Complex Numbers and Quadratic Equations

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Complex Numbers and Quadratic Equations — solved practice questions

8 JEE Mathematics questions with step-by-step solutions. Attempt each, then reveal the answer.

1. Q1easy

   The value of i^(2023), where i = sqrt(−1), is:

   • Ai
   • B−i
   • C1
   • D−1
   Show answer & solution

   Correct answer: (B) −i

   Powers of i cycle with period 4. Since 2023 = 4×505 + 3, i^2023 = i^3 = −i.

2. Q2easy

   The modulus of the complex number z = 3 − 4i is:

   • A5
   • B7
   • C1
   • D25
   Show answer & solution

   Correct answer: (A) 5

   |z| = sqrt(a^2 + b^2) = sqrt(3^2 + (−4)^2) = sqrt(9 + 16) = sqrt(25) = 5.

3. Q3easy

   If the sum of the roots of the quadratic equation 2x^2 − 5x + 3 = 0 is S and the product is P, then S + P equals:

   • A1
   • B5/2
   • C4
   • D3/2
   Show answer & solution

   Correct answer: (C) 4

   For ax^2 + bx + c = 0, S = −b/a = 5/2 and P = c/a = 3/2. Hence S + P = 5/2 + 3/2 = 4.

4. Q4medium

   The value of (1 + i)^8, where i = sqrt(−1), is:

   • A16i
   • B−16
   • C8
   • D16
   Show answer & solution

   Correct answer: (D) 16

   (1 + i)^2 = 2i, so (1 + i)^8 = (2i)^4 = 16 · i^4 = 16.

5. Q5medium

   If ω is a non-real cube root of unity, then the value of (1 − ω + ω^2)^3 is:

   • A−8
   • B8
   • C−8ω
   • D1
   Show answer & solution

   Correct answer: (A) −8

   Since 1 + ω + ω^2 = 0, we have 1 + ω^2 = −ω, so 1 − ω + ω^2 = −ω − ω = −2ω. Cubing: (−2ω)^3 = −8ω^3 = −8.

6. Q6medium

   The square roots of the complex number −3 + 4i are:

   • A±(2 + i)
   • B±(1 + 2i)
   • C±(2 − i)
   • D±(1 − 2i)
   Show answer & solution

   Correct answer: (B) ±(1 + 2i)

   Seek a + bi with (a + bi)^2 = −3 + 4i: a^2 − b^2 = −3 and 2ab = 4. Solving gives a = 1, b = 2, so the roots are ±(1 + 2i).

7. Q7medium

   If α and β are the roots of x^2 − 6x + 8 = 0, then the value of α^2 + β^2 is:

   • A36
   • B16
   • C20
   • D52
   Show answer & solution

   Correct answer: (C) 20

   α + β = 6 and αβ = 8. Then α^2 + β^2 = (α + β)^2 − 2αβ = 36 − 16 = 20.

8. Q8medium

   The values of k for which the quadratic equation x^2 − kx + 9 = 0 has equal (repeated) real roots are:

   • Ak = 6 only
   • Bk = 3
   • Ck = 9
   • Dk = ±6
   Show answer & solution

   Correct answer: (D) k = ±6

   Equal roots require discriminant zero: k^2 − 4·9 = 0, so k^2 = 36, giving k = ±6.

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