Mathematics · Class 12 Science
Ch 4Complex Numbers and Quadratic Equations — Class 12 Mathematics, concept-first.
In earlier classes, you studied linear equations in one and two variables, and quadratic equations in one variable — always working within the real number system. But the real numbers have a genuine limitation. Consider the simplest possible quadratic equation, . Rearranging gives .
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Complex Number Arithmetic
Imagine you're trying to solve . You know that no real number squared gives . The square of any real number is either zero or positive. So this equation has no real solution.
Most relevant Q&A
- Express the following in the form $a + ib$: $5i\left(-\dfrac{3}{5}i\right)$Free
- Express the following in the form $a + ib$: $i^{9} + i^{19}$Free
- Express the following in the form $a + ib$: $i^{-39}$Free
- Express the following in the form $a + ib$: $3(7 + i7) + i(7 + i7)$Preview
- Express the following in the form $a + ib$: $(1 - i) - (-1 + i6)$Preview
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
In earlier classes, you studied linear equations in one and two variables, and quadratic equations in one variable — always working within the real number system.
Complex Numbers
You already know that the equation has no real solution — no real number, when squared, gives . To solve such equations, we extend the real number system by introducing a new symbol.
Algebra of Complex Numbers
Complex numbers are added by adding their real parts and their imaginary parts separately. If and , then
Addition of Two Complex Numbers
When you add two complex numbers, you simply add their real parts together and their imaginary parts together. This is the most natural extension of addition from real numbers.
Difference of Two Complex Numbers
Subtraction of complex numbers is built directly from addition and the concept of the negative of a complex number. For any two complex numbers and , the difference is defined as:
Multiplication of Two Complex Numbers
When you multiply two complex numbers, you treat them like binomials in , but with one crucial rule: .
Division of Two Complex Numbers
Division of complex numbers is defined in a way that keeps the result a complex number. For any two complex numbers and , with , the quotient is defined as:
Power of i
The imaginary unit is defined by . From this single fact, every higher power of can be reduced to one of just four values: , , , or . The pattern repeats in a cycle of length 4.
The Square Roots of a Negative Real Number
We already know that . But notice that as well. So both and are square roots of . When we write the symbol , however, we mean only the principal square root, which is .
Identities
The first thing to understand is that the algebraic identities you know from real numbers are not just a coincidence.
+−Worked Examplesi3 questions
- Example 2Express the following in the form of $a + bi$: (i) $5i\left(-\dfrac{1}{8}i\right)$ (ii) $(-i)(2i)\left(-\dfrac{1}{8}i\right)^{3}$Free
- Example 3Express $(5 - 3i)^{3}$ in the form $a + ib$.Preview
- Example 4Express $\left(-\sqrt{3} + \sqrt{-2}\right)\left(2\sqrt{3} - i\right)$ in the form of $a + ib$.Preview
The Modulus and the Conjugate of a Complex Number
16 QEvery complex number carries two fundamental real-valued companions: its modulus and its conjugate. These two tools let us measure the "size" of a complex number and reflect it across the real axis, a…
+−Worked Examplesi2 questions
+−Exercise 4.1i14 questions
- Q1Express the following in the form $a + ib$: $5i\left(-\dfrac{3}{5}i\right)$Free
- Q2Express the following in the form $a + ib$: $i^{9} + i^{19}$Free
- Q3Express the following in the form $a + ib$: $i^{-39}$Free
- Q4Express the following in the form $a + ib$: $3(7 + i7) + i(7 + i7)$Preview
- Q5Express the following in the form $a + ib$: $(1 - i) - (-1 + i6)$Preview
- Q6Express the following in the form $a + ib$: $\left(\dfrac{1}{5} + i\dfrac{2}{5}\right) - \left(4 + i\dfrac{5}{2}\right)$Preview
- Q7Express the following in the form $a + ib$: $\left[\left(\dfrac{1}{3} + i\dfrac{7}{3}\right) + \left(4 + i\dfrac{1}{3}\right)\right] - \left…Preview
- Q8Express the following in the form $a + ib$: $(1 - i)^{4}$Preview
- Q9Express the following in the form $a + ib$: $\left(\dfrac{1}{3} + 3i\right)^{3}$Preview
- Q10Express the following in the form $a + ib$: $\left(-2 - \dfrac{1}{3}i\right)^{3}$Preview
- Q11Find the multiplicative inverse of $4 - 3i$.Preview
- Q12Find the multiplicative inverse of $\sqrt{5} + 3i$.Preview
- Q13Find the multiplicative inverse of $-i$.Preview
- Q14Express the following expression in the form of $a + ib$: $\dfrac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}\,i) - (\sqrt{3} - i\s…Preview
Argand Plane and Polar Representation
You already know that every ordered pair of real numbers corresponds to exactly one point in the XY-plane, and vice versa.
