mathematics

Western Cape Government - Western Cape Education Department - Directorate: Curriculum Fl - Mathematics Revision Booklet - Term 1 - Grade 11

Revision Program Purpose

The revision program is designed to assist students in revising the critical content and skills that are planned to be covered during the 1st term. The main purposes include:

• Preparing students to understand key mathematical concepts.
• Providing opportunities to establish the required standards for success in the National Curriculum Statement (NCS) examination.

Key Principles in Mastering Mathematics

To achieve mastery in Mathematics, students must adhere to the following principles:

1. The Importance of Process Over Result: The final answer is not the most important aspect in Mathematics; rather, a systematic and logical layout of every step of your working is crucial.
2. Overcoming Carelessness: Students should not accept errors due to carelessness as unavoidable. Carelessness can be addressed by checking one’s work, ensuring the correctness and validity of calculations step-by-step.
3. Avoiding Shortcuts: Students should not take shortcuts by omitting steps in their calculations, as this can lead to errors and misunderstanding.
4. Persistence in Problem Solving: Despair should not hinder progress in Mathematics. Students are encouraged to persist and continue trying until they understand the material, adopting a positive self-affirmation attitude ("I CAN!!!!!").
5. Practice Makes Perfect: Regular practice leads to improvement in mathematical skills.

Topics Covered in the Revision Program

The following topics will be covered in the Term 1 revision program:

• Exponents and Surds
• Equations and Inequalities
• Trigonometry

Topic: Exponents and Surds

Definitions and Key Concepts

An exponent, denoted as $a^n$, represents the product of the base $a$ multiplied by itself $n$ times, where $a > 0$ and $n ext{ is a natural number } (n \in \mathbb{N})$.

Important Laws of Exponents

1. Product of Powers: $a^m imes a^n = a^{m+n}$
2. Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
3. Power of a Power: $(a^m)^n = a^{mn}$
4. Power of a Product: $(ab)^m = a^m imes b^m$
5. Negative Exponent: $a^{-m} = \frac{1}{a^m}$
6. Exponent of Zero: $a^0 = 1$ for any $a
   eq 0$.

Surds

A root of a counting number that is an irrational number is called a surd. For example, if $\sqrt{a}$ is irrational, then it is termed as a surd.

Calculation Rules for Exponents

• The laws of exponents can only be used when the bases are the same.
• It is essential to make the bases the same when solving exponential equations.
  • If the variable is in the exponent, equate the exponents once the bases are the same.
  • If the variable is in the base, use the reciprocal of the exponent on both sides of the equation.
• For constants, if $x = a$ where $a$ is constant:
  • If $m$ is odd, there is only one solution.
  • If $m$ is even, there are two solutions (one positive and one negative).

Example Calculations (to be done without a calculator)

1. Determine the following:
   1.1 $9^{-1/2}$
   1.2 $(3^{-1}2^{-1})^{-2}$
   1.3 $(0.5)^{-3} + \sqrt{\sqrt{29}} - (0.125)$
   1.4 $\sqrt{48}$
   1.5 $32021^{3} + 32021$
   1.6 $\sqrt{\alpha - 1} \times \sqrt{a^2 - 2a + 1}$
   1.7 $\sqrt{2}\sqrt{5} - 4\sqrt{2}\sqrt{5} + 4$
   1.8 $\sqrt{(1 - \sqrt{2})^2} \cdot 3\sqrt{3} + 2\sqrt{2}$

2. Simplify the following and leave answers with positive exponents:
   2.1 $(-6x^2)^3 \times (-x^{-3})^{-2} \div (-3x^3)^2$
   2.2 $3^5 + 6^5 \cdot 4^{-5} - 7 + 1.14^{-1}$
   2.3 $9^{-1} \div 25^{-1} \cdot 6^{1.6}$
   2.4 $10^{-1} \cdot 15^{2x - 1 + 2x + 3}$
   2.5 $3^{2x + 1}$
   2.6 $5^{X + 2} \cdot 4.5 . 5^{-1} + 2.5^{X + 1}$
   2.7 $5^{1000} - 550^{2} + 24$
   2.8 $\sqrt{4a + 1} \cdot 3\sqrt{ga + 1}$
   2.9 If $3a = p$ and $2b = q$, express $2.9a - 3.8b$ in terms of $p$ and $q$.

3. Solve for $x$: