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NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 4 Complex Numbers and Quadratic Equations

NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 4 Complex Numbers and Quadratic Equations help students revise and apply the important concepts of complex numbers and quadratic equations. This exercise includes a variety of questions that test the understanding of operations on complex numbers and methods for solving quadratic equations.

Prepared according to the latest CBSE Class 11 Maths syllabus, this exercise focuses on concepts such as real and imaginary parts of complex numbers, operations on complex numbers, modulus and conjugate of a complex number, and solving quadratic equations using different methods. Practicing these problems helps students strengthen their problem-solving skills and build confidence for examinations.

NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 4 Complex Numbers and Quadratic Equations

Complex Numbers and Quadratic Equations

Medium

Q.

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.

\begin{array}{l} 1 . (5 i) (- \frac{3}{5} i) \\ 2. i^{9} + i^{19} \\ 3. i^{- 39} \\ 4. 3 (7 + i 7) + i (7 + i 7) \\ 5. (1 - i) - (- 1 + i 6) \\ 6. (\frac{1}{5} + i \frac{2}{5}) - (4 + i \frac{5}{2}) \\ 7. [(\frac{1}{3} + i \frac{7}{3}) + (4 + i \frac{1}{3})] - (- \frac{4}{3} + i) \\ 8. {(1 - i)}^{4} \\ 9 . {(\frac{1}{3} + 3 i)}^{3} \\ 10 . {(- 2 - \frac{1}{3} i)}^{3} \end{array}

Complex Numbers and Quadratic Equations

Difficult

Q.

If a + ib = \frac{{(x + iy)}^{2}}{{2x}^{2} +1} {, prove that a}^{2} {+b}^{2} = \frac{{(x^{2} +1)}^{2}}{{({2x}^{2} +1)}^{2}} .

Complex Numbers and Quadratic Equations

Medium

Q.

For any two complex numbers, z₁ and z₂, prove that
Re (z₁ z₂) = Re z₁ Re z₂ – Imz₁ Imz₂.

Complex Numbers and Quadratic Equations

Medium

Q.

Reduce (\frac{1}{1 - 4i} - \frac{2}{1 + i}) (\frac{3 - 4i}{5 + i}) to the standard form.

Complex Numbers and Quadratic Equations

Difficult

Q.

If x - iy = \sqrt{\frac{a - ib}{c - id}} prove that {(x^{2} + y^{2})}^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}} .

Complex Numbers and Quadratic Equations

Medium

Q.

\begin{array}{l} Convert the following in the polar form: \\ (i) \frac{1+7i}{{(2 - i)}^{2}} \\ (ii) \frac{1+3i}{1 - 2i} \end{array}

Complex Numbers and Quadratic Equations

Medium

Q.

Solve each of the equation in Exercises 6 to 9.

\begin{array}{l} 6 . 3 x^{2} - 4 x + \frac{20}{3} = 0 \\ 7 . x^{2} - 2 x + \frac{3}{2} = 0 \\ 8. 27 x^{2} - 10 x + 1 = 0 \\ 9. 21 x^{2} - 28 x + 10 = 0 \end{array}

Complex Numbers and Quadratic Equations

Medium

Q.

If z_{1} = 2 - i, z_{2} = 1 + i, find | \frac{z_{1} + z_{2} + 1}{z_{1} - z_{2} + 1} | .

Complex Numbers and Quadratic Equations

Medium

Q.

\begin{array}{l} {Let z}_{1} {= 2 - i,  z}_{2} = -2 + i. Find \\ (i) Re (\frac{z_{1} z_{2}}{{\bar{z}}_{1}}), (ii) Im (\frac{1}{z_{1} {\bar{z}}_{1}}) \end{array}

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: 1 + i

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the modulus and argument of the complex number \frac{1+2i}{1-3i} .

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the real numbers x and y if
(x – iy) (3 + 5i) is the conjugate of –6 – 24i.

Complex Numbers and Quadratic Equations

Medium

Q.