Miscellaneous Examples
Miscellaneous Exercise on Chapter 4
+−Miscellaneous Exercisei14 questions
- Q1Evaluate: $\left[i^{18} + \left(\dfrac{1}{i}\right)^{25}\right]^{3}$.Free
- Q2For any two complex numbers $z_1$ and $z_2$, prove that $\operatorname{Re}(z_1 z_2) = \operatorname{Re} z_1\, \operatorname{Re} z_2 - \opera…Free
- Q3Reduce $\left(\dfrac{1}{1 - 4i} - \dfrac{2}{1 + i}\right)\left(\dfrac{3 - 4i}{5 + i}\right)$ to the standard form.Free
- Q4If $x - iy = \sqrt{\dfrac{a - ib}{c - id}}$, prove that $(x^{2} + y^{2})^{2} = \dfrac{a^{2} + b^{2}}{c^{2} + d^{2}}$.Preview
- Q5If $z_1 = 2 - i$, $z_2 = 1 + i$, find $\left|\dfrac{z_1 + z_2 + 1}{z_1 - z_2 + 1}\right|$.Preview
- Q6If $a + ib = \dfrac{(x + i)^{2}}{2x^{2} + 1}$, prove that $a^{2} + b^{2} = \dfrac{(x^{2} + 1)^{2}}{(2x^{2} + 1)^{2}}$.Preview
- Q7Let $z_1 = 2 - i$, $z_2 = -2 + i$. Find (i) $\operatorname{Re}\left(\dfrac{z_1 z_2}{\bar{z}_1}\right)$, (ii) $\operatorname{Im}\left(\dfrac{…Preview
- Q8Find the real numbers $x$ and $y$ if $(x - iy)(3 + 5i)$ is the conjugate of $-6 - 24i$.Preview
- Q9Find the modulus of $\dfrac{1 + i}{1 - i} - \dfrac{1 - i}{1 + i}$.Preview
- Q10If $(x + iy)^{3} = u + iv$, then show that $\dfrac{u}{x} + \dfrac{v}{y} = 4(x^{2} - y^{2})$.Preview
- Q11If $\alpha$ and $\beta$ are different complex numbers with $|\beta| = 1$, then find $\left|\dfrac{\beta - \alpha}{1 - \bar{\alpha}\beta}\rig…Preview
- Q12Find the number of non-zero integral solutions of the equation $|1 - i|^{x} = 2^{x}$.Preview
- Q13If $(a + ib)(c + id)(e + if)(g + ih) = A + iB$, then show that $(a^{2} + b^{2})(c^{2} + d^{2})(e^{2} + f^{2})(g^{2} + h^{2}) = A^{2} + B^{2}…Preview
- Q14If $\left(\dfrac{1 + i}{1 - i}\right)^{m} = 1$, then find the least positive integral value of $m$.Preview
Summary
- A complex number is of the form , where and (so ). is the real part, the imaginary part. - Equality: iff and .
Exemplar Problems
Higher-order thinking / exemplar-style practice problems.
