Complex Numbers — Chapter Test | Class 11 IMO

Q1

An architect in Jaipur plots a garden's four corners at 1+i, -1+i, -1-i, and 1-i (in meters) on a complex grid. What is the area of the garden?

The points form a square centered at the origin with a side length of 2 meters. Area = 2 × 2 = 4 sq meters.

Q2

The distance of the point 3 + 4i from origin is:

Distance equals modulus.

Q3

In a logical puzzle, a word is coded by mapping its alphabetical rank n to iⁿ. What is the sum of the complex code for the word 'BAD' (where B=2, A=1, D=4)?

B=2 -> i², A=1 -> i¹, D=4 -> i⁴. Sum = -1 + i + 1 = i.

Q4

The argument of 1 + i is:

tan θ = 1.

Q5

If z = 2 − 2i, then arg(z) is:

Point is in fourth quadrant.

Q6

What is the principal argument of any positive real number?

Positive real numbers lie on the positive x-axis, which corresponds to an angle of 0 radians.

Q7

The point representing −2 − 5i lies in:

Both coordinates are negative.

Q8

A shopkeeper's profit function is given by P(x) = |3 + 4i|x - 20 (in ₹), where x is the number of items sold. If he sells 10 items, his profit is:

|3 + 4i| = 5. So P(x) = 5x - 20. For x = 10, P(10) = 5(10) - 20 = 50 - 20 = 30.

Q9

Find the principal argument of -1 + i√3.

The point is in the 2nd quadrant. α = tan⁻¹(√3/1) = π/3. Principal argument θ = π - α = π - π/3 = 2π/3.

Q10

What is the principal argument of any negative real number?

Negative real numbers lie on the negative x-axis. By convention, the principal argument for this is π.

Q11

What is the minimum value of |z - 1| + |z - 5| for any complex number z?

By the triangle inequality, |z - 1| + |5 - z| ≥ |(z - 1) + (5 - z)| = |4| = 4. The minimum distance occurs on the line segment between 1 and 5.

Q12

Calculate the value of i⁹ + i¹⁹.

i⁹ = i and i¹⁹ = i³. So, i + (-i) = 0.

Q13

Find the value of √(-16) × √(-25).

√(-16) = 4i and √(-25) = 5i. Multiplying them gives (4i)(5i) = 20i² = 20(-1) = -20.

Q14

Find the square roots of -i.

Let √(-i) = x + iy. x² - y² = 0, 2xy = -1. x² + y² = 1. Solving gives x = ±1/√2 and y = ∓1/√2. Roots are ±(1-i)/√2.

Q15

Find the multiplicative inverse of 3 - 4i.

Inverse = 1/(3-4i). Multiply numerator and denominator by 3+4i to get (3+4i)/(9+16) = (3+4i)/25.