Master Class 11 Math Chapter 1 Step-by-step Exercise 1.1 solutions
Class 11 Math Unlocked: Master Chapter 1 (Number Systems) & Iota Secrets
Complete Exercise 1.1 Solved Notes + Exam Style Quiz
Welcome to the Big League!
Assalam-o-Alaikum Students! Welcome to First Year Mathematics. If you think Math is only what you studied until Matric, get ready because today we are entering a new 'Universe'. Class 11 Chapter 1 isn't just about numbers; it’s about changing how you think. Today, we will find answers to questions that have been in your mind since childhood but were never explained.
Until now, you learned that the square root of a negative number (like $\sqrt{-4}$) is impossible, and we usually ignored it as a 'Math Error'. But today, we give that error a new name—$i$ (Iota). This is a world where numbers aren't just Real; they are 'Imaginary' too. We will learn how these two worlds combine to form Complex Numbers.
This chapter is the foundation for Engineering and Physics. I have designed this article so you feel like I am standing right in front of you, drawing every concept on the whiteboard. From the properties of Complex numbers to their modulus, we will cover everything easily. Are you ready for this mathematical adventure? Let’s dive in!
💡 The Power of Iota ($i$)
Remember these values; they are the heart of Chapter 1:
- $i = \sqrt{-1}$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
Exercise 1.1 - Step-by-Step Solutions
Q1: Simplify $(7, 9) + (3, -5)$
Teacher's Note: In $(a, b)$, '$a$' is Real and '$b$' is Imaginary. We add Real to Real and Imaginary to Imaginary.
Step: $(7+3, 9-5)$
Result: $(10, 4)$
MCQ 1: In $(x, y)$, which part is imaginary? (A) x (B) y Answer
(B) y
MCQ 2: The value of $(2, 3) + (0, 0)$ is? (A) $(0, 0)$ (B) $(2, 3)$ Answer
(B)
Q2: Multiply $(2, 6)$ and $(3, 7)$
Formula: $(ac - bd, ad + bc)$
Calculation: $(2\times3 - 6\times7, 2\times7 + 6\times3)$
Step: $(6 - 42, 14 + 18)$
Result: $(-36, 32)$
MCQ 3: Value of $i^2$ is? (A) 1 (B) -1 Answer
(B) -1
MCQ 4: Every real number is a complex number? (A) True (B) False Answer
(A) True
Q3: Simplify $\frac{2+3i}{4-i}$
Step 1 (Rationalize): Multiply numerator and denominator by conjugate $(4+i)$.
Step 2: $\frac{(2+3i)(4+i)}{(4-i)(4+i)} = \frac{8 + 2i + 12i + 3i^2}{16 - i^2}$
Step 3: Since $i^2 = -1$, we get $\frac{8 + 14i - 3}{16 + 1} = \frac{5 + 14i}{17}$
Result: $\frac{5}{17} + \frac{14}{17}i$
MCQ 5: Conjugate of $2+3i$ is? (A) $2-3i$ (B) $-2+3i$ Answer
(A)
MCQ 6: The product of a complex number and its conjugate is always? (A) Imaginary (B) Real Answer
(B) Real
Q4: Find Modulus of $3 - 4i$
Formula: $|z| = \sqrt{a^2 + b^2}$
Step: $|z| = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25}$
Result: $5$
MCQ 7: $|i|$ is equal to? (A) 1 (B) -1 Answer
(A) 1
MCQ 8: Modulus represents ___ from origin. (A) Angle (B) Distance Answer
(B) Distance
Q5: Show that $z + \bar{z}$ is a real number
Let $z = a + bi$, then $\bar{z} = a - bi$.
Addition: $(a + bi) + (a - bi) = a + a + bi - bi = 2a$.
Conclusion: Since $2a$ has no '$i$', it is a purely real number.
MCQ 9: $z - \bar{z}$ results in? (A) Real (B) Purely Imaginary Answer
(B)
MCQ 10: If $z = \bar{z}$, then z is? (A) Purely Real (B) Purely Imaginary Answer
(A)
🎉 You've Just Started Your College Math Journey!
Number Systems are the gateway to advanced calculus and algebra. Keep practicing these basics.
