus and conjugate in the Argand planeआर्गंड तल में मापांक और संयुग्मीModulus and conjugate in the Argand planeFig 4.2–4.3 · 4 tappable partsचित्र 4.2–4.3 · 4 टैप करने लायक हिस्सेFig 4.2–4.3 · 4 tappable partsOpen
Example 7Find the conjugate of \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}.उदाहरण 7\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} का संयुग्मी ज्ञात कीजिए।Example 7\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} ka conjugate nikaliye.
Solution — step by step (2)हल — चरण दर चरण (2)Hal — step by step (2)
  1. \frac{6 + 9i - 4i + 6}{2 - i + 4i + 2} = \frac{12 + 5i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i}
  2. = \frac{48 - 36i + 20i + 15}{25} = \frac{63 - 16i}{25}

✔ Conjugate = \frac{63}{25} + \frac{16}{25}iसंयुग्मी = \frac{63}{25} + \frac{16}{25}iConjugate = \frac{63}{25} + \frac{16}{25}i

Example 8If x + iy = \frac{a + ib}{a - ib}, prove that x^2 + y^2 = 1.उदाहरण 8यदि x + iy = \frac{a + ib}{a - ib}, तो सिद्ध कीजिए कि x^2 + y^2 = 1।Example 8Agar x + iy = \frac{a + ib}{a - ib}, to prove kijiye ki x^2 + y^2 = 1.
Solution — step by step (1)हल — चरण दर चरण (1)Hal — step by step (1)
  1. Quick way: take modulus on both sides.आसान तरीका: दोनों ओर मापांक लीजिए।Aasan tareeka: dono taraf modulus lijiye.
    x^2 + y^2 = |x + iy|^2 = \frac{|a + ib|^2}{|a - ib|^2} = \frac{a^2 + b^2}{a^2 + b^2} = 1
Rationalised bookIn the current NCERT book this chapter ends at the Argand plane — polar form and the "Quadratic Equations" section are no longer in it (the chapter title still says "and Quadratic Equations"). There is one exercise (4.1) plus the Miscellaneous Exercise.संशोधित पुस्तकवर्तमान NCERT पुस्तक में यह अध्याय आर्गंड तल पर समाप्त होता है — ध्रुवीय रूप और "द्विघातीय समीकरण" खंड अब इसमें नहीं हैं (अध्याय के शीर्षक में अब भी "द्विघातीय समीकरण" लिखा है)। एक प्रश्नावली (4.1) और विविध प्रश्नावली है।Rationalised bookCurrent NCERT book mein ye chapter Argand plane par khatam hota hai — polar form aur "Quadratic Equations" section ab isme nahi hain (title mein ab bhi "Quadratic Equations" likha hai). Ek exercise (4.1) aur Miscellaneous Exercise hai.

★ Yaad rakho — key formulas★ याद रखो — मुख्य सूत्र★ Yaad rakho — key formulas

  1. i^2 = -1; i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i.i^2 = -1; i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i।i^2 = -1; i^{4k} = 1, i^{4k+1} = i, i^{4k+2} = -1, i^{4k+3} = -i.
  2. (a + ib)(c + id) = (ac - bd) + i(ad + bc).(a + ib)(c + id) = (ac - bd) + i(ad + bc)।(a + ib)(c + id) = (ac - bd) + i(ad + bc).
  3. |a + ib| = \sqrt{a^2 + b^2}, \overline{a + ib} = a - ib, z\bar{z} = |z|^2.|a + ib| = \sqrt{a^2 + b^2}, \overline{a + ib} = a - ib, z\bar{z} = |z|^2।|a + ib| = \sqrt{a^2 + b^2}, \overline{a + ib} = a - ib, z\bar{z} = |z|^2.
  4. \frac{1}{z} = \frac{\bar{z}}{|z|^2} — to divide, multiply by the conjugate of the denominator.\frac{1}{z} = \frac{\bar{z}}{|z|^2} — भाग देने के लिए हर के संयुग्मी से गुणा कीजिए।\frac{1}{z} = \frac{\bar{z}}{|z|^2} — divide karne ke liye denominator ke conjugate se guna kijiye.
  5. |z_1 z_2| = |z_1||z_2|, \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} — often the fastest route in proofs.|z_1 z_2| = |z_1||z_2|, \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} — प्रमाणों में अक्सर सबसे तेज़ तरीका।|z_1 z_2| = |z_1||z_2|, \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} — proofs mein aksar sabse fast tareeka.