+−Show 52 questionsHide questions52 questions
- Q1For a positive integer $n$, find the value of $(1-i)^n\left(1-\dfrac{1}{i}\right)^n$.Free
- Q2Evaluate $\displaystyle\sum_{n=1}^{13}\left(i^n+i^{n+1}\right)$, where $n\in\mathbf{N}$.Free
- Q3If $\left(\dfrac{1+i}{1-i}\right)^3-\left(\dfrac{1-i}{1+i}\right)^3=x+iy$, then find $(x, y)$.Free
- Q4If $\dfrac{(1+i)^2}{2-i}=x+iy$, then find the value of $x+y$.Preview
- Q5If $\left(\dfrac{1-i}{1+i}\right)^{100}=a+ib$, then find $(a, b)$.Preview
- Q6If $(1+i)z=(1-i)\bar{z}$, then show that $z=-i\bar{z}$.Preview
- Q7If $z=x+iy$, then show that $z\bar{z}+2(z+\bar{z})+b=0$, where $b\in\mathbf{R}$, represents a circle.Preview
- Q8If the real part of $\dfrac{\bar{z}+2}{\bar{z}-1}$ is 4, then show that the locus of the point representing $z$ in the complex plane is a ci…Preview
- Q9Show that the complex number $z$, satisfying the condition $\arg\left(\dfrac{z-1}{z+1}\right)=\dfrac{\pi}{4}$ lies on a circle.Preview
- Q10Solve the equation $|z|=z+1+2i$.Preview
- Q11What is the conjugate of $\dfrac{2-i}{(1-2i)^2}$?Preview
- Q12If $|z_1|=|z_2|$, is it necessary that $z_1=z_2$?Preview
- Q13If $\dfrac{(a^2+1)^2}{2a-i}=x+iy$, what is the value of $x^2+y^2$?Preview
- Q14Find $\left|(1+i)\dfrac{(2+i)}{(3+i)}\right|$.Preview
- Q15Where does $z$ lie, if $\left|\dfrac{z-5i}{z+5i}\right|=1$.Preview
- Q16If $|z+1|=z+2(1+i)$, then find $z$.Preview
- Q17Show that $\left|\dfrac{z-2}{z-3}\right|=2$ represents a circle. Find its centre and radius.Preview
- Q18If $\dfrac{z-1}{z+1}$ is a purely imaginary number ($z\neq-1$), then find the value of $|z|$.Preview
- Q19If $|z_1|=1$ ($z_1\neq-1$) and $z_2=\dfrac{z_1-1}{z_1+1}$, then show that the real part of $z_2$ is zero.Preview
- Q20If $|z_1|=|z_2|=\ldots=|z_n|=1$, then show that $|z_1+z_2+z_3+\ldots+z_n|=\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}+\ldots+\dfrac{1…Preview
- Q21Solve the system of equations $\mathrm{Re}(z^2)=0$, $|z|=2$.Preview
- Q22Find the complex number satisfying the equation $z+\sqrt{2}\,|(z+1)|+i=0$.Preview
- Q23$\sin x+i\cos 2x$ and $\cos x-i\sin 2x$ are conjugate to each other for: (A) $x=n\pi$ (B) $x=\left(n+\dfrac{1}{2}\right)\dfrac{\pi}{2}$ (C)…Preview
- Q24The real value of $\alpha$ for which the expression $\dfrac{1-i\sin\alpha}{1+2i\sin\alpha}$ is purely real is: (A) $(n+1)\dfrac{\pi}{2}$ (B)…Preview
- Q25If $z=x+iy$ lies in the third quadrant, then $\dfrac{\bar{z}}{z}$ also lies in the third quadrant if: (A) $x>y>0$ (B) $x<y<0$ (C) $y<x<0$ (D…Preview
- Q26The value of $(z+3)(\bar{z}+3)$ is equivalent to: (A) $|z+3|^2$ (B) $|z-3|$ (C) $z^2+3$ (D) None of thesePreview
- Q27If $\left(\dfrac{1+i}{1-i}\right)^x=1$, then: (A) $x=2n+1$ (B) $x=4n$ (C) $x=2n$ (D) $x=4n+1$, where $n\in\mathbf{N}$Preview
