COMPLEX NUMBERS EAPCET PYQS

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Complex Numbers EAPCET & EAMCET PYQs

Complex Number (Introduction)

A complex number is an ordered pair of real numbers $(a, b)$ written as $z = a + ib$, where:

• $a$ is called the real part of $z$, denoted as $\text{Re}(z) = a$

• $b$ is called the imaginary part of $z$, denoted as $\text{Im}(z) = b$

• $i$ is the imaginary unit with the property $i^2 = -1$

Equality of Complex Numbers

Two complex numbers $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are equal if and only if:

$$z_1 = z_2 \iff a_1 = a_2 \text{ and } b_1 = b_2$$

Modulus (Absolute Value)

For a complex number $z = a + ib$, the modulus is: $|z| = \sqrt{a^2 + b^2}$

Conjugate of Complex Number

For $z = a + ib$, the conjugate is $\bar{z} = a – ib$

$$z \cdot \bar{z} = (a + ib)(a – ib) = a^2 + b^2 = |z|^2$$

Argand Plane

The complex number $z = a + ib$ is represented as point $(a, b)$ where the horizontal axis is the real axis and vertical axis is the imaginary axis.

Argument (Amplitude)

The argument of $z = a + ib \neq 0$ is the angle $\theta$ that the line from origin to $z$ makes with positive real axis:

$$\arg(z) = \theta \text{ where } a = |z|\cos\theta, \quad b = |z|\sin\theta$$

Principal Argument: $-\pi < \arg(z) \leq \pi$

Polar Form

Complex number $z = a + ib$ can be written as:

$$z = r(\cos\theta + i\sin\theta) = r \text{ cis } \theta$$

where $r = |z|$ and $\theta = \arg(z)$

Essential Formulas

De Moivre’s Theorem

If $z = r(\cos\theta + i\sin\theta)$, then

$$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$$

nth Roots of Unity

Roots of $z^n = 1$ are: $e^{2\pi i k/n}$, $k = 0,1,\ldots,n-1$

Cube Roots of Unity

$$1, \omega, \omega^2 \text{ where } \omega = e^{2\pi i/3}$$ $$1 + \omega + \omega^2 = 0$$ $$\omega^3 = 1$$

Modulus Properties

$$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$$ $$\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$$

Argument Formula

$$\arg(z_1 \cdot z_2) = \arg(z_1) + \arg(z_2)$$ $$\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) – \arg(z_2)$$

Powers of $i$

$$i^1 = i, \quad i^2 = -1$$ $$i^3 = -i, \quad i^4 = 1$$ $$i^{4n+r} = i^r$$

TS EAMCET 2025 – Topic-Wise PYQs (24 Questions)

🌟 Topic 1: Roots of Unity & Cube Roots

EAMCET 2025 Medium

Q1: If $\omega \neq 1$ is a cube root of unity, then one root among the $7^{th}$ roots of $(1 + \omega)$ is

A) $1 + \omega$

B) $1 – \omega$

C) $\omega – \omega^2$

D) $\omega/(\omega – \omega^2)$

✓ Ans: A) $1 + \omega$

EAMCET 2025 Hard

Q5: One of the roots of $(x + 1)^4 + 81 = 0$ is

A) $3(1+i)/\sqrt{2}$

B) $-(3+\sqrt{2}i)/\sqrt{2}$

C) $-(3+\sqrt{2}i)/\sqrt{2}$

D) $-(3+3i)/\sqrt{2}$

✓ Ans: B)

EAMCET 2025 Medium

Q13: If $a$ is root of $x^2 – x + 1 = 0$, find $\sum_{k=1}^{12} (a^k + 1/a^k)^3$

A) $-32$

B) $32$

C) $0$

D) $16$

✓ Ans: C) $0$

EAMCET 2025 Medium

Q22: If $\omega$ is a cube root of unity and $x = \omega^2 – \omega + 2$, then $x$ satisfies

A) $x^2 – 4x + 7 = 0$

B) $x^2 + 4x + 7 = 0$

C) $x^2 – 2x + 4 = 0$

D) $x^2 + 2x + 4 = 0$

✓ Ans: A) $x^2 – 4x + 7 = 0$

⚡ Topic 2: Powers, Exponentials & De Moivre’s

EAMCET 2025 Hard

Q6: $(1 – i\sqrt{3})^{2025} =$

A) $2^{2025}$

B) $2^{2026}$

C) $-2^{2025}$

D) $-2^{2026}$

✓ Ans: C) $-2^{2025}$

EAMCET 2025 Medium

Q8: $\left(\frac{1+i}{1-i}\right)^{228} =$

A) $-4((1-i)/(1+i))^{226}$

B) $4((1-i)/(1+i))^{226}$

C) $((1-i)/(1+i))^{228}$

D) $-((1-i)/(1+i))^{228}$

✓ Ans: C) $((1-i)/(1+i))^{228}$

EAMCET 2025 Hard

Q10: $(\sqrt{3}+i)^{10} + (\sqrt{3}-i)^{10} =$

A) $1024\sqrt{3}$

B) $1024$

C) $2048$

D) $512\sqrt{3}$

✓ Ans: B) $1024$

EAMCET 2025 Hard

Q18: If $n \neq 3K$, then $(\sqrt{3}+i)^{2n} + (\sqrt{3}-i)^{2n} =$

A) $(-1)^n 2^{2n+1}$

B) $(-1)^{n+1}2^{2n+1}$

C) $(-1)^{n+1}2^{2n}$

D) $(-1)^n 2^{2n}$

✓ Ans: C) $(-1)^{n+1}2^{2n}$

📐 Topic 3: Modulus, Amplitude & Argument

EAMCET 2025 Medium

Q3: If $|Z_1 – 3 – 4i| = 5$ and $|Z_2| = 15$, find sum of max and min values of $|Z_1 – Z_2|$

