128 practice questions on Complex Numbers and Quadratic Equations , sorted Easy → Hard. Try each one first, then open its answer page for the worked explanation. Want the full theory first? Read the Complex Numbers and Quadratic Equations notes .
Argand Plane: z = a + ib Re Im z = a+ib a (real part) b (imaginary part) θ = arg(z) |z| = length of the vector OZ = √(a²+b²); θ = angle OZ makes with the positive real axis A complex number z = a+ib is plotted as a point (or vector from the origin) on the Argand plane, with the real part along the horizontal axis and the imaginary part along the vertical; its modulus |z| is the vector's length, and its argument θ is the angle from the positive real axis.
Easy - 40 questions Q1.
What is the standard form of a quadratic equation?
A ax + b = 0B ax² + bx + c = 0C ax³ + bx + c = 0D a/x + b = 0Show answer & explanation →
Q2.
Solve: x² - 4 = 0
A x = 2, just one of the two valid rootsB x = -2, just the other valid root aloneC x = ±2D x = ±4, doubling the actual correct valuesShow answer & explanation →
Q4.
If D = 0, the roots of the quadratic equation are:
A Two distinct real rootsB One repeated real rootC Two imaginary rootsD No rootsShow answer & explanation →
Q12.
Quadratic equation with roots 3 and 5 is:
A x² - 8x + 15 = 0B x² + 8x + 15 = 0C x² - 15x + 8 = 0D x² + 15 = 0Show answer & explanation →
Q17.
Which of these is NOT a method to solve quadratic equations?
A FactorizationB Quadratic formulaC Completing the squareD Long division onlyShow answer & explanation →
Q20.
For the equation 4x² - 12x + 9 = 0, the nature of roots is:
A Two distinct real rootsB Two equal real rootsC No real rootsD One positive one negativeShow answer & explanation →
Medium - 40 questions Q43.
If roots are 2+√3 and 2-√3, form the equation.
A x² - 4x + 1 = 0B x² + 4x + 1 = 0C x² - 4x - 1 = 0D x² + 4x - 1 = 0Show answer & explanation →
Q49.
For what value of k does kx² + 6x + 1 = 0 have real roots?
A k > 9, obtained by reversing the discriminant inequalityB k less than or equal to 9C k < 0, requiring the leading coefficient to be negativeD k = 9 only, taken from setting the discriminant to zeroShow answer & explanation →
Q50.
If alpha and beta are roots of 2x² + x - 6 = 0, find alpha/beta + beta/alpha.
A 25/24B -25/24C 25/12D -13/12Show answer & explanation →
Q58.
If 1/alpha and 1/beta are roots of x² + px + q = 0, then alpha, beta are roots of:
A x² + px + q = 0B qx² + px + 1 = 0C px² + qx + 1 = 0D x² - px + q = 0Show answer & explanation →
Q60.
For a quadratic with negative discriminant, the parabola:
A Crosses x-axis twiceB Touches x-axis onceC Never crosses x-axisD Is undefinedShow answer & explanation →
Q68.
The locus of |z - 1| = 2 in the Argand plane is:
A A circle of radius 1 centred at (2,0)B A circle of radius 2 centred at (1,0)C A lineD A parabolaShow answer & explanation →
Q70.
De Moivre theorem: (cos θ + i sin θ)ⁿ =
A n cos θ + ni sin θB cos θⁿ + i sin θⁿC cos(θ/n) + i sin(θ/n)D cos nθ + i sin nθShow answer & explanation →
Q79.
The polar form of z = -i is:
A 1(cos 90° + i sin 90°)B 1(cos 270° + i sin 270°)C 1(cos 180° + i sin 180°)D cos(-90°) + i sin(-90°)Show answer & explanation →
Hard - 48 questions Q81.
If alpha and beta are roots of ax² + bx + c = 0, then alpha³ + beta³ =
A (3abc - b³)/a³B -(b³ - 3abc)/a³C (b³ - 3abc)/a³D 3abc/a³Show answer & explanation →
Q82.
The equation x² - 2px + q = 0 has two real roots r, s. For r² + s² to be minimized, what condition on p,q?
A p = 0B p² = qC q must be maximumD p must equal qShow answer & explanation →
Q87.
Both roots of x² + ax + b = 0 are real and exceed 2. Which must hold?
A a > -4 and b > 4B a < -4 and b > 4C a > 4 and b > 4D a < 4 and b < -4Show answer & explanation →
Q88.
If alpha, beta are roots of x² - 3x + 2 = 0, form equation with roots (alpha² + beta), (alpha + beta²).
A x² - 8x + 12 = 0B x² + 8x + 12 = 0C x² - 7x + 12 = 0D x² - 8x + 15 = 0Show answer & explanation →
Q91.
If alpha, beta are roots of x² - px + q = 0, find alpha<sup>4</sup> + beta<sup>4.</sup>
A (p²-2q)² - 2q²B p<sup>4</sup> - 4p²q + 2q²C (p²-q)² - q²D Both A and BShow answer & explanation →
Q92.
The equation x² - 2ax + a² - 1 = 0 has roots in (-2, 4). Range of a:
A -1 < a < 3B 1 < a < 3C -1 < a < 5D 0 < a < 4Show answer & explanation →
Q95.
If p, q are roots of x² + px + q = 0 (p not= 0), then:
A p = 1, q = -2B p = -2, q = 1C p = 1, q = 1D p = q = 0Show answer & explanation →
Q100.
The cube roots of unity ω satisfy x³ = 1. The non-real roots are:
A (−1 ± i√3)/2B (1 ± i√3)/2C (−1 ± i√3)/3D ±iShow answer & explanation →
Q103.
z satisfies |z - 3| = |z + 3|. The locus is:
A Circle of radius 3B Real axisC Imaginary axis (y-axis)D Line y = xShow answer & explanation →
Q113.
Use De Moivre theorem to find cos 3θ in terms of cos θ.
A 3cosθ - 4cos³θB cos³θ - sin²θC 3cos²θ - 1D 4cos³θ - 3cosθShow answer & explanation →
Q116.
If z = (√3 + i)/(1 - i), find |z| and arg(z).
A |z|=√2, arg=5π/12B |z|=√2, arg=7π/12C |z|=2, arg=5π/12D |z|=1, arg=5π/12Show answer & explanation →
Q121.
If z is a complex number with |z| = 1 and z ≠ ±1, then z/(1 + z²) is:
A Purely imaginaryB Purely realC Of modulus 1D Equal to the conjugate of zShow answer & explanation →
Q126.
The locus of z for which arg((z − 1)/(z + 1)) = π/2 is:
A The line x = 0B An arc of the circle x² + y² = 1C A parabolaD The entire real axisShow answer & explanation →
Q127.
The set of real k for which x² − 2kx + (k² + k − 5) = 0 has both roots less than 5 is:
A k > 4B k < 4C k = 4D all real kShow answer & explanation →
Q128.
x² + 2(k + 1)x + (9k − 5) = 0 has both roots negative when:
A k > 6 onlyB 5/9 < k ≤ 1 or k ≥ 6C k < 5/9D −1 < k < 5/9Show answer & explanation →