Exercise 1.2 Class 9 Math – Real Numbers |PDF Notes

If you’re looking for Exercise 1.2 Class 9 Math notes, you’ve come to the right spot! This page provides solved solutions for Exercise 1.2, designed to help you master topics like rationalizing denominators which is making denominators free of square roots, simplifying tricky expressions with exponents and radicals, and applying algebraic identities. Every problem is solved step-by-step so you can follow along easily.

The solutions are available in PDF format for free download, perfect for practicing at home, preparing for tests, or revising quickly before exams. Whether you’re stuck on a question or just want to check your answers, these notes will guide you through Exercise 1.2 smoothly. Start studying now and build your confidence for Chapter 1!

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Common Mistakes to Avoid in Exercise 1.2 Class 9 Math

1. Incorrect Application of Exponent Rules

Rule:
\( a^m \times a^n = a^{m+n} \)
This rule only applies when the base is the same.
Mistake Example:
\( 2^3 \times 3^2 \ne 6^5 \)
The bases are different (2 and 3), so you cannot add the exponents.

2. Partial Simplification

After rationalizing a denominator, make sure you simplify the expression completely.
Mistake Example:
\( \frac{52 – 13\sqrt{3}}{13} \rightarrow 4 – \sqrt{3} \)
Don’t stop at the unsimplified form. Divide each term in the numerator by 13 to simplify fully.

3. Ignoring Conjugate Pairs

When rationalizing expressions with square roots, students sometimes fail to multiply all terms in the numerator by the conjugate of the denominator.
Tip:
If the denominator is \( a + \sqrt{b} \), multiply both numerator and denominator by \( a – \sqrt{b} \), and vice versa. Be sure to apply multiplication to every term in the numerator.

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Key Definitions

Surd:

An irrational number expressed as a non-terminating, non-repeating root.
Examples: \( \sqrt{2}, \sqrt{3}, \sqrt{5} \)

Conjugate:

A pair of expressions like \( (a + \sqrt{b}) ) and ( (a – \sqrt{b}) \) used to rationalize denominators.

Rational Number:

A number that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \ne 0 \).

Indices:

The plural of index (or exponent). It tells how many times a number (called the base) is multiplied by itself.
Example: \( a^m \) means \( a \times a \times a \dots \) \(m–times\).

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Important Laws and Formulas

Laws of Radicals	Laws of Indices
\( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \) \( \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m \) \( \left( \sqrt[n]{a} \right)^{1/n} = \left( a^n \right)^{1/n} = a \)	\( a^m \cdot a^n = a^{m+n} \) \( (a^m)^n = a^{mn} \) \( (ab)^n = a^n b^n \) \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \) \( \frac{a^m}{a^n} = a^{m-n} \) \( a^0 = 1 \)

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📐 Real-Life Applications

Surds in Architecture & Carpentry:

• Calculating the exact length of rafters or beams when building a roof (using √2 for 45° angles).
• Measuring the diagonal length of rectangular tiles or screens (for example, TV screen size measured diagonally involves √(width² + height²)).

Radicals in Medicine & Biology:

• Calculating dosages or growth rates that involve square roots, such as in pharmacokinetics or modeling population growth.
• Estimating the surface area of organs or tumors where radical formulas appear in volume and area calculations.

Exponents in Computer Science & Technology:

• Understanding how data storage grows exponentially (e.g., bits and bytes doubling with each memory upgrade).
• Algorithms involving powers and roots for encryption, data compression, or signal processing.

Exponents in Finance & Economics:

Modeling population growth or inflation rates which follow exponential growth or decay. Calculating compound interest on savings or loans where the formula is A = P × (1 + r)ⁿ.

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Short Questions

Question 1: Define a surd. Give an example of a binomial surd.

Answer: A surd is an irrational number expressed as a root (e.g., \( \sqrt{5} )\). A binomial surd contains two terms, such as \( 1 + \sqrt{3} \).

Question 2: Rationalize \( \frac{7}{\sqrt{5} + \sqrt{3}} \).
Answer: Multiply numerator and denominator by the conjugate \( \sqrt{5} – \sqrt{3} \):
\[
\frac{7}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} – \sqrt{3}}{\sqrt{5} – \sqrt{3}} = \frac{7(\sqrt{5} – \sqrt{3})}{(\sqrt{5} + \sqrt{3})(\sqrt{5} – \sqrt{3})} = \frac{7(\sqrt{5} – \sqrt{3})}{5 – 3} = \frac{7(\sqrt{5} – \sqrt{3})}{2}
\]
Question 3: Simplify \( \left( \frac{27}{125} \right)^{-\frac{2}{3}} \).
Answer: Rewrite bases as powers:
\[
\left( \frac{3^{3}}{5^{3}} \right)^{-\frac{2}{3}} = \frac{5^{2}}{3^{2}} = \frac{25}{9}
\]
Question 4: Expand \( (3 + 2)(3 – 2) \). What algebraic identity does this demonstrate?
Answer:
\[
(3 + 2)(3 – 2) = 3^{2} – 2^{2} = 9 – 4 = 5
\]
Identity used: \( (a + b)(a – b) = a^{2} – b^{2} \).

Question 5: Explain why \( \sqrt{5} \times \sqrt{5} \) is rational, but \( 2 + \sqrt{3} \) is irrational.
Answer: \( \sqrt{5} \times \sqrt{5} = 5 \), which is rational.
\( 2 + \sqrt{3} \) cannot be simplified to a fraction and remains irrational.

Question 6: Simplify \( 3^{n+2} \times 9^{n-1} \).
Answer: Rewrite 9 as \( 3^{2} \):
\[
3^{n+2} \times (3^{2})^{n-1} = 3^{n+2} \times 3^{2n-2} = 3^{(n+2) + (2n-2)} = 3^{3n}
\]
Question 7: If \( x = 2 + \sqrt{5} \), find \( x^{2} + \frac{1}{x^{2}} \).
Answer:
\[
x^{2} = (2 + \sqrt{5})^{2} = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}
\]
\[
\frac{1}{x} = \frac{1}{2 + \sqrt{5}} \times \frac{2 – \sqrt{5}}{2 – \sqrt{5}} = 2 – \sqrt{5}
\]
\[
\Rightarrow \frac{1}{x^{2}} = (2 – \sqrt{5})^{2} = 4 – 4\sqrt{5} + 5 = 9 – 4\sqrt{5}
\]
Sum:
\[
x^{2} + \frac{1}{x^{2}} = (9 + 4\sqrt{5}) + (9 – 4\sqrt{5}) = 18
\]
Question 8: Calculate the diagonal length of a square with side \( 5\sqrt{2} \) cm. Express your answer as a surd.
Answer:
Diagonal length \(= \text{side} \times \sqrt{2} = 5\sqrt{2} \times \sqrt{2} = 5 \times 2 = 10 \) cm.

We hope these 9 class math notes for Exercise 1.2 helped you understand the important concepts of real numbers, including rationalization and simplification. To keep improving, don’t stop here—explore more exercises from Chapter 1 to strengthen your foundation:

• ✅ Exercise 1.1 – Real Numbers | Solved Notes PDF
• ✅ Exercise 1.3 – Real Numbers | Solved Notes PDF

Also, if you need the full book for reference or revision, download it here:

• 📘 Class 9 Math Textbook – Punjab Board PDF

Keep practicing and stay confident. If you find these notes helpful, share them with your friends and continue learning with notesofmath.com!

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Video Lecture

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