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Class 11 Mathematics

Chapter 4 Solutions — Complex Numbers and Quadratic Equations

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Overview

Step-by-step NCERT solutions for Complex Numbers and Quadratic Equations (Chapter 4, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Complex Numbers and Quadratic Equations textbook chapter.

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What these solutions cover

All 28 questions in Complex Numbers and Quadratic Equations are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 4.1

  1. Express the complex number (5i)(−3/5 i) in the form a + ib.
  2. Express i^9 + i^19 in the form a + ib.
  3. Express i^(−39) in the form a + ib.
  4. Express 3(7 + i7) + i(7 + i7) in the form a + ib.
  5. Express (1 − i) − (−1 + i6) in the form a + ib.
  6. Express (1/5 + i 2/5)(−4 + i 5/2) in the form a + ib.
  7. Express {(1/3 + i√3) + (4 + i/3)} − (−4/3 + i) in the form a + ib.
  8. Express (1 − i)^4 in the form a + ib.
  9. Express (1/3 + 3i)^3 in the form a + ib.
  10. Express (−2 − i/3)^3 in the form a + ib.
  11. Find the multiplicative inverse of 4 − 3i.
  12. Find the multiplicative inverse of √5 + 3i.
  13. Find the multiplicative inverse of −i.
  14. Express the following expression in the form a + ib: (3 + i√5)(3 − i√5) / [(√3 + √2 i) − (√3 − i√2)]

Miscellaneous Exercise

  1. Evaluate: [i^18 + (1/i)^25]^3.
  2. For any two complex numbers z_1 and z_2, prove that Re(z_1 z_2) = Re z_1 Re z_2 − Im z_1 Im z_2.
  3. Reduce [1/(1 − 4i) − 2/(1 + i)] × [(3 − 4i)/(5 + i)] to the standard form.
  4. If x − iy = √[(a − ib)/(c − id)], prove that (x² + y²)² = (a² + b²)/(c² + d²).
  5. If z_1 = 2 − i, z_2 = 1 + i, find |(z_1 + z_2 + 1)/(z_1 − z_2 + 1)|.
  6. If a + ib = (x + i)²/(2x² + 1), prove that a² + b² = (x² + 1)²/(2x² + 1)².
  7. Let z_1 = 2 − i, z_2 = −2 + i. Find
    • (i) Re(z_1 z_2 / z̄_1),
    • (ii) Im(1/(z_1 z̄_1)).
  8. Find the real numbers x and y if (x − iy)(3 + 5i) is the conjugate of −6 − 24i.
  9. Find the modulus of (1 + i)/(1 − i) − (1 − i)/(1 + i).
  10. If (x + iy)^3 = u + iv, then show that u/x + v/y = 4(x² − y²).
  11. If α and β are different complex numbers with |β| = 1, then find |(β − α)/(1 − ᾱβ)|.
  12. Find the number of non-zero integral solutions of the equation |1 − i|^x = 2^x.
  13. If (a + ib)(c + id)(e + if)(g + ih) = A + iB, then show that (a² + b²)(c² + d²)(e² + f²)(g² + h²) = A² + B².
  14. If [(1 + i)/(1 − i)]^m = 1, then find the least positive integral value of m.
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More NCERT Solutions for Class 11

Read the Complex Numbers and Quadratic Equations textbook chapter / PDF, or browse all NCERT Class 11 Mathematics solutions.

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