📘 Unit 1: Complex Numbers – Exercise 1.1 (Solved)
Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska. Based on PECTAA 2026 syllabus (National Curriculum 2023).
📖 What's Inside: This exercise covers powers of iota (ι), simplification of complex expressions, and solving complex equations. Each problem is presented with step-by-step solutions as per the official PECTAA 2026 Mathematics curriculum. Perfect for Punjab Boards (Lahore, Gujranwala, Multan, etc.) and all BISE boards across Pakistan.
📚 Related Resources – Unit 1: Complex Numbers
Complex numbers extend real number system. Includes iota powers, conjugates, and applications.
📖 Exercise 1.1 – Solved Problems
1 Simplify: \( \iota^{5} \)
\[ \iota^{5} = \iota^{4} \cdot \iota = (1) \cdot \iota = \iota \]
Since \( \iota^2 = -1 \), we have \( \iota^4 = (\iota^2)^2 = (-1)^2 = 1 \). Therefore \( \iota^5 = \iota \).
2 Simplify: \( \iota^{16} \)
\[ \iota^{16} = (\iota^{2})^{8} = (-1)^{8} = 1 \]
Using \( \iota^2 = -1 \), we get \( \iota^{16} = (\iota^2)^8 = (-1)^8 = 1 \).
3 Simplify: \( (-\iota)^{-19} \)
\[ (-\iota)^{-19} = \frac{1}{(-\iota)^{19}} = \frac{1}{-\iota^{19}} = -\frac{1}{\iota^{19}} \]
\[ \iota^{19} = \iota^{16} \cdot \iota^{3} = 1 \cdot (-\iota) = -\iota \]
\[ -\frac{1}{-\iota} = \frac{1}{\iota} = \frac{\iota}{\iota^2} = \frac{\iota}{-1} = -\iota \]
Thus \( (-\iota)^{-19} = -\iota \).
4 Simplify: \( \iota^{11} + \iota^{5} \)
\[ \iota^{11} = \iota^{10} \cdot \iota = (\iota^2)^5 \cdot \iota = (-1)^5 \cdot \iota = -\iota,\quad \iota^{5} = \iota \]
\[ \iota^{11} + \iota^{5} = -\iota + \iota = 0 \]
5 Evaluate: \( (\iota^{4} + \iota^{3} + \iota^{2} + \iota)^{2} \)
\[ \iota^{4} = 1,\ \iota^{3} = -\iota,\ \iota^{2} = -1,\ \iota = \iota \]
\[ 1 + (-\iota) + (-1) + \iota = (1 - 1) + (-\iota + \iota) = 0 \]
\[ (0)^{2} = 0 \]
6 Simplify: \( \left(\frac{\iota^{8}}{\iota^{5}}\right)^{-5} \)
\[ \frac{\iota^{8}}{\iota^{5}} = \iota^{8-5} = \iota^{3} \]
\[ (\iota^{3})^{-5} = \iota^{-15} = \frac{1}{\iota^{15}},\quad \iota^{15} = \iota^{12} \cdot \iota^{3} = 1 \cdot (-\iota) = -\iota \]
\[ \frac{1}{-\iota} = -\frac{1}{\iota} = -\frac{\iota}{\iota^2} = -\frac{\iota}{-1} = \iota \]
7 Simplify: \( \iota^{13} \times \iota^{29} \)
\[ \iota^{13} \times \iota^{29} = \iota^{13+29} = \iota^{42} \]
\[ \iota^{42} = (\iota^{2})^{21} = (-1)^{21} = -1 \]
8 Write in terms of \( \iota \): \( 2 + \sqrt{-4} \)
\[ 2 + \sqrt{-4} = 2 + \sqrt{4 \cdot (-1)} = 2 + 2\sqrt{-1} = 2 + 2\iota \]
(ii) \( 3 - \sqrt{-7} \)
\[ 3 - \sqrt{-7} = 3 - \sqrt{7 \cdot (-1)} = 3 - \sqrt{7}\iota \]
(iii) \( \frac{2}{5} + \frac{\sqrt{-16}}{5} \)
\[ \frac{2}{5} + \frac{4\iota}{5} = \frac{2 + 4\iota}{5} \]
(iv) \( \sqrt{2} - \sqrt{-3} \)
\[ \sqrt{2} - \sqrt{3}\iota \]
9 Find \( x \) and \( y \): \( (2x+5) + (y-3)\iota = 1 + 2\iota \)
\[ 2x + 5 = 1 \implies x = -2,\quad y - 3 = 2 \implies y = 5 \]
(ii) \( (3x+2) - (4-y)\iota = 5 + 3\iota \)
\[ 3x + 2 = 5 \implies x = 1,\quad -(4-y) = 3 \implies y = 7 \]
(iii) \( (2+\iota)x + (1-2\iota)y = 3 + 4\iota \)
\[ (2x + y) + (x - 2y)\iota = 3 + 4\iota \implies 2x+y=3,\ x-2y=4 \]
\[ \text{Solving: } x = 2,\ y = -1 \]
(iv) \( (1-\iota)x + (2+\iota)y = 4 - \iota \)
\[ (x+2y) + (-x+y)\iota = 4 - \iota \implies x+2y=4,\ -x+y=-1 \]
\[ \text{Solving: } x = 2,\ y = 1 \]
(v) \( (3x-1) + (2y-3)\iota = 8 + 7\iota \)
\[ 3x - 1 = 8 \implies x = 3,\quad 2y - 3 = 7 \implies y = 5 \]
📐 Key Formulas – Complex Numbers
Iota Powers: \( \iota^2 = -1,\ \iota^3 = -\iota,\ \iota^4 = 1 \)
General Power: \( \iota^{4n} = 1,\ \iota^{4n+1} = \iota,\ \iota^{4n+2} = -1,\ \iota^{4n+3} = -\iota \)
Complex Number Form: \( z = a + b\iota \) where \( a,b \in \mathbb{R} \)
💡 Exam Tip:
For board exams, remember that powers of iota repeat every 4 steps. Always reduce the exponent modulo 4 to find the simplified value. These solutions follow the PECTAA 2026 pattern and are prepared by Subject Specialist Muhammad Tayyab.
Unit 1✨ Complex Numbers (Current Chapter)
Unit 2Quadratic Equations and Inequalities
Unit 3Matrices and Determinants
Unit 4Functions and Graphs
Unit 5Algebraic Fractions
Unit 6Vectors in Plane
Unit 7Trigonometry
Unit 8Chords and Arcs of a Circle
Unit 9Tangent and Angles of a Circle
Unit 10Practical Geometry of Circles
Unit 11Information Handling
Unit 12Probability
📖 Complete syllabus coverage for Class 10 Mathematics (PECTAA 2026) – Units 1 to 12
Created by Hira Science Academy | Aligned with PECTAA 2026 Syllabus