A) $75$

B) $30$

C) $35$

D) $20$

✓ Ans: B) $30$

EAMCET 2025 Hard

Q9: Number of real values of $(-1 – \sqrt{3}i)^{1/4}$

A) $0$

B) $1$

C) $2$

D) $3$

✓ Ans: C) $2$

EAMCET 2025 Medium

Q12: Amplitude of $\frac{\sqrt{3}+0i-\sqrt{ai}}{(-1+i-i)}$

A) $\pi/2$

B) $\pi/3$

C) $-5\pi/12$

D) $-\pi/6$

✓ Ans: D) $-\pi/6$

EAMCET 2025 Hard

Q19: If $|Z| = 2, Z_1 = (2/e)e^{i\alpha}$ and $\theta = \text{amp}(Z)$, find $\frac{Z_1^2-Z_1}{Z_1+Z_1^2}$

A) $2^n i \tan(n\theta + n\alpha)$

B) $i\tan(n\theta – n\alpha)$

C) $i\tan(n\theta + n\alpha)$

D) $\tan(n\theta + n\alpha)$

✓ Ans: C) $i\tan(n\theta + n\alpha)$

🎯 Topic 4: Locus & Geometric Interpretation

EAMCET 2025 Medium

Q2: If $Z = r(\cos\theta + i\sin\theta)$ is solution of $x^3 = i$, find $i^8(\cos\theta + i\sin\theta)^2$

A) $\sqrt{3}/2 + i/2$

B) $1$

C) $-i$

D) $-\sqrt{3}/2 + i/2$

✓ Ans: C) $-i$

EAMCET 2025 Hard

Q7: If $z = x + iy$ and $\text{Amp}((z-3)/(z-2i)) = -\pi/2$, locus of P is

A) Circle $x^2 + y^2 – 3x – 2y = 0$

B) Arc of circle (excluding points)

C) Arc not containing origin

D) Circle not containing $(0,2)$

✓ Ans: B) Arc of circle x² + y² – 3x – 2y = 0 (excluding points)

EAMCET 2025 Hard

Q11: If $z = x + iy$ and $\text{Im}[(z-3)/(z+3i)] = 0$, locus of P is

A) $x^2 + y^2 – 3x + 3y = 0$

B) $2xy – 3x + 3y = 9$

C) $x – y – 3 = 0$, $(x,y) \neq (0,-3)$

D) $x + y + 3 = 0$, $(x,y) \neq (0,-2)$

✓ Ans: C) x – y – 3 = 0, (x,y) ≠ (0,-3)

EAMCET 2025 Hard

Q23: If $z = x + iy$ and $\text{Arg}((z-3i)/(z+3i)) = \pi/4$, locus of P is

A) $x^2 + y^2 – y – 6 = 0$

B) $x^2 + y^2 – x – y – 6 = 0$

C) $x^2 + y^2 + 5x – y – 6 = 0$, $(x,y) \neq (0,-)$

D) $x^2 + y^2 + x – y – 6 = 0$

✓ Ans: C) x² + y² + 5x – y – 6 = 0, (x,y) ≠ (0,-)

🔍 Topic 5: Complex Equations & Operations

EAMCET 2025 Medium

Q4: Eight vertices of regular octagon: $z_j = 1/(x_{j-2i})$, $j=1,2,…,8$. Radius of circumcircle is

A) $1/4$

B) $i/4$

C) $i$

D) $2$

✓ Ans: A) 1/4

EAMCET 2025 Hard

Q15: One value of $\sqrt{24 – 70i} + \sqrt{-24 + 70i}$ is

A) $2 + 12i$

B) $12 – 2i$

C) $-12 + 2i$

D) $-12 – 2i$

✓ Ans: D) -12 – 2i

EAMCET 2025 Medium

Q16: If $a = |\bar{a}|$; $b = |\bar{b}|$ then $(\bar{a}/a – \bar{a}/b)^2 =$

A) $((\bar{a}-\bar{b})/a^2b^2)^2$

B) $((\bar{a}-b)/ab)^2$

C) $((b\bar{a}-\bar{a}b)/ab)^2$

D) $((\bar{a}\bar{a}-\bar{b}\bar{b})/a^2b^2)^2$

✓ Ans: B) ((ā-b)/ab)²

EAMCET 2025 Hard

Q21: Product of all values of $(\sqrt{3} – i)^{1/3}$

A) $8$

B) $-8$

C) $8i$

D) $-8i$

✓ Ans: B) -8

EAMCET 2025 Hard

Q24: If $\frac{2i+3i}{i-2} – \frac{4i-3}{3i+4i} = x + iy$, then $3x + y =$

A) $4$

B) $-4$

C) $-2$

D) $2$

✓ Ans: B) -4

📊 Topic 6: Quadrants & Regions

EAMCET 2025 Medium

Q17: In Argand plane, no value of $\sqrt{1 – i\sqrt{3}}$ lies in

A) First quadrant

B) Second quadrant

C) Third quadrant

D) Fourth quadrant

✓ Ans: A) First quadrant

🎲 Topic 7: Mixed & Advanced Problems

EAMCET 2025 Hard

Q14: Set of $\theta$ such that $\frac{1-i\cos\theta}{1+2i\sin\theta}$ is purely imaginary

