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
| Power | Value | Power | Value |
|---|---|---|---|
| i⁰ | 1 | i⁴ | 1 |
| i¹ | i | i⁵ | i |
| i² | −1 | i⁶ | −1 |
| i³ | −i | i⁷ | −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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Frequently Asked Questions
What is the value of i⁵ in complex numbers?
i⁵ = i, since powers of i repeat in a cycle of 4 (i⁴ = 1).
How do you find x and y when two complex numbers are equal?
Equate the real parts to each other and the imaginary parts to each other, then solve the resulting equations.
What is Exercise 1.1 of Class 10 Maths Chapter 1 about?
It covers simplifying powers of iota (i), converting square roots of negative numbers into terms of i, and solving for unknowns using the equality of complex numbers.
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