- Q28A real value of $x$ satisfies the equation $\left(\dfrac{3-4ix}{3+4ix}\right)=\alpha-i\beta$ $(\alpha,\beta\in\mathbf{R})$ if $\alpha^2+\bet…Preview
- Q29Which of the following is correct for any two complex numbers $z_1$ and $z_2$? (A) $|z_1z_2|=|z_1||z_2|$ (B) $\arg(z_1z_2)=\arg(z_1)\cdot\ar…Preview
- Q30The point represented by the complex number $2-i$ is rotated about origin through an angle $\dfrac{\pi}{2}$ in the clockwise direction, the…Preview
- Q31Let $x, y\in\mathbf{R}$, then $x+iy$ is a non real complex number if: (A) $x=0$ (B) $y=0$ (C) $x\neq0$ (D) $y\neq0$Preview
- Q32If $a+ib=c+id$, then: (A) $a^2+c^2=0$ (B) $b^2+c^2=0$ (C) $b^2+d^2=0$ (D) $a^2+b^2=c^2+d^2$Preview
- Q33The complex number $z$ which satisfies the condition $\left|\dfrac{i+z}{i-z}\right|=1$ lies on: (A) circle $x^2+y^2=1$ (B) the $x$-axis (C)…Preview
- Q34If $z$ is a complex number, then: (A) $|z^2|>|z|^2$ (B) $|z^2|=|z|^2$ (C) $|z^2|<|z|^2$ (D) $|z^2|\geq|z|^2$Preview
- Q35$|z_1+z_2|=|z_1|+|z_2|$ is possible if: (A) $z_2=\bar{z}_1$ (B) $z_2=\dfrac{1}{z_1}$ (C) $\arg(z_1)=\arg(z_2)$ (D) $|z_1|=|z_2|$Preview
- Q36The real value of $\theta$ for which the expression $\dfrac{1+i\cos\theta}{1-2i\cos\theta}$ is a real number is: (A) $n\pi+\dfrac{\pi}{4}$ (…Preview
- Q37If $f(z)=\dfrac{7-z}{1-z^2}$, where $z=1+2i$, then $|f(z)|$ is: (A) $\dfrac{|z|}{2}$ (B) $|z|$ (C) $2|z|$ (D) none of these.Preview
- Q38For any two complex numbers $z_1, z_2$ and any real numbers $a, b$, $\;|az_1-bz_2|^2+|bz_1+az_2|^2=\;$ _____.Preview
- Q39The value of $\sqrt{-25}\times\sqrt{-9}$ is _____.Preview
- Q40The number $\dfrac{(1-i)^3}{1-i^3}$ is equal to _____.Preview
- Q41The sum of the series $i+i^2+i^3+\ldots$ upto 1000 terms is _____.Preview
- Q42Multiplicative inverse of $1+i$ is _____.Preview
- Q43If $z_1$ and $z_2$ are complex numbers such that $z_1+z_2$ is a real number, then $z_2=$ _____.Preview
- Q44If $|z+4|\leq3$, then the greatest and least values of $|z+1|$ are _____ and _____.Preview
- Q45If $\left|\dfrac{z-2}{z+2}\right|=\dfrac{\pi}{6}$, then the locus of $z$ is _____.Preview
- Q46The order relation is defined on the set of complex numbers.Preview
- Q47For any complex number $z$ the minimum value of $|z|+|z-1|$ is 1.Preview
- Q48The locus represented by $|z-1|=|z-i|$ is a line perpendicular to the join of $(1, 0)$ and $(0, 1)$.Preview
- Q49If $z$ is a complex number such that $z\neq0$ and $\mathrm{Re}(z)=0$, then $\mathrm{Im}(z^2)=0$.Preview
- Q50The inequality $|z-4|<|z-2|$ represents the region given by $x>3$.Preview
- Q512 is not a complex number.Preview
- Q52Match the statements of Column A and Column B. Column A: (a) The polar form of $i+\sqrt{3}$ is; (b) The amplitude of $-1+\sqrt{-3}$ is; (c)…Preview