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JEE Mains Maths · Formula sheet

Complex Numbers formulas

16 formulas and 16 common traps for JEE Mains Maths Complex Numbers, grouped by subtopic.

Full notes with worked examples

Algebra of Complex Numbers

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Real and imaginary parts

Dividing complex numbers

a+ibc+id=(ac+bd)+i(bc−ad)c2+d2\frac{a+ib}{c+id}=\frac{(ac+bd)+i(bc-ad)}{c^2+d^2}

Equations in z and its conjugate

Conjugate identities

zzˉ=∣z∣2,z+zˉ=2 Re(z),z−zˉ=2i Im(z)z\bar z=|z|^2,\quad z+\bar z=2\,\mathrm{Re}(z),\quad z-\bar z=2i\,\mathrm{Im}(z)

Modulus rules and bounds

Modulus of a sum

∣a±b∣2=∣a∣2+∣b∣2±2 Re(abˉ)|a\pm b|^2=|a|^2+|b|^2\pm2\,\mathrm{Re}(a\bar b)

Common traps

Count every angle in the interval

A condition such as cos⁡2θ=12\cos^2\theta=\frac12 has several solutions in an interval like [−π,2π][-\pi,2\pi]. List them all before adding, and check that no denominator vanishes at any of them.

Squaring can add a root

In x2+y2=x+k\sqrt{x^2+y^2}=x+k, squaring needs x+k≥0x+k\ge0. Check each answer in the original equation, since a modulus is never negative.

A bound must be reached

The triangle inequality gives a bound, and it is the answer only if some zz attains it. Check that the extreme case points the numbers the same way and still satisfies every condition.

Polar Form, Argument and De Moivre

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Polar form and the principal argument

Polar form

z=r(cos⁡θ+isin⁡θ)=reiθz=r(\cos\theta+i\sin\theta)=re^{i\theta}

Powers by De Moivre's theorem

De Moivre's theorem

(reiθ)n=rneinθ(re^{i\theta})^n=r^ne^{in\theta}

Rotation and triangles

Angle at a vertex

z3−z1z2−z1=∣z3−z1∣∣z2−z1∣ eiθ\frac{z_3-z_1}{z_2-z_1}=\frac{|z_3-z_1|}{|z_2-z_1|}\,e^{i\theta}

Common traps

The inverse tangent is not the argument

tan⁡−1yx\tan^{-1}\frac yx lands in (−π2,π2)\left(-\frac\pi2,\frac\pi2\right). For a point with x<0x<0, add or subtract π\pi to reach the right quadrant.

Reduce the angle, and raise the modulus

201π6\frac{201\pi}6 is not a principal argument: take away 32π32\pi first. And a base of modulus 2 raised to the 21st power carries a factor 2212^{21} that is easy to drop.

Turn about the right point

zeiαze^{i\alpha} turns about the origin only. For a turn about aa, subtract aa, rotate, then add aa back; clockwise uses e−iαe^{-i\alpha}.

Roots of Unity

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Cube roots of unity

The two facts

ω3=1,1+ω+ω2=0\omega^3=1,\qquad1+\omega+\omega^2=0

The nth roots of unity

Roots of unity

zn=1 ⇒ z=e2πik/n,k=0,1,…,n−1z^n=1\ \Rightarrow\ z=e^{2\pi ik/n},\quad k=0,1,\dots,n-1

Common traps

Count the multiples of 3

In a sum over n=1n=1 to NN, the terms with 3∣n3\mid n behave differently, and there are ⌊N/3⌋\lfloor N/3\rfloor of them. Miscounting them by one is the usual slip.

The root 1 is missing

xn−1+⋯+1=0x^{n-1}+\dots+1=0 has only the n−1n-1 roots other than 1. So the sum of their kkth powers is n−1n-1 when n∣kn\mid k, and −1-1 otherwise.

