NET's complex number questions ask for powers of i, conjugate arithmetic, the modulus, or a switch between Cartesian and polar form. Most questions need one identity and ten seconds of arithmetic. The rest need De Moivre's theorem, which turns a cube into a multiplication of angles.
Key facts
- Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
- Complex Numbers is unit 1 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
- Our chapter map estimates about 6 MCQs from this chapter. NUST does not publish chapter counts
Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map
What the chapter covers
The chapter introduces the imaginary unit i, with i^2 = -1, and the complex number z = a + bi. You learn to add, multiply and divide complex numbers, plot them on the Argand plane, take conjugates and moduli, and convert between Cartesian form a + bi and polar form r(cos θ + i sin θ).
It is the first unit of the Class 11 book, so many students studied it a long time ago and have forgotten the shortcuts. That makes it a good chapter to revise late: the content is small and the marks come back fast.
Our practice set for this chapter also carries questions on the properties of real numbers and inequalities. Those are definition questions and take one read.
Formulas
Powers of i repeat every 4:
i^1 = i i^2 = -1 i^3 = -i i^4 = 1
i^n = i^(n mod 4) works for negative n too
For z = a + bi:
conjugate z̄ = a - bi
modulus |z| = sqrt(a^2 + b^2)
z z̄ = |z|^2
z + z̄ = 2a z - z̄ = 2bi
1/z = z̄ / |z|^2
|z1 z2| = |z1| |z2|
|z1 / z2| = |z1| / |z2|
Polar form and De Moivre:
z = r(cos θ + i sin θ), r = |z|, tan θ = b/a (check the quadrant)
(cos θ + i sin θ)^n = cos nθ + i sin nθ
Results worth memorising because they appear inside other questions:
(1 + i)^2 = 2i (1 - i)^2 = -2i
(1 + i)/(1 - i) = i (1 - i)/(1 + i) = -i
square roots of i: ±(1 + i)/sqrt(2)
Worked MCQs
Five questions taken from the types in our practice bank. We checked each by a second method.
1. Find i^-27 + i^24.
-27 = 4(-7) + 1, so i^-27 = i^1 = i
24 = 4(6), so i^24 = 1
answer: 1 + i
2. Find |(1 + i)/(3 - i)|.
Do not divide. Take the moduli separately:
|1 + i| = sqrt(2), |3 - i| = sqrt(10)
sqrt(2)/sqrt(10) = 1/sqrt(5) = sqrt(5)/5
3. Find the smallest positive k with ((1 + i)/(1 - i))^k = 1.
The fraction is i, from the list above. i^k = 1 first happens at k = 4.
4. Find the imaginary part of (2 + 3i)^3.
(2 + 3i)^2 = 4 + 12i - 9 = -5 + 12i
(-5 + 12i)(2 + 3i) = -10 - 15i + 24i + 36i^2 = -46 + 9i
The imaginary part is 9. Strictly, it is the real number 9, not 9i. Some options write 9i, so read the question and the options together.
5. Find the real part of (sqrt(3) + i)^3.
Expanding works but De Moivre is faster. Here r = 2 and θ = 30°:
(sqrt(3) + i)^3 = 2^3 (cos 90° + i sin 90°) = 8i
The real part is 0. Check by expansion: (sqrt(3) + i)^2 = 2 + 2 sqrt(3) i, and multiplying by sqrt(3) + i gives 2 sqrt(3) + 2i + 6i - 2 sqrt(3) = 8i.
Common traps
- Imaginary part includes i. It does not. For
a + bi, the imaginary part is b. - Argument in the wrong quadrant. For
-1 - i,tan θ = 1, but θ is -135° (or 225°), not 45°. Plot the point before you trust the calculator. - Negative powers of i.
i^-1 = -i, not i. Usen mod 4with a positive remainder: -1 becomes 3. - Modulus of a sum.
|z1 + z2|is not|z1| + |z2|in general. They are equal only when the two points lie on the same ray from the origin, as with1 + iand2 + 2i. - Pure imaginary vs real.
z + z̄ = 0means z is pure imaginary.z - z̄ = 0means z is real. The options swap them.
How to prepare
Write the powers of i, the conjugate rules and the three memorised results on one card. Do a 20-question timed set. Then spend one session on polar form and De Moivre, since that is where the longer questions come from.
Complex numbers come back in matrices and determinants, where some determinants contain i, and in trigonometric identities through De Moivre. The NET mathematics guide puts all the chapters in order.
FAQ
How do I find a large power of i quickly?
Divide the power by 4 and keep the remainder. Remainder 0 gives 1, 1 gives i, 2 gives -1, and 3 gives -i.
What is the modulus of a complex number?
For a + bi it is sqrt(a^2 + b^2), the distance of the point from the origin on the Argand plane.
Is De Moivre's theorem in the NET syllabus?
The unit covers polar form, and De Moivre's theorem is how you raise a polar number to a power. Our practice bank has questions on it, so learn it for integer powers.