A) $\{n\pi + (-1)^n\pi/4, n \in \mathbb{Z}\}$

B) $\{n\pi/2 + (-1)^n\pi/4, n \in \mathbb{Z}\}$

C) $\{n\pi + (-1)^n\pi/5, n \in \mathbb{Z}\}$

D) $\{2n\pi \pm \pi/4, n \in \mathbb{Z}\}$

✓ Ans: B) {nπ/2 + (-1)^n π/4, n ∈ Z}

EAMCET 2025 Hard

Q20: $\omega$ is cube root of unity, $|Z-1| \leq 2$ and $|\omega Z – 1 – \omega^2| = r$ have no common solution. Then

A) $0 \leq r \leq 4$

B) $r = $ only $|i/\omega|$

C) $r > 4$

D) $1 < r < 2$

✓ Ans: C) r > 4

📚 EAPCET: Basic Operations (4 Questions)

EAPCET 2024 Easy

Q1: If $z_1 = 2 + 3i$ and $z_2 = 1 – 2i$, find $z_1 + z_2$

A) $3 + i$

B) $3 – i$

C) $-3 + i$

D) $3 + 5i$

✓ Ans: A) 3 + i

EAPCET 2023 Easy

Q2: Simplify $(2 + i)(3 – 2i)$

A) $6 – 2i$

B) $8 – i$

C) $8 + i$

D) $6 + 4i$

✓ Ans: B) 8 – i

EAPCET 2022 Medium

Q3: Compute $(1 + i)^2 + (1 – i)^2$

A) $0$

B) $2$

C) $-2$

D) $4i$

✓ Ans: A) 0

EAPCET 2024 Medium

Q4: Find $\frac{2 + 3i}{1 – i}$

A) $-1/2 + 5i/2$

B) $1/2 – 5i/2$

C) $2 + 3i$

D) $-1 + 5i$

✓ Ans: A) -1/2 + 5i/2

📐 EAPCET: Modulus & Conjugate (8 Questions)

EAPCET 2024 Easy

Q5: Find $|3 + 4i|$

A) $5$

B) $7$

C) $25$

D) $1$

✓ Ans: A) 5

EAPCET 2023 Easy

Q6: If $|z| = 2$ and $z = a + bi$, find $a^2 + b^2$

A) $2$

B) $4$

C) $8$

D) $16$

✓ Ans: B) 4

EAPCET 2024 Easy

Q7: Find conjugate of $5 – 3i$

A) $5 + 3i$

B) $-5 – 3i$

C) $3 + 5i$

D) $-5 + 3i$

✓ Ans: A) 5 + 3i

EAPCET 2023 Easy

Q8: If $z = 2 + i$, find $z \cdot \bar{z}$

A) $3$

B) $4$

C) $5$

D) $6$

✓ Ans: C) 5

EAPCET 2022 Medium

Q9: If $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$ where $|z_1| = 3, |z_2| = 4$

A) $7$

B) $12$

C) $1$

D) $7/3$

✓ Ans: B) 12

EAPCET 2021 Medium

Q10: If $|z + \bar{z}| = 10$, find Re(z)

A) $5$

B) $10$

C) $20$

D) $2.5$

✓ Ans: A) 5

EAPCET 2020 Medium

Q11: If $z – \bar{z} = 4i$, find Im(z)

A) $2$

B) $-2$

C) $4$

D) $8$

✓ Ans: A) 2

EAPCET 2019 Medium

Q12: If $|z_1/z_2| = 2$ and $|z_1| = 6$, find $|z_2|$

A) $3$

B) $12$

C) $2$

D) $4$

✓ Ans: A) 3

⚡ EAPCET: Powers of i & Arguments (8 Questions)

