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Exercise 1.1 Class 10 Maths Chapter 1 (Complex Numbers) — complete step-by-step solutions for the new book. Free PDF download, Al Huda Science Academy.

Exercise 1.1 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 1.1, Chapter 1 (Complex Numbers) for Class 10 Mathematics — New Book. It covers simplifying powers of i, writing expressions in terms of i, and finding unknown values x and y by comparing real and imaginary parts.

Q.1 — Simplify the Following

(i) i⁵

i⁵ = i⁴ × i = 1 × i = i

(ii) i¹⁶

i¹⁶ = (i⁴)⁴ = 1⁴ = 1

(iii) (−i)⁻¹⁹

(−i)⁻¹⁹ = (−i)¹⁶ × (−i)³ = 1 × (−i)³ = −i

(iv) 27i⁻²⁶

27i⁻²⁶ = 27 × 1/i²⁶ = 27 × 1/i² = 27 × (−1) = −27

(v) i¹¹ + i⁵

i¹¹ + i⁵ = −i + i = 0

(vi) (i⁴ + i³ + i² + i)²

Inside bracket = 1 − i − 1 + i = 0 → (0)² = 0

(vii) (i⁸/i⁷)⁻⁵

(i¹)⁻⁵ = 1/i⁵ = i

(viii) i¹³ × i²⁹

i¹³ × i²⁹ = i⁴² = −1

Q.2 — Write in Terms of i

(i) 2 + √−4

= 2 + i·2 = 2 + 2i

(ii) 3 − √−7

= 3 − i√7

(iii) 2/5 + √−16/√5

= (2 + 4i)/5

(iv) √2 − √−3

= √2 − i√3

Q.3 — Find the Values of x and y

(i) (2x + 5) + (y − 3)i = 1 + 2i

x = −2, y = 5

(ii) (3x + 2) − (4 − y)i = 5 + 3i

x = 1, y = 7

(iii) (2 + i)x + (1 − 2i)y = 3 + 4i

x = 2, y = −1

(iv) (1 − i)x + (2 + i)y = 4 − i

x = 2, y = 1

(v) (3x − 1) + (2y − 3)i = 8 + 7i

x = 3, y = 5

Key Facts — Powers of i

Index rules: iᵐ·iⁿ = iᵐ⁺ⁿ, iᵐ/iⁿ = iᵐ⁻ⁿ, (iᵐ)ⁿ = iᵐⁿ, i⁻ⁿ = 1/iⁿ

Square roots of negatives: √(−a) = i√a (for a > 0)

Conjugate rules: if z = a + bi, then z̄ = a − bi, z + z̄ = 2a, z − z̄ = 2bi

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