⚠ Common mistakes⚠ आम गलतियाँ⚠ Common mistakes

  • \sqrt{-4}\sqrt{-9} = \sqrt{36} = 6 — wrong. Write 2i \cdot 3i = 6i^2 = -6.\sqrt{-4}\sqrt{-9} = \sqrt{36} = 6 — गलत। 2i \cdot 3i = 6i^2 = -6 लिखिए।\sqrt{-4}\sqrt{-9} = \sqrt{36} = 6 — galat. 2i \cdot 3i = 6i^2 = -6 likhiye.
  • Saying Im(3 - 4i) = -4i. The imaginary part is the real number -4.Im(3 - 4i) = -4i कहना। काल्पनिक भाग वास्तविक संख्या -4 है।Im(3 - 4i) = -4i kehna. Imaginary part real number -4 hai.
  • |a + ib| = a + b or \sqrt{a^2 - b^2} — both wrong; it is \sqrt{a^2 + b^2}.|a + ib| = a + b या \sqrt{a^2 - b^2} — दोनों गलत; यह \sqrt{a^2 + b^2} है।|a + ib| = a + b ya \sqrt{a^2 - b^2} — dono galat; ye \sqrt{a^2 + b^2} hai.
  • Forgetting that i^2 = -1 changes the sign when expanding (a + ib)(c + id): the real part is ac - bd, not ac + bd.(a + ib)(c + id) खोलते समय i^2 = -1 से चिह्न बदलना भूल जाना: वास्तविक भाग ac - bd है, ac + bd नहीं।(a + ib)(c + id) kholte waqt i^2 = -1 se sign badalna bhool jaana: real part ac - bd hai, ac + bd nahi.
  • Using the remainder wrongly for negative powers: i^{-1} = -i, not i.ऋणात्मक घातों में गलती: i^{-1} = -i, न कि i।Negative powers mein galti: i^{-1} = -i, na ki i.

✎ NCERT exercises — solutions✎ NCERT प्रश्नावली — हल✎ NCERT exercises — solutions

All 28 questions of the NCERT exercises. Tap a question to see the solution — try it yourself first!NCERT प्रश्नावली के सभी 28 प्रश्न। हल देखने के लिए प्रश्न पर टैप करें — पहले स्वयं हल करके देखें!NCERT exercises ke saare 28 questions. Hal dekhne ke liye question par tap kijiye — pehle khud try kijiye!

Exercise 4.1प्रश्नावली 4.1Exercise 4.1 · 14 Q

Exercises 11–13: find the multiplicative inverse.प्रश्न 11–13: गुणनात्मक प्रतिलोम ज्ञात कीजिए।Exercises 11–13: find the multiplicative inverse.

Q1. (5i)\left(-\frac{3}{5}i\right)
  1. = -3i^2 = 3 = 3 + i0
Q2. i^9 + i^{19}
  1. i^9 = i^{8} \cdot i = i,\ i^{19} = i^{16} \cdot i^3 = -i \Rightarrow i - i = 0 = 0 + i0
Q3. i^{-39}
  1. i^{-39} = \frac{1}{i^{39}} = \frac{1}{i^{36} \cdot i^3} = \frac{1}{-i} = \frac{i}{-i^2} = i = 0 + i1
Q4. 3(7 + 7i) + i(7 + 7i)
  1. 21 + 21i + 7i + 7i^2 = 21 - 7 + 28i = 14 + 28i
Q5. (1 - i) - (-1 + 6i)
  1. 1 - i + 1 - 6i = 2 - 7i
Q6. \left(\frac{1}{5} + \frac{2}{5}i\right) - \left(4 + \frac{5}{2}i\right)
  1. \left(\frac{1}{5} - 4\right) + \left(\frac{2}{5} - \frac{5}{2}\right)i = -\frac{19}{5} - \frac{21}{10}i
Q7. \left[\left(\frac{1}{3} + \frac{7}{3}i\right) + \left(4 + \frac{1}{3}i\right)\right] - \left(-\frac{4}{3} + i\right)
  1. \text{Re} = \frac{1}{3} + 4 + \frac{4}{3} = \frac{17}{3},\quad \text{Im} = \frac{7}{3} + \frac{1}{3} - 1 = \frac{5}{3}