Quadratic Equations with Complex Roots

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Roots, sum and product

Sum and product of the roots

α+β=−ba,αβ=ca\alpha+\beta=-\frac ba,\qquad\alpha\beta=\frac ca

Powers of the roots

Recurrence for power sums

Sn+2=p Sn+1+q Sn(α2=pα+q)S_{n+2}=p\,S_{n+1}+q\,S_n\qquad(\alpha^2=p\alpha+q)

Common traps

Conjugate roots need real coefficients

For z2−(3+i)z+(2+2i)=0z^2-(3+i)z+(2+2i)=0 the roots are 22 and 1+i1+i, not a conjugate pair. Use the conjugate root only when every coefficient is real.

Which root is alpha

When a question fixes Im(α)>Im(β)\mathrm{Im}(\alpha)>\mathrm{Im}(\beta), expressions like αn−βn\alpha^n-\beta^n or αβ\frac\alpha\beta depend on the choice. Name the roots before computing.

Lines and Circles in the Complex Plane

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Straight lines

Line in complex form

aˉz+azˉ+c=0(c∈R)\bar az+a\bar z+c=0\quad(c\in\mathbb R)

Circles

Circle in complex form

zzˉ+αˉz+αzˉ+d=0: centre −α, r=∣α∣2−dz\bar z+\bar\alpha z+\alpha\bar z+d=0:\ \text{centre }-\alpha,\ r=\sqrt{|\alpha|^2-d}

Common traps

Expand before trusting a sign

z(1+i)+zˉ(1−i)=2(x−y)z(1+i)+\bar z(1-i)=2(x-y), while z(1−i)+zˉ(1+i)=2(x+y)z(1-i)+\bar z(1+i)=2(x+y). Expanding once with z=x+iyz=x+iy avoids picking the wrong half-plane or line.

Divide before reading the centre

After squaring a ratio of distances the x2+y2x^2+y^2 term has a coefficient like 3 or 8. Divide through by it first; reading the centre from the undivided equation gives the wrong point.

Arcs and Conic Loci

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Real, imaginary or fixed-argument quotients

Radius of the arc

arg⁡z−az−b=θ ⇒ r=∣a−b∣2sin⁡θ\arg\frac{z-a}{z-b}=\theta\ \Rightarrow\ r=\frac{|a-b|}{2\sin\theta}

Ellipses, hyperbolas and parabolas

Ellipse from two foci

∣z−z1∣+∣z−z2∣=2a>∣z1−z2∣: b2=a2−c2, c=12∣z1−z2∣|z-z_1|+|z-z_2|=2a>|z_1-z_2|:\ b^2=a^2-c^2,\ c=\tfrac12|z_1-z_2|

Common traps

A fixed argument gives one arc

arg⁡z−az−b=θ\arg\frac{z-a}{z-b}=\theta is one arc, not the whole circle; the other arc has the argument θ−π\theta-\pi. Before counting intersections or measuring distances, check that the point found lies on the right arc.

Compare the constant with the focal distance

A sum of distances is an ellipse only when the constant exceeds the distance between the two points. Check 2k2k against ∣a−b∣|a-b| before using aa, bb and cc.

Regions and Extreme Distances

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Greatest and least distances

Distance from a point to a circle

∣∣c−a∣−r∣≤∣z−c∣≤∣c−a∣+r(∣z−a∣=r)\big||c-a|-r\big|\le|z-c|\le|c-a|+r\quad(|z-a|=r)

Regions, areas and lattice points

Sector and segment

sector=12r2φ,segment=12r2(φ−sin⁡φ)\text{sector}=\tfrac12r^2\varphi,\qquad\text{segment}=\tfrac12r^2(\varphi-\sin\varphi)

Common traps

Is the extreme point still in the region?

For a disk cut by a line, the farthest or nearest point of the whole disk may lie on the removed side. Then the answer sits at an end of the chord or at the foot of a perpendicular on the line.

Which side of the bisector

∣z−a∣<∣z−b∣|z-a|<|z-b| is the side that contains aa. Test one point, such as aa itself, rather than guessing from the sign of an expanded inequality.

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