Class 10 Mathematics Unit 1: Complex Numbers – Full Unit Test

Full Unit Test | Unit 1: Complex Numbers | Objective Type

This Class 10 Mathematics Full Unit Test is based on Unit 1: Complex Numbers. It contains 15 objective-type multiple-choice questions covering important concepts including complex numbers, conjugates, modulus, additive identity, additive inverse and powers of i.

Test Information

• Class: 10th
• Subject: Mathematics
• Unit: Unit 1 – Complex Numbers
• Test Type: Full Unit Test
• Question Type: Objective / MCQs
• Time Allowed: 20 Minutes
• Maximum Marks: 15
• Total MCQs: 15

Instructions

Four possible answers A, B, C and D to each question are given. Choose the answer which you think is correct and fill the corresponding circle in the answer book with a marker or ink pen. Cutting or filling two or more circles will result in zero marks in that question.

Class 10 Mathematics Unit 1 – Complex Numbers MCQs

1. z z̄ =
   • (A) 0
   • (B) 1
   • (C) |z|²
   • (D) None of these
2. (1 + i)(1 + i) =
   • (A) 2i
   • (B) 2 + 2i
   • (C) −2i
   • (D) 2 − 2i
3. i² + i⁴ =
   • (A) −1
   • (B) 0
   • (C) 1
   • (D) 2
4. Real part of (2 − 3i)(2 + 3i) is:
   • (A) −3
   • (B) 1
   • (C) 4
   • (D) 13
5. What is additive inverse of 5 − 2i?
   • (A) 5 + 2i
   • (B) −5 − 2i
   • (C) 5 − 2i
   • (D) −5 + 2i
6. If z = 4 + 4i, then z + z̄ = :
   • (A) 8
   • (B) 8 + 8i
   • (C) 8i
   • (D) 0
7. Z = x + iy is called a complex number where i =
   • (A) −1
   • (B) −√−1
   • (C) √−1
   • (D) None of these
8. If Z = −i, then |Z| =
   • (A) −1
   • (B) 1
   • (C) i
   • (D) Does not exist
9. If z = 5 + 4i, then |z| = :
   • (A) 9
   • (B) 25
   • (C) 41
   • (D) √41
10. If Z = 3 − 2i then Z̄ =
    • (A) 3 − 2i
    • (B) −3 + 2i
    • (C) 3 + 2i
    • (D) −3 − 2i
11. If Z = 1/(1 + i), then Z̄ =
    • (A) (1 + i)/2
    • (B) (1 − i)/2
    • (C) 1/(1 − i)
    • (D) None of these
12. If (2 + i)i = x + 2i, then x =
    • (A) −1
    • (B) 0
    • (C) 1
    • (D) i
13. Conjugate of a complex number Z = x + iy is defined by Z̄ =
    • (A) x + iy
    • (B) x − iy
    • (C) −x + iy
    • (D) −x − iy
14. The number 0 = 0 + 0i is called
    • (A) Additive identity
    • (B) Multiplicative identity
    • (C) Additive inverse
    • (D) None of these
15. If Z = 2 + 3i then 4Z =
    • (A) 8 + 3i
    • (B) 2 + 12i
    • (C) 8 + 12i
    • (D) 2 + 3i

Class 10 Mathematics | Unit 1: Complex Numbers | Subjective Type

This post contains important Class 10 Mathematics Unit 1: Complex Numbers subjective questions, including short questions, long questions, complex-number operations, conjugates, modulus, inverses, real and imaginary parts, and properties of complex numbers.

Test Information

• Class: 10th
• Subject: Mathematics
• Unit: Unit 1 – Complex Numbers
• Type: Subjective
• Time Allowed: 2 Hours 10 Minutes
• Maximum Marks: 60

Part-I

Question 2:

Write short answers to any six (06) questions.  (2 × 6 = 12)

1. Simplify i⁻²⁵.
2. Simplify (i⁸ / i⁵)⁻⁵.
3. Write in terms of i: √2 − √−3.
4. Find the values of x and y:
   (3x + 2) − (4 − y)i = 5 + 3i.
5. If z₁ = 2 + 3i and z₂ = 4 + 7i, then find z₁z₂.
6. If z = −7 − 4i, then show that z + 0 = z.
7. Find the multiplicative inverse of 4 − 3i.
8. If z₁ = 2 + 5i, z₂ = 1 − 3i and z₃ = 2 + i, then verify that:
   z₁ + z₂ = z₂ + z₁.
9. Show that |z| = |−z| = |z̄| = |−z̄|.

