Complex Numbers and Quadratic Equations Questions for NDA

Basics of Complex Numbers

A complex number is written as z = a + ib, where a is the real part and b is the imaginary part, and i is the imaginary unit defined as square root of -1. Think of it as a point (a, b) on a 2D plane called the Argand plane, where the x-axis is real and y-axis is imaginary. Two complex numbers are equal only if both real parts are equal AND both imaginary parts are equal.

• i = sqrt(-1), i squared = -1, i cubed = -i, i to power 4 = 1. After every 4 powers, the cycle repeats.
• Real part of z = Re(z) = a. Imaginary part = Im(z) = b.
• Conjugate of z = a + ib is z* = a - ib. Conjugate flips the sign of imaginary part only.
• |z| = modulus = sqrt(a squared + b squared). This is the distance from origin on Argand plane.
• z times z* = |z| squared = a squared + b squared. Very useful in division.

Argument and Polar Form

The argument (angle) of a complex number z = a + ib is the angle theta that the line from origin to point (a, b) makes with the positive x-axis. The polar form rewrites z as r(cos theta + i sin theta) where r = |z|. NDA frequently asks you to convert forms or find the argument of expressions like (1+i)/(1-i).

• arg(z) = arctan(b/a). But adjust the quadrant based on signs of a and b.
• Principal argument lies between -pi and pi (inclusive).
• Polar form: z = r(cos theta + i sin theta), also written as r cis theta.
• arg(z1 times z2) = arg(z1) + arg(z2). arg(z1/z2) = arg(z1) - arg(z2).
• Modulus of product: |z1 times z2| = |z1| times |z2|.

Cube Roots of Unity

The three cube roots of 1 are: 1, omega, and omega squared, where omega = (-1 + i*sqrt(3))/2. These appear frequently in NDA questions involving algebraic identities and factorization. The two properties of omega are so important that you must memorize them the night before the exam.

• 1 + omega + omega squared = 0. This is the golden identity.
• omega cubed = 1. Powers of omega repeat with period 3.
• omega and omega squared are conjugates of each other.
• omega squared = (-1 - i*sqrt(3))/2.
• Common use: factoring a cubed + b cubed + c cubed - 3abc = (a+b+c)(a + b*omega + c*omega^2)(a + b*omega^2 + c*omega).

Quadratic Equations and Nature of Roots

A quadratic equation is ax^2 + bx + c = 0 with a not equal to 0. The discriminant D = b^2 - 4ac decides the nature of roots. NDA tests this both directly (find D) and indirectly (if roots are equal, find k). Always remember: complex roots always appear in conjugate pairs when coefficients are real.

• D > 0: two distinct real roots. D = 0: two equal real roots. D < 0: two complex conjugate roots.
• Sum of roots = -b/a. Product of roots = c/a. Memorize as S = -b/a, P = c/a.
• If alpha and beta are roots, equation is: x^2 - (alpha+beta)x + alpha*beta = 0.
• For real coefficients, complex roots come in pairs: if p+iq is a root, then p-iq is also a root.
• If one root is 2+3i and coefficients are real, the other root must be 2-3i.

• Forgetting the quadrant adjustment when finding argument: arctan(b/a) always gives angle in first or fourth quadrant only. If a < 0, add or subtract pi to get correct quadrant.
• Mixing up 1 + omega + omega^2 = 0 with omega^3 = 0. omega^3 equals 1, not 0. The sum of all three cube roots equals 0.
• In division of complex numbers, students multiply only the numerator by conjugate and forget to divide by |denominator|^2. Always multiply both top and bottom by conjugate of denominator.
• When D < 0 and coefficients are real, assuming roots are unrelated. They are always conjugates. NDA uses this to set up equations for unknown coefficients.
• Confusing modulus with argument. Modulus is a positive real number (magnitude). Argument is an angle in radians. Do not mix them up in polar form questions.

• i-cycle mnemonic: I In Negative Isolation — i, -1, -i, 1. Four steps, then repeat. Remainder on dividing power by 4 gives your answer.
• Omega rule: ONE plus OMEGA plus OMEGA-SQUARE = ZERO. Think of it as three brothers whose sum is always zero, and each cubed becomes ONE.
• For Sum and Product of roots, remember SP formula: S = -b/a (negative of middle over first), P = c/a (last over first). Mnemonic: NMA — Negative Middle over A, Last over A.
• Argand plane = Address plane. Real part is the street number (x), imaginary part is the floor number (y). Modulus is your straight-line distance from the origin building.

• NDA typically asks 3 to 4 questions from this combined topic. Cube roots of unity and argument/modulus questions appear almost every year — prioritize these over obscure subtopics.
• Questions on omega are very fast to solve if you remember 1+omega+omega^2=0. A typical question gives a big expression like (1+omega^2)^5 and expects simplification — the answer is almost always -1 or 1 or omega.
• Quadratic equations in NDA often ask: given one complex root, find the equation or a coefficient. Always use the conjugate-pair rule for real coefficients — it cuts the work in half.
• Polar form and De Moivre questions are medium difficulty. If time is short in the exam, skip De Moivre and return to it — modulus and argument questions are faster and more frequent.
• Do not waste time deriving the quadratic formula in the exam. Memorize D = b^2 - 4ac and the sum/product formulas. Most NDA quadratic questions can be solved with these alone without finding actual roots.