EAPCET 2024 Easy

Q13: Find $i^{37}$

A) $i$

B) $-i$

C) $1$

D) $-1$

✓ Ans: A) i

EAPCET 2023 Easy

Q14: Simplify $i^{100} + i^{101}$

A) $1 + i$

B) $1 – i$

C) $2$

D) $0$

✓ Ans: A) 1 + i

EAPCET 2024 Medium

Q15: Find $\arg(1 + i)$

A) $\pi/6$

B) $\pi/4$

C) $\pi/3$

D) $\pi/2$

✓ Ans: B) π/4

EAPCET 2023 Medium

Q16: Find $\arg(-1 – i)$

A) $3\pi/4$

B) $-3\pi/4$

C) $\pi/4$

D) $-\pi/4$

✓ Ans: B) -3π/4

EAPCET 2022 Hard

Q17: If $\arg(z) = \pi/3$ and $|z| = 2$, find $z$

A) $1 + i\sqrt{3}$

B) $2 + 2i$

C) $\sqrt{3} + i$

D) $2 + i\sqrt{3}$

✓ Ans: A) 1 + i√3

EAPCET 2021 Medium

Q18: Find $\sum_{k=0}^{3} i^k$

A) $0$

B) $1$

C) $i$

D) $1 + i$

✓ Ans: A) 0

EAPCET 2020 Medium

Q19: Find $i^{2018}$

A) $-1$

B) $1$

C) $i$

D) $-i$

✓ Ans: B) 1

EAPCET 2020 Medium

Q20: Find $\arg(i)$

A) $0$

B) $\pi/2$

C) $\pi$

D) $3\pi/2$

✓ Ans: B) π/2

🌀 EAPCET: Polar Form & Roots (8 Questions)

EAPCET 2024 Medium

Q21: Express $z = 1 + i$ in polar form

A) $\sqrt{2}$ cis $\pi/4$

B) $2$ cis $\pi/3$

C) $\sqrt{2}$ cis $\pi/6$

D) $2$ cis $\pi/4$

✓ Ans: A) √2 cis π/4

EAPCET 2023 Medium

Q22: Convert $2$ cis $\pi/3$ to rectangular form

A) $1 + i\sqrt{3}$

B) $\sqrt{3} + i$

C) $2 + 2i$

D) $1 – i\sqrt{3}$

✓ Ans: A) 1 + i√3

EAPCET 2021 Hard

Q23: Express $-2 – 2i$ in polar form

A) $2\sqrt{2}$ cis $5\pi/4$

B) $2$ cis $\pi$

C) $4$ cis $3\pi/4$

D) $2\sqrt{2}$ cis $3\pi/4$

✓ Ans: A) 2√2 cis 5π/4

EAPCET 2024 Hard

Q24: Find square roots of $3 + 4i$

A) $\pm(2 + i)$

B) $\pm(1 + 2i)$

C) $\pm(2 – i)$

D) $\pm(1 – 2i)$

✓ Ans: A) ±(2 + i)

EAPCET 2023 Hard

Q25: If $z^2 = -1$, find $z$

A) $\pm i$

B) $\pm 1$

C) $\pm(1 + i)$

D) No solution

✓ Ans: A) ±i

EAPCET 2022 Hard

Q26: Find $\sqrt{i}$

A) $\pm(1+i)/\sqrt{2}$

B) $\pm(1-i)/\sqrt{2}$

C) $\pm 1$

D) $\pm i$

✓ Ans: A) ±(1+i)/√2

EAPCET 2020 Hard

Q27: Find cube roots of unity

A) $1, \omega, \omega^2$

B) $1, i, -i$

C) $1, 1, 1$

D) $-1, i, -i$

✓ Ans: A) 1, ω, ω²

EAPCET 2019 Medium

Q28: If $\omega$ is cube root of unity, find $1 + \omega + \omega^2$

A) $0$

B) $1$

C) $3$

D) $\omega$

✓ Ans: A) 0

🔮 EAPCET: Cube Roots of Unity Properties (4 Questions)

EAPCET 2024 Medium

Q29: If $\omega = e^{2\pi i/3}$, find $\omega^3$

A) $0$

B) $1$

C) $\omega$

D) $-1$

✓ Ans: B) 1

EAPCET 2023 Medium

Q30: If $\omega$ is cube root of unity, find $\omega^2$

A) $\bar{\omega}$

B) $1$

C) $\omega^{-1}$

D) All of above

✓ Ans: D) All of above

EAPCET 2022 Hard

Q31: Evaluate $\omega^{100} + \omega^{200}$ where $\omega$ is cube root of unity

A) $0$

B) $1$

C) $-1$

D) $2$

✓ Ans: B) 1

EAPCET 2020 Hard

Q32: If $\omega$ is cube root of unity, then $\omega^{3n} =$

A) $0$

B) $1$

C) $\omega^n$

D) $-1$

✓ Ans: B) 1

🎯 EAPCET: Locus & Geometry (4 Questions)

EAPCET 2024 Hard

Q33: If $|z – 1| = 1$, the locus of $z$ is

A) Circle with center 1, radius 1

B) Straight line

C) Parabola

D) Ellipse

✓ Ans: A) Circle with center 1, radius 1

EAPCET 2023 Hard

Q34: Find all complex numbers $z$ such that $z^2 = \bar{z}$

A) $0, 1, \omega, \omega^2$

B) $0, 1$

C) $0, 1, -1$

D) $0$

✓ Ans: B) 0, 1

EAPCET 2022 Hard

Q35: If $z_1 = 1 + 2i$ and $z_2 = 3 – 4i$, find $|z_1 + z_2| – |z_1 – z_2|$

A) $0$

B) $2$

C) $4$

D) $6$

✓ Ans: A) 0

EAPCET 2020 Hard

Q36: If $z_1 \cdot z_2 = |z_1| \cdot |z_2|$, then $z_1$ and $z_2$ are

A) Conjugates

B) In same direction (collinear)

C) Perpendicular

D) Equal

✓ Ans: B) In same direction (collinear)

