Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations – Extra Questions with Answers

Powers of i cycle through just four values, and these questions use that pattern alongside modulus and conjugate rules to work through complex number arithmetic.

Last Updated: September 23, 2026

How to Approach the HOTS Questions in Complex Numbers and Quadratic Equations

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Very Short Answer Questions (1 mark)

Q1. Simplify i⁵ (i to the power 9).
Ans: i⁵ = i⁴×i = 1×i = i.

Q2. Find the modulus of 3+4i.
Ans: √(3²+4²) = √25 = 5.

Q3. Add (2+3i) and (4−i).
Ans: 6+2i.

Q4. What is the discriminant of x²+x+1=0?
Ans: 1²−4(1)(1) = −3.

Q5. Is the number 5 (with no i term) a complex number?
Ans: Yes, since it can be written as 5+0i.

Short Answer Questions (2–3 marks)

Q6. Multiply (2+3i) and (1−2i), showing your work.
Ans: (2+3i)(1−2i) = 2(1)+2(−2i)+3i(1)+3i(−2i) = 2−4i+3i−6i² = 2−i−6(−1) = 2−i+6 = 8−i.

Q7. Solve x²+4=0 for x, expressing the answer in terms of i.
Ans: x²=−4, so x=±√(−4)=±2i.

Q8. Find the roots of x²−2x+5=0 using the quadratic formula.
Ans: D = (−2)²−4(1)(5) = 4−20 = −16. x = [2 ± √(−16)]/2 = [2 ± 4i]/2 = 1 ± 2i.

Higher-Order Thinking / Application Questions

Q9. If z = 3+4i, verify that z × z̄ = |z|², showing all steps.
Ans: z̄ = 3−4i. z×z̄ = (3+4i)(3−4i) = 3²−(4i)² = 9−16i² = 9−16(−1) = 9+16 = 25. |z|² = (√(3²+4²))² = (√25)² = 25. Both equal 25, confirming z×z̄ = |z|².

Q10. A quadratic equation has one root 2+3i. Using the fact that complex roots of a real-coefficient quadratic always occur in conjugate pairs, find the other root and construct the original quadratic equation.
Ans: The other root must be the conjugate, 2−3i. Sum of roots = (2+3i)+(2−3i) = 4. Product of roots = (2+3i)(2−3i) = 4−9i² = 4+9 = 13. The quadratic equation (in the form x²−(sum)x+(product)=0) is x²−4x+13=0.

Frequently Asked Questions

How would you find the modulus and argument of the complex number 1 + i root3?
The modulus is root of (1+3) = 2, and the argument is arctan of root3 over 1, which equals 60 degrees or pi over 3 radians.

If the roots of a quadratic equation are 2+3i and 2-3i, how would you form the equation?
Sum of roots = 4 and product of roots = 4+9 = 13, so the equation is x squared minus 4x plus 13 = 0.

Written by Satish

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