JEE 2026 – Mathematics
Complex Numbers and Quadratic Equations
1. Need for Complex Numbers
The real number system cannot solve equations like $x^2+1=0$.
To overcome this limitation, mathematics introduces a new number $i$ such that $i^2=-1$.
2. Definition and Basic Form
A complex number is $z=a+ib$, where $a,b\in\mathbb{R}$.
| Term | Meaning |
|---|---|
| $a$ | Real part $\Re(z)$ |
| $b$ | Imaginary part $\Im(z)$ |
3. Equality of Complex Numbers
$a+ib=c+id \iff a=c \text{ and } b=d$
4. Algebra of Complex Numbers
| Operation | Result |
|---|---|
| Addition | $(a+ib)+(c+id)$ |
| Subtraction | $(a+ib)-(c+id)$ |
| Multiplication | $(a+ib)(c+id)$ |
5. Conjugate and Modulus
$\bar z=a-ib,\quad |z|=\sqrt{a^2+b^2}$
Important identities: $z+\bar z=2a$, $z\bar z=|z|^2$
6. Argand Plane
A complex number $z=a+ib$ corresponds to a point $(a,b)$ on a plane.
Real axis → horizontal, Imaginary axis → vertical.
7. Argument and Polar Form
$\tan\theta=\dfrac{b}{a}$
$z=r(\cos\theta+i\sin\theta)$
$z=r(\cos\theta+i\sin\theta)$
Principal argument: $-\pi<\theta\le\pi$
8. Euler’s Formula
$e^{i\theta}=\cos\theta+i\sin\theta$
9. Powers of $i$
| Power | Value |
|---|---|
| $i^1$ | $i$ |
| $i^2$ | $-1$ |
| $i^3$ | $-i$ |
| $i^4$ | $1$ |
10. Quadratic Equations
Standard form: $ax^2+bx+c=0,\; a\ne0$
11. Quadratic Formula
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
12. Discriminant and Nature of Roots
$D=b^2-4ac$
| $D$ | Nature of Roots |
|---|---|
| $>0$ | Real and distinct |
| $=0$ | Real and equal |
| $<0$ | Complex conjugate |
13. Complex Roots of Quadratic Equation
If $D<0$: $x=\dfrac{-b}{2a}\pm i\dfrac{\sqrt{4ac-b^2}}{2a}$
14. Relations Between Roots and Coefficients
$\alpha+\beta=-\dfrac{b}{a}$
$\alpha\beta=\dfrac{c}{a}$
$\alpha\beta=\dfrac{c}{a}$
15. Formation and Location of Roots
Using sum and product of roots, we can form equations and determine whether roots lie
between two numbers or on a particular side of the number line.
15 Most Important JEE Questions with Solutions
Q1. If $|z|=5$ and $\Re(z)=3$, find $|z-\bar z|$.
$z=3+ib,\; |z|^2=9+b^2=25\Rightarrow b=4$.
$z-\bar z=2ib=8i\Rightarrow |z-\bar z|=8$.
Q2. Find $\left|\dfrac{1+i}{1-i}\right|$.
$\dfrac{|1+i|}{|1-i|}=\dfrac{\sqrt2}{\sqrt2}=1$.
Q3. Solve $x^2+4x+13=0$.
$D=16-52=-36$. Roots: $x=-2\pm3i$.
Q4. If $z+\dfrac1z=2$, find $z$.
$z^2-2z+1=0\Rightarrow (z-1)^2=0\Rightarrow z=1$.
Q5. Find $\arg(-1+i)$.
Point lies in second quadrant. $\theta=\pi-\tan^{-1}(1)=\dfrac{3\pi}{4}$.
Q6. If roots of $x^2-5x+13=0$ are $\alpha,\beta$, find $|\alpha|$.
$\alpha\beta=13\Rightarrow |\alpha|^2=13\Rightarrow |\alpha|=\sqrt{13}$.
Q7. Find $i^{100}$.
$100\equiv0\pmod4\Rightarrow i^{100}=1$.
Q8. If $|z-1|=|z+1|$, find $\Re(z)$.
$z=x+iy$. Distance equality gives $x=0$.
Q9. Find nature of roots of $x^2-2(k+1)x+k^2+1=0$.
$D=4(k+1)^2-4(k^2+1)=8k$. Roots real if $k\ge0$.
Q10. If $z=i^{27}+i^{28}$, find $z$.
$i^{27}=i^3=-i,\; i^{28}=i^4=1\Rightarrow z=1-i$.
Q11. If $z\bar z=25$ and $\Re(z)=4$, find $\Im(z)$.
$16+b^2=25\Rightarrow b=\pm3$.
Q12. Solve $x^2-2x+5=0$.
$D=4-20=-16$. Roots: $x=1\pm2i$.
Q13. Find $|1+2i|+|2-3i|$.
$|1+2i|=\sqrt5,\; |2-3i|=\sqrt{13}$. Sum $=\sqrt5+\sqrt{13}$.
Q14. If $\alpha,\beta$ are roots of $x^2+2x+10=0$, find $\alpha^2+\beta^2$.
$\alpha+\beta=-2,\;\alpha\beta=10$.
$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=4-20=-16$.
Q15. If $|z|=|z-4|$, find locus of $z$.
Points equidistant from $(0,0)$ and $(4,0)$.
Locus: perpendicular bisector $x=2$.