✔ \frac{17}{3} + \frac{5}{3}i

Q8. (1 - i)^4
  1. (1 - i)^2 = 1 - 2i + i^2 = -2i \Rightarrow (1 - i)^4 = (-2i)^2 = 4i^2 = -4 = -4 + i0
Q9. \left(\frac{1}{3} + 3i\right)^3
  1. = \frac{1}{27} + 3 \cdot \frac{1}{9} \cdot 3i + 3 \cdot \frac{1}{3} \cdot 9i^2 + 27i^3
  2. = \frac{1}{27} + i - 9 - 27i = -\frac{242}{27} - 26i
Q10. \left(-2 - \frac{1}{3}i\right)^3
  1. = -\left(2 + \frac{i}{3}\right)^3 = -\left[8 + 3 \cdot 4 \cdot \frac{i}{3} + 3 \cdot 2 \cdot \frac{i^2}{9} + \frac{i^3}{27}\right]
  2. = -\left[8 + 4i - \frac{2}{3} - \frac{i}{27}\right] = -\frac{22}{3} - \frac{107}{27}i
Q11. 4 - 3i
  1. \frac{\bar{z}}{|z|^2} = \frac{4 + 3i}{16 + 9} = \frac{4}{25} + \frac{3}{25}i
Q12. \sqrt{5} + 3i
  1. \frac{\sqrt{5} - 3i}{5 + 9} = \frac{\sqrt{5}}{14} - \frac{3}{14}i
Q13. -i
  1. \frac{1}{-i} = \frac{i}{-i^2} = i = 0 + i1
Q14. Express \frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}\,i) - (\sqrt{3} - i\sqrt{2})} in the form a + ib.\frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}\,i) - (\sqrt{3} - i\sqrt{2})} को a + ib के रूप में व्यक्त कीजिए।Express \frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}\,i) - (\sqrt{3} - i\sqrt{2})} in the form a + ib.
  1. \text{Num} = 9 - 5i^2 = 14,\quad \text{Den} = 2\sqrt{2}\,i\text{अंश} = 9 - 5i^2 = 14,\quad \text{हर} = 2\sqrt{2}\,i\text{Numerator} = 9 - 5i^2 = 14,\quad \text{Denominator} = 2\sqrt{2}\,i
  2. \frac{14}{2\sqrt{2}\,i} = \frac{7}{\sqrt{2}\,i} \cdot \frac{-i}{-i} = \frac{-7i}{\sqrt{2}} = 0 - \frac{7\sqrt{2}}{2}i

Miscellaneous Exercise on Chapter 4अध्याय 4 पर विविध प्रश्नावलीMiscellaneous Exercise on Chapter 4 · 14 Q

