Take timed practice tests on Complex Numbers for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
Ready
1. The value of i⁴⁰ is:
Explanation: Powers of i cycle with period 4: i¹=i, i²=−1, i³=−i, i⁴=1. Since 40 is divisible by 4, i⁴⁰ = 1.
2. If z = 3 + 4i, then |z| is:
Explanation: |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
3. The conjugate of z = 2 − 5i is:
Explanation: The conjugate of a + bi is a − bi. So conjugate of 2 − 5i is 2 + 5i.
4. The argument (principal value) of the complex number −1 + i is:
Explanation: −1 + i lies in the second quadrant. tan θ = 1/(−1) → reference angle = π/4. Principal argument = π − π/4 = 3π/4.
5. If z₁ = 2 + 3i and z₂ = 1 − i, then z₁ + z₂ is:
Explanation: Add real and imaginary parts: (2+1) + (3+(−1))i = 3 + 2i.
6. (1 + i)² equals:
Explanation: (1+i)² = 1 + 2i + i² = 1 + 2i − 1 = 2i.
7. 1/(1 + i) in standard form is:
Explanation: Multiply numerator and denominator by conjugate: 1/(1+i) × (1−i)/(1−i) = (1−i)/(1−i²) = (1−i)/2.
8. The modulus and argument of z = −√3 + i are:
Explanation: |z| = √(3+1) = 2. The point (−√3, 1) is in Q2. tan θ_ref = 1/√3 → θ_ref = π/6. Argument = π − π/6 = 5π/6.
9. Using De Moivre's theorem, (cos 60° + i sin 60°)³ equals:
Explanation: By De Moivre: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Here: cos 180° + i sin 180° = −1 + 0i = −1.
10. The cube roots of unity are 1, ω, ω². Then ω² + ω + 1 equals:
Explanation: The three cube roots of unity satisfy x³ = 1, i.e., x³ − 1 = 0, i.e., (x−1)(x²+x+1) = 0. So ω and ω² satisfy x²+x+1 = 0, meaning 1 + ω + ω² = 0.
11. If z = (1+i)/(1−i), then z⁴ is:
Explanation: z = (1+i)/(1−i) × (1+i)/(1+i) = (1+i)²/2 = 2i/2 = i. So z⁴ = i⁴ = 1.
12. If |z − 1| = |z + 1|, then z lies on:
Explanation: |z−1| = |z+1| means z is equidistant from 1 and −1 on the real axis. The locus is the perpendicular bisector of the segment joining 1 and −1, which is the imaginary axis (x = 0).
13. If arg(z) = π/3, then arg(z̄) is:
Explanation: If z = r(cos θ + i sin θ), then z̄ = r(cos θ − i sin θ) = r(cos(−θ) + i sin(−θ)). So arg(z̄) = −arg(z) = −π/3.
14. For complex numbers z₁ and z₂, |z₁ + z₂| ≤ |z₁| + |z₂| is known as:
Explanation: The triangle inequality for complex numbers states |z₁ + z₂| ≤ |z₁| + |z₂|, analogous to the triangle inequality in vectors.
15. If z² + z + 1 = 0, then z equals:
Explanation: z² + z + 1 = 0 → z = (−1 ± √(1−4))/2 = (−1 ± i√3)/2. These are precisely the non-real cube roots of unity ω and ω².
16. The complex number z satisfying |z − 2| = |z − 2i| lies on:
Explanation: Let z = x + iy. |z−2|² = (x−2)² + y². |z−2i|² = x² + (y−2)². Setting equal: (x−2)² + y² = x² + (y−2)² → −4x + 4 = −4y + 4 → x = y. So z lies on y = x.
17. If ω is a primitive cube root of unity, then (1 + ω − ω²)³ is:
Explanation: Since 1 + ω + ω² = 0, we have 1 + ω = −ω². So 1 + ω − ω² = −ω² − ω² = −2ω². Then (−2ω²)³ = −8ω⁶ = −8(ω³)² = −8(1)² = −8.
18. If z₁ = 2(cos 30° + i sin 30°) and z₂ = 3(cos 60° + i sin 60°), then |z₁z₂| is:
Explanation: |z₁z₂| = |z₁||z₂| = 2 × 3 = 6. The modulus of a product equals the product of moduli.
19. The fourth roots of 16 are:
Explanation: We need z⁴ = 16. |z|⁴ = 16 → |z| = 2. Arguments: 2kπ/4 for k = 0,1,2,3 → 0, π/2, π, 3π/2. So roots: 2, 2i, −2, −2i.
20. If |z| = 1 and z ≠ ±1, then all values of z/(1 − z²) lie on:
Explanation: Let z = cos θ + i sin θ (|z|=1). Then 1 − z² = 1 − cos 2θ − i sin 2θ = 2sin²θ − 2i sinθ cosθ = 2sinθ(sinθ − i cosθ). And z/(1−z²) = (cosθ + i sinθ)/(2sinθ(sinθ − i cosθ)). Numerator × conjugate of denominator divided by |denominator|²: this yields a purely imaginary number. So values lie on the imaginary axis.
21. If z = x + iy satisfies |z + 1| = 2|z − 1|, then the locus is:
Explanation: |z+1|² = 4|z−1|². (x+1)² + y² = 4[(x−1)² + y²]. x² + 2x + 1 + y² = 4x² − 8x + 4 + 4y². 3x² + 3y² − 10x + 3 = 0 → x² + y² − (10/3)x + 1 = 0. This is a circle (coefficient of x² = coefficient of y²).
22. The sum of a complex number and its conjugate is:
Explanation: If z = a + bi, then z + z̄ = (a + bi) + (a − bi) = 2a, which is always real.
23. If z = cos θ + i sin θ, then z + 1/z equals:
Explanation: 1/z = cos(−θ) + i sin(−θ) = cos θ − i sin θ. So z + 1/z = 2cos θ. (This is Euler's formula result.)
24. If α and β are roots of z² − z + 1 = 0, then α¹⁰¹ + β¹⁰¹ is:
Explanation: Roots of z²−z+1=0 are z = (1±i√3)/2 = e^(±iπ/3), which are primitive 6th roots of unity ω and ω̄ where ω = e^(iπ/3). α¹⁰¹ = e^(101iπ/3). 101 = 6×16 + 5, so e^(101iπ/3) = e^(5iπ/3) = cos(300°) + i sin(300°) = 1/2 − i√3/2. Similarly β¹⁰¹ = 1/2 + i√3/2. Sum = 1. But checking: roots are e^(±iπ/3), period 6. 101 mod 6 = 5. α⁵ + β⁵ = 2cos(5π/3) = 2(1/2) = 1. So answer is 1.