Complex Numbers EAPCET & EAMCET PYQs

Complex Number (Introduction)

A complex number is an ordered pair of real numbers $(a, b)$ written as $z = a + ib$, where:

• $a$ is called the real part of $z$, denoted as $\text{Re}(z) = a$

• $b$ is called the imaginary part of $z$, denoted as $\text{Im}(z) = b$

• $i$ is the imaginary unit with the property $i^2 = -1$

Equality of Complex Numbers

Two complex numbers $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are equal if and only if:

$$z_1 = z_2 \iff a_1 = a_2 \text{ and } b_1 = b_2$$

Modulus (Absolute Value)

For a complex number $z = a + ib$, the modulus is: $|z| = \sqrt{a^2 + b^2}$

Conjugate of Complex Number

For $z = a + ib$, the conjugate is $\bar{z} = a – ib$

$$z \cdot \bar{z} = (a + ib)(a – ib) = a^2 + b^2 = |z|^2$$

Argand Plane

The complex number $z = a + ib$ is represented as point $(a, b)$ where the horizontal axis is the real axis and vertical axis is the imaginary axis.

Argument (Amplitude)

The argument of $z = a + ib \neq 0$ is the angle $\theta$ that the line from origin to $z$ makes with positive real axis:

$$\arg(z) = \theta \text{ where } a = |z|\cos\theta, \quad b = |z|\sin\theta$$

Principal Argument: $-\pi < \arg(z) \leq \pi$

Polar Form

Complex number $z = a + ib$ can be written as:

$$z = r(\cos\theta + i\sin\theta) = r \text{ cis } \theta$$

where $r = |z|$ and $\theta = \arg(z)$

Essential Formulas

De Moivre’s Theorem

If $z = r(\cos\theta + i\sin\theta)$, then

$$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$$

nth Roots of Unity

Roots of $z^n = 1$ are: $e^{2\pi i k/n}$, $k = 0,1,\ldots,n-1$

Cube Roots of Unity

$$1, \omega, \omega^2 \text{ where } \omega = e^{2\pi i/3}$$ $$1 + \omega + \omega^2 = 0$$ $$\omega^3 = 1$$

Modulus Properties

$$|z_1 \cdot z_2| = |z_1| \cdot |z_2|$$ $$\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$$

Argument Formula

$$\arg(z_1 \cdot z_2) = \arg(z_1) + \arg(z_2)$$ $$\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) – \arg(z_2)$$

Powers of $i$

$$i^1 = i, \quad i^2 = -1$$ $$i^3 = -i, \quad i^4 = 1$$ $$i^{4n+r} = i^r$$

TS EAMCET 2025 – Topic-Wise PYQs (24 Questions)

🌟 Topic 1: Roots of Unity & Cube Roots

EAMCET 2025 Medium

Q1: If $\omega \neq 1$ is a cube root of unity, then one root among the $7^{th}$ roots of $(1 + \omega)$ is

A) $1 + \omega$

B) $1 – \omega$

C) $\omega – \omega^2$

D) $\omega/(\omega – \omega^2)$

✓ Ans: A) $1 + \omega$

EAMCET 2025 Hard

Q5: One of the roots of $(x + 1)^4 + 81 = 0$ is

A) $3(1+i)/\sqrt{2}$

B) $-(3+\sqrt{2}i)/\sqrt{2}$

C) $-(3+\sqrt{2}i)/\sqrt{2}$

D) $-(3+3i)/\sqrt{2}$

✓ Ans: B)

EAMCET 2025 Medium

Q13: If $a$ is root of $x^2 – x + 1 = 0$, find $\sum_{k=1}^{12} (a^k + 1/a^k)^3$

A) $-32$

B) $32$

C) $0$

D) $16$

✓ Ans: C) $0$

EAMCET 2025 Medium

Q22: If $\omega$ is a cube root of unity and $x = \omega^2 – \omega + 2$, then $x$ satisfies

A) $x^2 – 4x + 7 = 0$

B) $x^2 + 4x + 7 = 0$

C) $x^2 – 2x + 4 = 0$

D) $x^2 + 2x + 4 = 0$

✓ Ans: A) $x^2 – 4x + 7 = 0$

⚡ Topic 2: Powers, Exponentials & De Moivre’s

EAMCET 2025 Hard

Q6: $(1 – i\sqrt{3})^{2025} =$

A) $2^{2025}$

B) $2^{2026}$

C) $-2^{2025}$

D) $-2^{2026}$

✓ Ans: C) $-2^{2025}$

EAMCET 2025 Medium

Q8: $\left(\frac{1+i}{1-i}\right)^{228} =$

A) $-4((1-i)/(1+i))^{226}$

B) $4((1-i)/(1+i))^{226}$

C) $((1-i)/(1+i))^{228}$

D) $-((1-i)/(1+i))^{228}$

✓ Ans: C) $((1-i)/(1+i))^{228}$

EAMCET 2025 Hard

Q10: $(\sqrt{3}+i)^{10} + (\sqrt{3}-i)^{10} =$

A) $1024\sqrt{3}$

B) $1024$

C) $2048$

D) $512\sqrt{3}$

✓ Ans: B) $1024$

EAMCET 2025 Hard

Q18: If $n \neq 3K$, then $(\sqrt{3}+i)^{2n} + (\sqrt{3}-i)^{2n} =$

A) $(-1)^n 2^{2n+1}$

B) $(-1)^{n+1}2^{2n+1}$

C) $(-1)^{n+1}2^{2n}$

D) $(-1)^n 2^{2n}$

✓ Ans: C) $(-1)^{n+1}2^{2n}$

📐 Topic 3: Modulus, Amplitude & Argument

EAMCET 2025 Medium

Q3: If $|Z_1 – 3 – 4i| = 5$ and $|Z_2| = 15$, find sum of max and min values of $|Z_1 – Z_2|$

