Exercise 1.3 Class 10 Maths Chapter 1 (Complex Numbers) — complete step-by-step solutions for the new book. Free PDF download, Al Huda Science Academy.
Exercise 1.3 Class 10 Maths — Chapter 1 (Complex Numbers) Full Solution
By Sir Usman Adal | Al Huda Science Academy | New Book (10th Mathematics)
The complete, step-by-step solution of Exercise 1.3, Chapter 1 (Complex Numbers) for Class 10 Mathematics — New Book. It covers finding the modulus of complex numbers, verifying conjugate identities, and evaluating real and imaginary parts of a product.
Q.1 — Find the Modulus of the Following Complex Numbers
(i) 4 + 3i
|z| = √(4² + 3²) = √25 = 5
(ii) −5 − 4i
|z| = √((−5)² + (−4)²) = √41 = √41
(iii) 3/5 − (4/5)i
|z| = √(9/25 + 16/25) = √1 = 1
(iv) −√2 − √3 i
|z| = √(2 + 3) = √5
Q.2 — If z₁ = 2 + 7i and z₂ = 4 − 3i, Verify That
(i) conjugate(z₁ + z₂) = conjugate(z₁) + conjugate(z₂)
Both sides = 6 − 4i ✓ Verified
(ii) conjugate(z₁z₂) = conjugate(z₁)·conjugate(z₂)
Both sides = 29 − 22i ✓ Verified
(iii) conjugate(z₁/z₂) = conjugate(z₁)/conjugate(z₂)
Both sides = (−13 − 34i)/25 ✓ Verified
Q.3 — If z = 5 − 2i, Verify That
(i) conjugate(conjugate(z)) = z
5 − 2i = 5 − 2i ✓ Verified
(ii) |z| = |conjugate(z)|
Both equal √29 ✓ Verified
(iii) |z| = |−z|
Both equal √29 ✓ Verified
(iv) z·conjugate(z) = |z|²
z·z̄ = 25 + 4 = 29, and |z|² = (√29)² = 29 ✓ Verified
(v) |z| = |−conjugate(z)|
Both equal √29 ✓ Verified
Q.4 — If z = 4 − 3i, Verify That |z| = |−z| = |conjugate(z)| = |−conjugate(z)|
All four values equal 5 ✓ Verified
Q.5 — If z₁ = 2 + 3i, z₂ = −1 + i, Evaluate
z₁z₂ = (2 + 3i)(−1 + i) = −2 + 2i − 3i + 3i² = −5 − i
(i) Re(z₁z₂) = −5
(ii) Im(z₁z₂) = −1
Key Facts — Exercise 1.3
Modulus: if z = x + iy, then |z| = √(x² + y²)
Conjugate: if z = x + iy, then z̄ = x − iy
Basic identities:
- conjugate(conjugate(z)) = z
- conjugate(z₁ + z₂) = conjugate(z₁) + conjugate(z₂)
- conjugate(z₁z₂) = conjugate(z₁)·conjugate(z₂)
- conjugate(z₁/z₂) = conjugate(z₁)/conjugate(z₂)
- |z| = |z̄| = |−z| = |−z̄|
- z·z̄ = |z|²
Powers of i: i² = −1, i³ = −i, i⁴ = 1, i⁵ = i, i⁶ = −1
Algebraic rules: (a+b)+c = a+(b+c), a+b = b+a, a(bc) = (ab)c, a(b+c) = ab+ac, a(b−c) = ab−ac, a² − b² = (a+b)(a−b)
Watch Full Class 10 Exercise 1.3 Math with Concepts
Frequently Asked Questions
How do you find the modulus of a complex number?
For z = x + iy, the modulus is |z| = √(x² + y²) — the square root of the sum of the squares of the real and imaginary parts.
What is the relationship between a complex number and its conjugate?
A complex number z = x + iy and its conjugate z̄ = x − iy always have the same modulus, and their product z·z̄ equals |z|², which is always a real number.
Why is z times its conjugate always a real number?
Because (x + iy)(x − iy) = x² + y² (the i² term cancels to a positive real value), the imaginary parts always eliminate each other in this product.
What is Exercise 1.3 of Class 10 Maths Chapter 1 about?
It covers finding the modulus of complex numbers and verifying key identities involving conjugates, such as conjugate of a sum, product, and quotient, plus evaluating real and imaginary parts of complex expressions.
Download and Watch Full Chapter 1, 10th Class Maths