Question 3

Write short answers to any six (06) questions.  (2 × 6 = 12)

1. Find the modulus of the following complex number: −5 − 4i.
2. If z₁ = 2 + 7i and z₂ = 4 − 3i, then verify that:
   z̄₁z̄₂ = z̄₁ z̄₂.
3. If z = 4 − 3i, then verify that:
   |z| = |−z| = |z̄| = |−z̄|.
4. If z = 5 − 2i, then verify that:
   z = z̄̄.
5. If z₁ = 2 + 3i, z₂ = −1 + i, then evaluate:
   Re(z₁z₂).
6. Find the real and imaginary parts of (4 + 3i)⁻¹.
7. If z = 5 − 2i, then verify that:
   z z̄ = |z|².
8. Prove that:
   Re(z) = (z + z̄) / 2.
9. Find the real and imaginary parts of the following complex number:
   (8 − 3i)².

Question 4

Write short answers to any six (06) questions.  (2 × 6 = 12)

1. Is “0” a complex number? Explain.
2. Simplify: i¹³ × i¹¹.
3. Find additive and multiplicative inverse of z = 8 + 9i.
4. If z₁ = 3 + 4i and z₂ = 2 + 3i, then verify that:
   z̄₁ + z̄₂ = z̄₁ + z̄₂.
5. If z₁ = 5 + 4i, z₂ = 3 + 2i, then find:
   z₁ / z₂.
6. Find the real and imaginary parts of:
   z = (2 + 7i)⁻¹.
7. Solve (3 − 4i)(a + bi) = 1 + 0i and find the values of a and b.
8. Define the modulus (absolute value) of a complex number.
9. Define a complex number.

Part-II

Note: Attempt any two (02) questions.  (2 × 8 = 16)

Question 5

(a) Find the values of x and y.

(1 − i)x + (2 + i)y = 4 − i

(b) Simplify:

[(1 + 2i) / (3 − i)](2 + i)

Question 6

(a) Simplify and write in the form a + bi:

(3 − 5i) ÷ (2 − 4i)

(b) If z₁ = 2 + 5i, z₂ = 1 − 3i and z₃ = 2 + i, then verify that:

(z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

Question 7

(a) If (2x + iy)(1 − i) = 4 + 2i, then find the values of x and y.

(b) Find the values of a and b, if:

(a + bi)(1 + 3i) = −8 + 11i

Part-III

Note: Attempt any one (01) question.  (1 × 8 = 8)

Question 8

(a) If z₁ = 5 + 4i and z₂ = 3 + 2i, then prove that:

overline(z₁ / z₂) = z̄₁ / z̄₂

(b) Solve the following simultaneous linear equations with complex coefficients for w and z:

2z + (3 + i)w = 9 − i

−iz − iw = −1 + i

Question 9

(a) If z₁ = 2 + 7i and z₂ = 4 − 3i, then verify that:

overline(z₁ / z₂) = z̄₁ / z̄₂

(b) Find the real and imaginary parts of the following complex number:

[(3 + 2i) / (4 + 3i)]⁻¹

Important Topics Covered

• Complex numbers
• Real and imaginary parts
• Modulus of a complex number
• Conjugate of a complex number
• Additive identity and additive inverse
• Multiplicative inverse
• Multiplication and division of complex numbers
• Powers of i
• Properties of complex numbers
• Real and imaginary parts of reciprocal complex numbers
• Associative and commutative properties
• Simultaneous linear equations with complex coefficients

Class 10 Mathematics Unit 1 – Complex Numbers

These subjective questions are useful for Class 10 Mathematics Unit 1 preparation. Students can use them for school tests, chapter-wise practice, revision, homework, and examination preparation. The questions cover both short-answer and long-answer concepts from the chapter Complex Numbers.

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Answer Key

Important Topics Covered

• Complex numbers
• Conjugate of a complex number
• Modulus of complex numbers
• Real and imaginary parts
• Additive inverse
• Additive identity
• Multiplication of complex numbers
• Powers of i
• Properties of conjugates

Class 10 Mathematics Test Preparation

This Class 10 Mathematics Unit 1 Full Unit Test is useful for students preparing for chapter-wise assessments, school tests, board examinations and objective-type Mathematics practice. Students can use these MCQs to revise the fundamental concepts of Complex Numbers.

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