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Mathematics · Class 12 Science

Ch 1Complex Numbers — Class 12 Mathematics, concept-first.

In earlier years you solved linear equations in one and two variables, and then quadratic equations of the form using the quadratic formula. That formula asks you to compute — and it works perfectly as long as the discriminant is non-negative. But look at the simplest possible case: , i.e. .

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Complex Numbers

Complex numbers extend the real number system so that every quadratic equation, even one like x²+1=0 with a negative discriminant, has a solution. Built formally as ordered pairs (a,b) of real numbers and written in…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

Introduction

In earlier years you solved linear equations in one and two variables, and then quadratic equations of the form using the quadratic formula.

1.1

Why We Need Complex Numbers — Definition as an Ordered Pair

Every real number, when squared, gives a result that is zero or positive. That simple fact means an equation like , i.e.

1.2

The a + ib Form: Real Part, Imaginary Part, and the Symbol i

Ordered pairs are precise but clumsy to write, so the next step is to recover the familiar notation from them — and to show it means exactly the same thing.

1.3

Conjugate, Multiplicative Inverse, and Square Roots of a Complex Number

The conjugate of , written , is obtained by flipping the sign of the imaginary part: . Geometrically (as the next sections will make precise) this is a mirror reflection across the real axis, but alge…

1.4

Modulus and Amplitude (Argument) of a Complex Number

So far complex numbers have been purely algebraic objects. This section attaches a size and a direction to each one, which is what eventually lets us picture them as points in a plane.

1.5

The Argand Plane: Picturing Complex Numbers and Their Operations Geometrically

Gauss popularised the idea, now standard, of plotting a complex number as the point in a plane with perpendicular axes: the horizontal axis (the real axis) carries the real numbers, and the vertical a…

1.6

Loci of Complex Numbers in the Argand Plane

A very useful skill is translating a condition written in terms of (such as ) into the familiar Cartesian equation of a curve — and conversely, recognising standard curves when they appear disguised a…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 19 questions19 questions
  1. Q1Find the complex conjugate of $(2 + 5i)(-4 + 6i)$.Preview
  2. Q2If $x + iy = \operatorname{cis}\alpha \cdot \operatorname{cis}\beta$, then find the value of $x^2 + y^2$.Preview
  3. Q3If $A, B, C$ are angles of a triangle such that $x = \operatorname{cis} A$, $y = \operatorname{cis} B$, $z = \operatorname{cis} C$, then fin…Preview
  4. Q4If $z = 2 - 3i$, then show that $z^2 - 4z + 13 = 0$.Preview
  5. Q5If $x + iy = \dfrac{1}{1 + \cos\theta + i\sin\theta}$, then show that $4x^2 - 1 = 0$.Preview
  6. Q6Find the complex conjugate of $(3 + 4i)(2 - 3i)$.Preview
  7. Q7Write $z = -\sqrt{3} + i$ in modulus-amplitude form.Preview
  8. Q8Show that the points in the Argand plane represented by the complex numbers $-2 + 7i$, $-\dfrac{3}{2} + \dfrac{1}{2}i$, $4 - 3i$, $\dfrac{7}…Preview
  9. Q9Find the square root of the complex number $7 + 24i$.Preview
  10. Q10If $z_1 = -1$ and $z_2 = i$, then find $\mathrm{Arg}\left(\dfrac{z_1}{z_2}\right)$.Preview
  11. Q11If $x + iy = \dfrac{1}{1 + \cos\theta + i\sin\theta}$, then show that $4x^2 - 1 = 0$.Preview
  12. Q12Write the complex number $\dfrac{4+3i}{(2+3i)(4-3i)}$ in the form $a+ib$.Preview
  13. Q13Write $z = -\sqrt{7} + i\sqrt{21}$ in the polar form.Preview
  14. Q14If $x+iy = \dfrac{1}{1+\cos\theta + i\sin\theta}$, then show that $4x^2 - 1 = 0$.Preview
  15. Q15Find a square root of the complex number $3+4i$.Preview
  16. Q16If the Arg $\bar{z}_1$ and Arg $z_2$ are $\frac{\pi}{5}$ and $\frac{\pi}{3}$ respectively, then find (Arg $z_1$ + Arg $z_2$).Preview
  17. Q17Show that the four points in the Argand Plane represented by the complex numbers $2+i$, $4+3i$, $2+5i$, $3i$ are the vertices of a square.Preview
  18. Q18Find the multiplicative inverse of $7 + 24i$.Preview
  19. Q19Show that the four points in the Argand plane represented by the complex numbers $2 + i,\ 4 + 3i,\ 2 + 5i,\ 3i$ are the vertices of a square…Preview