Find the modulus of \frac{1 + i}{1 – i} - \frac{1 - i}{1 + i} .

Complex Numbers and Quadratic Equations

Medium

Q.

{If (x + iy)}^{3} = u + iv, then show that \frac{u}{x} + \frac{v}{y} = 4 (x^{2} {– y}^{2}) .

Complex Numbers and Quadratic Equations

Difficult

Q.

If α and β are different complex numbers with |β| =1, then find |\frac{β-α}{1- \bar{α} β}| .

Complex Numbers and Quadratic Equations

Medium

Q.

\begin{array}{l} Find the number of non-zero integral solutions of the \\ equation {|1 - i|}^{x} {= 2}^{x} . \end{array}

Complex Numbers and Quadratic Equations

Medium

Q.

If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that
(a² + b²) (c² + d²) (e² + f²) (g² + h²) = A² + B²

Complex Numbers and Quadratic Equations

Medium

Q.

Evaluate : {[i^{18} + {(\frac{1}{i})}^{25}]}^{3}

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: i

Complex Numbers and Quadratic Equations

Medium

Q.

Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.

11. 4 – 3i
12.

\sqrt{5} + 3 i

13. – i

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
x² + 3x + 5 = 0

Complex Numbers and Quadratic Equations

Medium

Q.

\begin{array}{l} Express the following expression in the form of a+ib: \\ \frac{(3+i \sqrt{5}) (3-i \sqrt{5})}{(\sqrt{3} + \sqrt{2} i) - (\sqrt{3} - \sqrt{2} i)} \end{array}

Complex Numbers and Quadratic Equations

Medium

Q.

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.

\begin{array}{l} 1 . z = - 1 - i \sqrt{3} \\ 2 . z = - \sqrt{3} + i \end{array}

Complex Numbers and Quadratic Equations

Medium

Q.

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:

3. 1 – i
4. – 1 + i
5. – 1 – i
6. – 3
7.

\sqrt{3} + i

8. i

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
x² + 3 = 0

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
2x² + x + 1 = 0

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
x² + 3x + 9 = 0

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
– x² + x – 2 = 0

Complex Numbers and Quadratic Equations

Easy

Q.

Solve the following equation:
x² – x + 2 = 0

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: –i

Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

\sqrt{2} x^{2} + x + \sqrt{2} = 0

Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

\sqrt{3} x^{2} - \sqrt{2} x + 3 \sqrt{3} = 0

Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:

x^{2} + x + \frac{1}{\sqrt{2}} = 0

Complex Numbers and Quadratic Equations

Medium

Q.

Solve the following equation:
$x^{2} + \frac{x}{\sqrt{2}} + 1 = 0$

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: – 15 – 8i

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: – 8 – 6i.

Complex Numbers and Quadratic Equations

Difficult

Q.

Find the square root of the following: 1 – i

Complex Numbers and Quadratic Equations

Medium

Q.

If {(\frac{1 + i}{1 - i})}^{m} =1, then find the least positive integral value of m.

NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 4 Complex Numbers and Quadratic Equations

The solutions are explained in a clear, step-by-step format so students can easily understand the methods and apply the formulas correctly while solving problems.

FAQs – Class 11 Maths Miscellaneous Exercise Chapter 4 Complex Numbers and Quadratic Equations

Q1. What is the focus of the Miscellaneous Exercise in Chapter 4?
The Miscellaneous Exercise focuses on revising the concepts of complex numbers, their properties, and methods of solving quadratic equations.

Q2. What topics are covered in this exercise?
The exercise includes problems related to complex numbers, modulus and conjugate, operations on complex numbers, and solutions of quadratic equations.

Q3. Why is this exercise important for exams?
This exercise combines all the important concepts of the chapter and helps students practice different types of problems that are commonly asked in examinations.

Q4. How can students prepare effectively for this exercise?
Students should revise all formulas related to complex numbers and quadratic equations, practice solving numerical problems, and carefully apply the correct methods while solving questions.