Q1. Evaluate \left[i^{18} + \left(\frac{1}{i}\right)^{25}\right]^3.मान ज्ञात कीजिए: \left[i^{18} + \left(\frac{1}{i}\right)^{25}\right]^3।Evaluate \left[i^{18} + \left(\frac{1}{i}\right)^{25}\right]^3.
  1. i^{18} = i^{16} i^2 = -1,\quad \left(\frac{1}{i}\right)^{25} = (-i)^{25} = -i^{25} = -i
  2. (-1 - i)^3 = -(1 + i)^3 = -(1 + 3i + 3i^2 + i^3) = -(-2 + 2i) = 2 - 2i
Q2. For any complex z_1, z_2 prove Re(z_1z_2) = Re z_1 Re z_2 − Im z_1 Im z_2.किन्हीं सम्मिश्र z_1, z_2 के लिए सिद्ध कीजिए Re(z_1z_2) = Re z_1 Re z_2 − Im z_1 Im z_2।For any complex z_1, z_2 prove Re(z_1z_2) = Re z_1 Re z_2 − Im z_1 Im z_2.
  1. z_1 = x_1 + iy_1,\ z_2 = x_2 + iy_2 \Rightarrow z_1z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
  2. \text{Re}(z_1z_2) = x_1x_2 - y_1y_2 = \text{Re}\,z_1\,\text{Re}\,z_2 - \text{Im}\,z_1\,\text{Im}\,z_2
Q3. Reduce \left(\frac{1}{1 - 4i} - \frac{2}{1 + i}\right)\left(\frac{3 - 4i}{5 + i}\right) to standard form.\left(\frac{1}{1 - 4i} - \frac{2}{1 + i}\right)\left(\frac{3 - 4i}{5 + i}\right) को मानक रूप में बदलिए।Reduce \left(\frac{1}{1 - 4i} - \frac{2}{1 + i}\right)\left(\frac{3 - 4i}{5 + i}\right) to standard form.
  1. \frac{1}{1 - 4i} = \frac{1 + 4i}{17},\ \frac{2}{1 + i} = 1 - i \Rightarrow \text{first bracket} = \frac{1 + 4i - 17 + 17i}{17} = \frac{-16 + 21i}{17}\frac{1}{1 - 4i} = \frac{1 + 4i}{17},\ \frac{2}{1 + i} = 1 - i \Rightarrow \text{पहला कोष्ठक} = \frac{1 + 4i - 17 + 17i}{17} = \frac{-16 + 21i}{17}\frac{1}{1 - 4i} = \frac{1 + 4i}{17},\ \frac{2}{1 + i} = 1 - i \Rightarrow \text{pehla bracket} = \frac{1 + 4i - 17 + 17i}{17} = \frac{-16 + 21i}{17}
  2. \frac{3 - 4i}{5 + i} = \frac{(3 - 4i)(5 - i)}{26} = \frac{11 - 23i}{26}
  3. \frac{(-16 + 21i)(11 - 23i)}{442} = \frac{-176 + 368i + 231i + 483}{442} = \frac{307 + 599i}{442}
Q4. If x - iy = \sqrt{\frac{a - ib}{c - id}}, prove (x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}.यदि x - iy = \sqrt{\frac{a - ib}{c - id}}, तो सिद्ध कीजिए (x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}।If x - iy = \sqrt{\frac{a - ib}{c - id}}, prove (x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}.
  1. (x - iy)^2 = \frac{a - ib}{c - id} \Rightarrow |x - iy|^4 = \frac{|a - ib|^2}{|c - id|^2}
  2. (x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}
Q5. If z_1 = 2 - i,\ z_2 = 1 + i, find \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}\right|.यदि z_1 = 2 - i,\ z_2 = 1 + i, तो \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}\right| ज्ञात कीजिए।If z_1 = 2 - i,\ z_2 = 1 + i, find \left|\frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}\right|.
  1. z_1 + z_2 + 1 = 4,\quad z_1 - z_2 + 1 = 2 - 2i
  2. \frac{|4|}{|2 - 2i|} = \frac{4}{2\sqrt{2}} = \sqrt{2}
Q6. If a + ib = \frac{(x + i)^2}{2x^2 + 1}, prove a^2 + b^2 = \frac{(x^2 + 1)^2}{(2x^2 + 1)^2}.यदि a + ib = \frac{(x + i)^2}{2x^2 + 1}, तो सिद्ध कीजिए a^2 + b^2 = \frac{(x^2 + 1)^2}{(2x^2 + 1)^2}।If a + ib = \frac{(x + i)^2}{2x^2 + 1}, prove a^2 + b^2 = \frac{(x^2 + 1)^2}{(2x^2 + 1)^2}.