A) $75$

B) $30$

C) $35$

D) $20$

✓ Ans: B) $30$

EAMCET 2025 Hard

Q9: Number of real values of $(-1 – \sqrt{3}i)^{1/4}$

A) $0$

B) $1$

C) $2$

D) $3$

✓ Ans: C) $2$

EAMCET 2025 Medium

Q12: Amplitude of $\frac{\sqrt{3}+0i-\sqrt{ai}}{(-1+i-i)}$

A) $\pi/2$

B) $\pi/3$

C) $-5\pi/12$

D) $-\pi/6$

✓ Ans: D) $-\pi/6$

EAMCET 2025 Hard

Q19: If $|Z| = 2, Z_1 = (2/e)e^{i\alpha}$ and $\theta = \text{amp}(Z)$, find $\frac{Z_1^2-Z_1}{Z_1+Z_1^2}$

A) $2^n i \tan(n\theta + n\alpha)$

B) $i\tan(n\theta – n\alpha)$

C) $i\tan(n\theta + n\alpha)$

D) $\tan(n\theta + n\alpha)$

✓ Ans: C) $i\tan(n\theta + n\alpha)$

🎯 Topic 4: Locus & Geometric Interpretation

EAMCET 2025 Medium

Q2: If $Z = r(\cos\theta + i\sin\theta)$ is solution of $x^3 = i$, find $i^8(\cos\theta + i\sin\theta)^2$

A) $\sqrt{3}/2 + i/2$

B) $1$

C) $-i$

D) $-\sqrt{3}/2 + i/2$

✓ Ans: C) $-i$

EAMCET 2025 Hard

Q7: If $z = x + iy$ and $\text{Amp}((z-3)/(z-2i)) = -\pi/2$, locus of P is

A) Circle $x^2 + y^2 – 3x – 2y = 0$

B) Arc of circle (excluding points)

C) Arc not containing origin

D) Circle not containing $(0,2)$

✓ Ans: B) Arc of circle x² + y² – 3x – 2y = 0 (excluding points)

EAMCET 2025 Hard

Q11: If $z = x + iy$ and $\text{Im}[(z-3)/(z+3i)] = 0$, locus of P is

A) $x^2 + y^2 – 3x + 3y = 0$

B) $2xy – 3x + 3y = 9$

C) $x – y – 3 = 0$, $(x,y) \neq (0,-3)$

D) $x + y + 3 = 0$, $(x,y) \neq (0,-2)$

✓ Ans: C) x – y – 3 = 0, (x,y) ≠ (0,-3)

EAMCET 2025 Hard

Q23: If $z = x + iy$ and $\text{Arg}((z-3i)/(z+3i)) = \pi/4$, locus of P is

A) $x^2 + y^2 – y – 6 = 0$

B) $x^2 + y^2 – x – y – 6 = 0$

C) $x^2 + y^2 + 5x – y – 6 = 0$, $(x,y) \neq (0,-)$

D) $x^2 + y^2 + x – y – 6 = 0$

✓ Ans: C) x² + y² + 5x – y – 6 = 0, (x,y) ≠ (0,-)

🔍 Topic 5: Complex Equations & Operations

EAMCET 2025 Medium

Q4: Eight vertices of regular octagon: $z_j = 1/(x_{j-2i})$, $j=1,2,…,8$. Radius of circumcircle is

A) $1/4$

B) $i/4$

C) $i$

D) $2$

✓ Ans: A) 1/4

EAMCET 2025 Hard

Q15: One value of $\sqrt{24 – 70i} + \sqrt{-24 + 70i}$ is

A) $2 + 12i$

B) $12 – 2i$

C) $-12 + 2i$

D) $-12 – 2i$

✓ Ans: D) -12 – 2i

EAMCET 2025 Medium

Q16: If $a = |\bar{a}|$; $b = |\bar{b}|$ then $(\bar{a}/a – \bar{a}/b)^2 =$

A) $((\bar{a}-\bar{b})/a^2b^2)^2$

B) $((\bar{a}-b)/ab)^2$

C) $((b\bar{a}-\bar{a}b)/ab)^2$

D) $((\bar{a}\bar{a}-\bar{b}\bar{b})/a^2b^2)^2$

✓ Ans: B) ((ā-b)/ab)²

EAMCET 2025 Hard

Q21: Product of all values of $(\sqrt{3} – i)^{1/3}$

A) $8$

B) $-8$

C) $8i$

D) $-8i$

✓ Ans: B) -8

EAMCET 2025 Hard

Q24: If $\frac{2i+3i}{i-2} – \frac{4i-3}{3i+4i} = x + iy$, then $3x + y =$

A) $4$

B) $-4$

C) $-2$

D) $2$

✓ Ans: B) -4

📊 Topic 6: Quadrants & Regions

EAMCET 2025 Medium

Q17: In Argand plane, no value of $\sqrt{1 – i\sqrt{3}}$ lies in

A) First quadrant

B) Second quadrant

C) Third quadrant

D) Fourth quadrant

✓ Ans: A) First quadrant

🎲 Topic 7: Mixed & Advanced Problems

EAMCET 2025 Hard

Q14: Set of $\theta$ such that $\frac{1-i\cos\theta}{1+2i\sin\theta}$ is purely imaginary

