Mathematics 3 March 2024 Common Te NSC QUESTION 1 Indicate whether each of the following numbers is rational or irrational: \( 1.1 .1 \sqrt{15} \) 1.1.2 2,5 \( 1.1 .3 \quad \sqrt[3]{8} \) 1.1.4 2,3333333.... .2 Write down TWO rational numbers between \( \sqrt{2} \) and \( \sqrt{10} \) (Show your working). . 3 Write down \( 0,11111111 \ldots \) as a fraction. . 4 Factorise the following expressions fully; 1.4.1 \( x^{3}-8 \) 1.4.2 \( 2 x^{3}+x^{2}-6 x-3 \) 1.4.3 \( \quad x^{4}-16 \) 5 Simplify the following expressions fully; 1.5.1 \( (2 x-1)\left(x^{2}-2 x+1\right) \) 1.5.2 \( (3 x+5)^{2} \) 1.5.3 \( \frac{2^{x}-2^{x-2}}{2^{x+1}-2^{x}} \) 1.5.4 \( \frac{3}{a-4}+\frac{2}{a+3}-\frac{21}{a^{2}-a-12} \)

Let's solve the problem step by step. ### 1. Indicate whether each of the following numbers is rational or irrational: 1.1.1 \( \sqrt{15} \) - **Rational or Irrational**: Irrational - **Explanation**: The square root of a non-perfect square (like 15) is irrational. 1.1.2 \( 2.5 \) - **Rational or Irrational**: Rational - **Explanation**: This number can be expressed as a fraction \( \frac{25}{10} \). 1.1.3 \( \sqrt[3]{8} \) - **Rational or Irrational**: Rational - **Explanation**: \( \sqrt[3]{8} = 2 \), which is a whole number and can be expressed as \( \frac{2}{1} \). 1.1.4 \( 2.3333333... \) - **Rational or Irrational**: Rational - **Explanation**: This number can be expressed as a fraction \( \frac{7}{3} \). ### 1.2 Write down TWO rational numbers between \( \sqrt{2} \) and \( \sqrt{10} \) (Show your working). - **Calculating \( \sqrt{2} \) and \( \sqrt{10} \)**: - \( \sqrt{2} \approx 1.414 \) - \( \sqrt{10} \approx 3.162 \) - **Choosing Rational Numbers**: - Two rational numbers between these values could be \( 2 \) and \( 3 \). ### 1.3 Write down \( 0.11111111 \ldots \) as a fraction. Let \( x = 0.11111111 \ldots \). - Multiply both sides by 10: - \( 10x = 1.11111111 \ldots \) - Subtract the first equation from the second: - \( 10x - x = 1.11111111 \ldots - 0.11111111 \ldots \) - \( 9x = 1 \) - \( x = \frac{1}{9} \) ### 1.4 Factorise the following expressions fully: 1.4.1 \( x^{3}-8 \) - This is a difference of cubes: - \( x^{3} - 2^{3} = (x - 2)(x^{2} + 2x + 4) \) 1.4.2 \( 2x^{3} + x^{2} - 6x - 3 \) - Grouping: - \( (2x^{3} + x^{2}) + (-6x - 3) \) - Factor out common terms: - \( x^{2}(2x + 1) - 3(2x + 1) \) - Factor by grouping: - \( (2x + 1)(x^{2} - 3) \) 1.4.3 \( x^{4} - 16 \) - This is a difference of squares: - \( (x^{2} - 4)(x^{2} + 4) \) - Further factor \( x^{2} - 4 \): - \( (x - 2)(x + 2)(x^{2} + 4) \) ### 1.5 Simplify the following expressions fully: 1.5.1 \( (2x - 1)(x^{2} - 2x + 1) \) - Recognize \( x^{2} - 2x + 1 = (x - 1)^{2} \): - \( (2x - 1)(x - 1)^{2} \) 1.5.2 \( (3x + 5)^{2} \) - Use the square of a binomial: - \( 9x^{2} + 30x + 25 \) 1.5.3 \( \frac{2^{x} - 2^{x-2}}{2^{x+1} - 2^{x}} \) - Simplifying the numerator: - \( 2^{x} - \frac{2^{x}}{4} = 2^{x}(1 - \frac{1}{4}) = 2^{x} \cdot \frac{3}{4} \) - Simplifying the denominator: - \( 2^{x+1} - 2^{x} = 2^{x}(2 - 1) = 2^{x} \) - Final expression: - \( \frac{\frac{3}{4} \cdot 2^{x}}{2^{x}} = \frac{3}{4} \) 1.5.4 \( \frac{3}{a-4} + \frac{2}{a+3} - \frac{21}{a^{2}-a-12} \) - Factor the denominator: - \( a^{2} - a - 12 = (a - 4)(a + 3) \) - Rewrite the third term: - \( \frac{21}{(a - 4)(a + 3)} \) - Combine the fractions: - \( \frac{3(a + 3) + 2(a - 4) - 21}{(a - 4)(a + 3)} \) - Simplifying the numerator: - \( 3a + 9 + 2a - 8 - 21 = 5a - 20 \) - Final expression: - \( \frac{5(a - 4)}{(a - 4)(a + 3)} = \frac{5}{a + 3} \) (for \( a \neq 4 \)) This concludes the solution to the problem.