  1. a^2 + b^2 = |a + ib|^2 = \frac{|x + i|^4}{(2x^2 + 1)^2} = \frac{(x^2 + 1)^2}{(2x^2 + 1)^2}
Q7. Let z_1 = 2 - i,\ z_2 = -2 + i. Find (i) Re\left(\frac{z_1z_2}{\bar{z}_1}\right) (ii) Im\left(\frac{1}{z_1\bar{z}_1}\right).मान लीजिए z_1 = 2 - i,\ z_2 = -2 + i। ज्ञात कीजिए (i) Re\left(\frac{z_1z_2}{\bar{z}_1}\right) (ii) Im\left(\frac{1}{z_1\bar{z}_1}\right)।Let z_1 = 2 - i,\ z_2 = -2 + i. Find (i) Re\left(\frac{z_1z_2}{\bar{z}_1}\right) (ii) Im\left(\frac{1}{z_1\bar{z}_1}\right).
  1. z_1z_2 = (2 - i)(-2 + i) = -4 + 2i + 2i - i^2 = -3 + 4i,\quad \bar{z}_1 = 2 + i
  2. \frac{-3 + 4i}{2 + i} = \frac{(-3 + 4i)(2 - i)}{5} = \frac{-6 + 3i + 8i + 4}{5} = \frac{-2 + 11i}{5} \Rightarrow \text{Re} = -\frac{2}{5}
  3. z_1\bar{z}_1 = |z_1|^2 = 5 \Rightarrow \frac{1}{5} \text{ is real} \Rightarrow \text{Im} = 0
Q8. Find real x, y if (x - iy)(3 + 5i) is the conjugate of -6 - 24i.वास्तविक x, y ज्ञात कीजिए यदि (x - iy)(3 + 5i), -6 - 24i का संयुग्मी है।Find real x, y if (x - iy)(3 + 5i) is the conjugate of -6 - 24i.
  1. (x - iy)(3 + 5i) = (3x + 5y) + i(5x - 3y) = -6 + 24i
  2. 3x + 5y = -6,\ 5x - 3y = 24 \Rightarrow 34x = 102 \Rightarrow x = 3,\ y = -3
Q9. Find the modulus of \frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}.\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i} का मापांक ज्ञात कीजिए।Find the modulus of \frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}.
  1. \frac{1 + i}{1 - i} = \frac{(1 + i)^2}{2} = i,\quad \frac{1 - i}{1 + i} = -i \Rightarrow i - (-i) = 2i,\ |2i| = 2
Q10. If (x + iy)^3 = u + iv, show that \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2).यदि (x + iy)^3 = u + iv, तो दर्शाइए \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2)।If (x + iy)^3 = u + iv, show that \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2).
  1. (x + iy)^3 = (x^3 - 3xy^2) + i(3x^2y - y^3) \Rightarrow u = x^3 - 3xy^2,\ v = 3x^2y - y^3
  2. \frac{u}{x} + \frac{v}{y} = (x^2 - 3y^2) + (3x^2 - y^2) = 4(x^2 - y^2)
Q11. If α, β are different complex numbers with |\beta| = 1, find \left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right|.यदि α, β भिन्न सम्मिश्र संख्याएँ हैं और |\beta| = 1, तो \left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right| ज्ञात कीजिए।If α, β are different complex numbers with |\beta| = 1, find \left|\frac{\beta - \alpha}{1 - \bar{\alpha}\beta}\right|.
  1. Use \bar{\beta}\beta = |\beta|^2 = 1 and factor \beta out of the denominator.\bar{\beta}\beta = |\beta|^2 = 1 का उपयोग कर हर में से \beta बाहर निकालिए।\bar{\beta}\beta = |\beta|^2 = 1 use karke denominator se \beta bahar nikaliye.
    1 - \bar{\alpha}\beta = \beta\bar{\beta} - \bar{\alpha}\beta = \beta\,\overline{(\beta - \alpha)} \Rightarrow |1 - \bar{\alpha}\beta| = |\beta|\,|\beta - \alpha| = |\beta - \alpha|