A) $\{n\pi + (-1)^n\pi/4, n \in \mathbb{Z}\}$

B) $\{n\pi/2 + (-1)^n\pi/4, n \in \mathbb{Z}\}$

C) $\{n\pi + (-1)^n\pi/5, n \in \mathbb{Z}\}$

D) $\{2n\pi \pm \pi/4, n \in \mathbb{Z}\}$

✓ Ans: B) {nπ/2 + (-1)^n π/4, n ∈ Z}

EAMCET 2025 Hard

Q20: $\omega$ is cube root of unity, $|Z-1| \leq 2$ and $|\omega Z – 1 – \omega^2| = r$ have no common solution. Then

A) $0 \leq r \leq 4$

B) $r = $ only $|i/\omega|$

C) $r > 4$

D) $1 < r < 2$

✓ Ans: C) r > 4

📚 EAPCET: Basic Operations (4 Questions)

EAPCET 2024 Easy

Q1: If $z_1 = 2 + 3i$ and $z_2 = 1 – 2i$, find $z_1 + z_2$

A) $3 + i$

B) $3 – i$

C) $-3 + i$

D) $3 + 5i$

✓ Ans: A) 3 + i

EAPCET 2023 Easy

Q2: Simplify $(2 + i)(3 – 2i)$

A) $6 – 2i$

B) $8 – i$

C) $8 + i$

D) $6 + 4i$

✓ Ans: B) 8 – i

EAPCET 2022 Medium

Q3: Compute $(1 + i)^2 + (1 – i)^2$

A) $0$

B) $2$

C) $-2$

D) $4i$

✓ Ans: A) 0

EAPCET 2024 Medium

Q4: Find $\frac{2 + 3i}{1 – i}$

A) $-1/2 + 5i/2$

B) $1/2 – 5i/2$

C) $2 + 3i$

D) $-1 + 5i$

✓ Ans: A) -1/2 + 5i/2

📐 EAPCET: Modulus & Conjugate (8 Questions)

EAPCET 2024 Easy

Q5: Find $|3 + 4i|$

A) $5$

B) $7$

C) $25$

D) $1$

✓ Ans: A) 5

EAPCET 2023 Easy

Q6: If $|z| = 2$ and $z = a + bi$, find $a^2 + b^2$

A) $2$

B) $4$

C) $8$

D) $16$

✓ Ans: B) 4

EAPCET 2024 Easy

Q7: Find conjugate of $5 – 3i$

A) $5 + 3i$

B) $-5 – 3i$

C) $3 + 5i$

D) $-5 + 3i$

✓ Ans: A) 5 + 3i

EAPCET 2023 Easy

Q8: If $z = 2 + i$, find $z \cdot \bar{z}$

A) $3$

B) $4$

C) $5$

D) $6$

✓ Ans: C) 5

EAPCET 2022 Medium

Q9: If $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$ where $|z_1| = 3, |z_2| = 4$

A) $7$

B) $12$

C) $1$

D) $7/3$

✓ Ans: B) 12

EAPCET 2021 Medium

Q10: If $|z + \bar{z}| = 10$, find Re(z)

A) $5$

B) $10$

C) $20$

D) $2.5$

✓ Ans: A) 5

EAPCET 2020 Medium

Q11: If $z – \bar{z} = 4i$, find Im(z)

A) $2$

B) $-2$

C) $4$

D) $8$

✓ Ans: A) 2

EAPCET 2019 Medium

Q12: If $|z_1/z_2| = 2$ and $|z_1| = 6$, find $|z_2|$

A) $3$

B) $12$

C) $2$

D) $4$

✓ Ans: A) 3

⚡ EAPCET: Powers of i & Arguments (8 Questions)

EAPCET 2024 Easy

Q13: Find $i^{37}$

A) $i$

B) $-i$

C) $1$

D) $-1$

✓ Ans: A) i

EAPCET 2023 Easy

Q14: Simplify $i^{100} + i^{101}$

A) $1 + i$

B) $1 – i$

C) $2$

D) $0$

✓ Ans: A) 1 + i

EAPCET 2024 Medium

Q15: Find $\arg(1 + i)$

A) $\pi/6$

B) $\pi/4$

C) $\pi/3$

D) $\pi/2$

✓ Ans: B) π/4

EAPCET 2023 Medium

Q16: Find $\arg(-1 – i)$

A) $3\pi/4$

B) $-3\pi/4$

C) $\pi/4$

D) $-\pi/4$

✓ Ans: B) -3π/4

EAPCET 2022 Hard

Q17: If $\arg(z) = \pi/3$ and $|z| = 2$, find $z$

A) $1 + i\sqrt{3}$

B) $2 + 2i$

C) $\sqrt{3} + i$

D) $2 + i\sqrt{3}$

✓ Ans: A) 1 + i√3

EAPCET 2021 Medium

Q18: Find $\sum_{k=0}^{3} i^k$

A) $0$

B) $1$

C) $i$

D) $1 + i$

✓ Ans: A) 0

EAPCET 2020 Medium

Q19: Find $i^{2018}$

A) $-1$

B) $1$

C) $i$

D) $-i$

✓ Ans: B) 1

EAPCET 2020 Medium

Q20: Find $\arg(i)$

A) $0$

B) $\pi/2$

C) $\pi$

D) $3\pi/2$

✓ Ans: B) π/2

🌀 EAPCET: Polar Form & Roots (8 Questions)

EAPCET 2024 Medium

Q21: Express $z = 1 + i$ in polar form

A) $\sqrt{2}$ cis $\pi/4$

B) $2$ cis $\pi/3$

C) $\sqrt{2}$ cis $\pi/6$

D) $2$ cis $\pi/4$

✓ Ans: A) √2 cis π/4

EAPCET 2023 Medium

Q22: Convert $2$ cis $\pi/3$ to rectangular form

A) $1 + i\sqrt{3}$

B) $\sqrt{3} + i$

C) $2 + 2i$

D) $1 – i\sqrt{3}$

✓ Ans: A) 1 + i√3

EAPCET 2021 Hard

Q23: Express $-2 – 2i$ in polar form

A) $2\sqrt{2}$ cis $5\pi/4$

B) $2$ cis $\pi$

C) $4$ cis $3\pi/4$

D) $2\sqrt{2}$ cis $3\pi/4$

✓ Ans: A) 2√2 cis 5π/4

EAPCET 2024 Hard

Q24: Find square roots of $3 + 4i$

A) $\pm(2 + i)$

B) $\pm(1 + 2i)$

C) $\pm(2 – i)$

D) $\pm(1 – 2i)$

✓ Ans: A) ±(2 + i)

EAPCET 2023 Hard

Q25: If $z^2 = -1$, find $z$

A) $\pm i$

B) $\pm 1$

C) $\pm(1 + i)$

D) No solution

✓ Ans: A) ±i

EAPCET 2022 Hard

Q26: Find $\sqrt{i}$

A) $\pm(1+i)/\sqrt{2}$

B) $\pm(1-i)/\sqrt{2}$

C) $\pm 1$

D) $\pm i$

✓ Ans: A) ±(1+i)/√2

EAPCET 2020 Hard

Q27: Find cube roots of unity

A) $1, \omega, \omega^2$

B) $1, i, -i$

C) $1, 1, 1$

D) $-1, i, -i$

✓ Ans: A) 1, ω, ω²

EAPCET 2019 Medium

Q28: If $\omega$ is cube root of unity, find $1 + \omega + \omega^2$

A) $0$

B) $1$

C) $3$

D) $\omega$

✓ Ans: A) 0

🔮 EAPCET: Cube Roots of Unity Properties (4 Questions)

EAPCET 2024 Medium

Q29: If $\omega = e^{2\pi i/3}$, find $\omega^3$

A) $0$

B) $1$

C) $\omega$

D) $-1$

✓ Ans: B) 1

EAPCET 2023 Medium

Q30: If $\omega$ is cube root of unity, find $\omega^2$

A) $\bar{\omega}$

B) $1$

C) $\omega^{-1}$

D) All of above

✓ Ans: D) All of above

EAPCET 2022 Hard

Q31: Evaluate $\omega^{100} + \omega^{200}$ where $\omega$ is cube root of unity

A) $0$

B) $1$

C) $-1$

D) $2$

✓ Ans: B) 1

EAPCET 2020 Hard

Q32: If $\omega$ is cube root of unity, then $\omega^{3n} =$

A) $0$

B) $1$

C) $\omega^n$

D) $-1$

✓ Ans: B) 1

🎯 EAPCET: Locus & Geometry (4 Questions)

EAPCET 2024 Hard

Q33: If $|z – 1| = 1$, the locus of $z$ is

A) Circle with center 1, radius 1

B) Straight line

C) Parabola

D) Ellipse

✓ Ans: A) Circle with center 1, radius 1

EAPCET 2023 Hard

Q34: Find all complex numbers $z$ such that $z^2 = \bar{z}$

A) $0, 1, \omega, \omega^2$

B) $0, 1$

C) $0, 1, -1$

D) $0$

✓ Ans: B) 0, 1

EAPCET 2022 Hard

Q35: If $z_1 = 1 + 2i$ and $z_2 = 3 – 4i$, find $|z_1 + z_2| – |z_1 – z_2|$

A) $0$

B) $2$

C) $4$

D) $6$

✓ Ans: A) 0

EAPCET 2020 Hard

Q36: If $z_1 \cdot z_2 = |z_1| \cdot |z_2|$, then $z_1$ and $z_2$ are

A) Conjugates

B) In same direction (collinear)

C) Perpendicular

D) Equal

✓ Ans: B) In same direction (collinear)