✔ The value is 1.मान 1 है।Value 1 hai.

Q12. Find the number of non-zero integral solutions of |1 - i|^x = 2^x.|1 - i|^x = 2^x के अशून्य पूर्णांकीय हलों की संख्या ज्ञात कीजिए।Find the number of non-zero integral solutions of |1 - i|^x = 2^x.
  1. (\sqrt{2})^x = 2^x \Rightarrow 2^{x/2} = 2^x \Rightarrow \frac{x}{2} = x \Rightarrow x = 0

✔ Only x = 0, so the number of non-zero integral solutions is 0.केवल x = 0, अतः अशून्य पूर्णांकीय हलों की संख्या 0 है।Sirf x = 0, isliye non-zero integral solutions ki sankhya 0 hai.

Q13. If (a + ib)(c + id)(e + if)(g + ih) = A + iB, show that (a^2 + b^2)(c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2.यदि (a + ib)(c + id)(e + if)(g + ih) = A + iB, तो दर्शाइए (a^2 + b^2)(c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2।If (a + ib)(c + id)(e + if)(g + ih) = A + iB, show that (a^2 + b^2)(c^2 + d^2)(e^2 + f^2)(g^2 + h^2) = A^2 + B^2.
  1. Take modulus of both sides and square, using |z_1z_2| = |z_1||z_2|.दोनों ओर मापांक लेकर वर्ग कीजिए, |z_1z_2| = |z_1||z_2| का उपयोग कर।Dono taraf modulus lekar square kijiye, |z_1z_2| = |z_1||z_2| use karke.
    |a + ib|^2|c + id|^2|e + if|^2|g + ih|^2 = |A + iB|^2
Q14. If \left(\frac{1 + i}{1 - i}\right)^m = 1, find the least positive integer m.यदि \left(\frac{1 + i}{1 - i}\right)^m = 1, तो m का न्यूनतम धन पूर्णांक मान ज्ञात कीजिए।If \left(\frac{1 + i}{1 - i}\right)^m = 1, find the least positive integer m.
  1. \frac{1 + i}{1 - i} = \frac{(1 + i)^2}{2} = \frac{2i}{2} = i \Rightarrow i^m = 1 \Rightarrow m = 4

🎯 Quiz — 12 questions🎯 क्विज़ — 12 प्रश्न🎯 Quiz — 12 questions

Tap an option. The answer and a short reason appear at once.किसी विकल्प पर टैप करें। उत्तर और छोटा कारण तुरंत दिखेगा।Kisi option par tap kijiye. Answer aur chhota reason turant dikhega.

1. i⁵⁰ equalsi⁵⁰ बराबर हैi⁵⁰ barabar hai

2. The imaginary part of 3 − 4i is3 − 4i का काल्पनिक भाग है3 − 4i ka imaginary part hai

3. (2 + 3i)(2 − 3i) =

4. |3 − 4i| =

5. The multiplicative inverse of i isi का गुणनात्मक प्रतिलोम हैi ka multiplicative inverse hai

6. √(−9) × √(−4) =

7. The conjugate of (1 + i)² is(1 + i)² का संयुग्मी है(1 + i)² ka conjugate hai

8. If x + iy = (3 + 2i)(1 − i) then x isयदि x + iy = (3 + 2i)(1 − i) तो x हैAgar x + iy = (3 + 2i)(1 − i) to x hai

9. (1 + i)/(1 − i) equals(1 + i)/(1 − i) बराबर है(1 + i)/(1 − i) barabar hai

10. In the Argand plane the conjugate of z isआर्गंड तल में z का संयुग्मी हैArgand plane mein z ka conjugate hai

11. If 4x + i(3x − y) = 3 − 6i (x, y real), then y =यदि 4x + i(3x − y) = 3 − 6i (x, y वास्तविक), तो y =Agar 4x + i(3x − y) = 3 − 6i (x, y real), to y =

12. |z₁z₂| equals|z₁z₂| बराबर है|z₁z₂| barabar hai

← Ch 3Ch 5 →

Found a mistake?कोई गलती दिखी?Koi galti dikhi? Report a mistakeगलती बताइएGalti bataiye — we fix confirmed errors quickly.— पक्की गलतियाँ हम जल्दी ठीक करते हैं।— confirm galtiyan hum jaldi theek karte hain.

These are Simple Learning Hub's own study notes and solutions, written to follow the NCERT Class 11 Mathematics textbook (Reprint 2026-27) section by section; read them with the official NCERT chapter. Diagrams are original SLH drawings.ये Simple Learning Hub के अपने स्टडी नोट्स हैं, जो NCERT कक्षा 11 गणित की किताब (2026-27) के हर अनुभाग के साथ लिखे गए हैं; इन्हें NCERT के आधिकारिक अध्याय के साथ पढ़िए। डायग्राम SLH के अपने बनाए चित्र हैं।Ye Simple Learning Hub ke apne study notes hain, jo NCERT Class 11 Maths ki kitaab (2026-27) ke har section ke saath likhe gaye hain; inhe NCERT ke official chapter ke saath padhiye. Diagrams SLH ki apni drawings hain.