Grade 9 mathematics June exam paper KwaZulu-Natal

1.1 Step 1: Understand the definition of a rational number. A rational number can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Step 2: Evaluate each option. A. $\sqrt{2}$ is an irrational number because its decimal representation is non-terminating and non-repeating. B. $\pi$ is an irrational number because its decimal representation is non-terminating and non-repeating. C. $0,333...$ is a repeating decimal, which can be written as $\frac{1}{3}$. This is a rational number. D. $\sqrt{-4}$ is an imaginary number ($2i$), not a real number, and therefore not a rational number in the set of real numbers.

The correct option is C.

1.2 Step 1: Evaluate $2^0$. Any non-zero number raised to the power of $0$ is $1$. $2^0 = 1$ Step 2: Evaluate $2^1$. Any number raised to the power of $1$ is itself. $2^1 = 2$ Step 3: Add the results. $2^0 + 2^1 = 1 + 2 = 3$

The correct option is D.

1.3 Step 1: List the multiples of $6$. Multiples of $6$: $6, 12, 18, 24, 30, ...$ Step 2: List the multiples of $8$. Multiples of $8$: $8, 16, 24, 32, 40, ...$ Step 3: Identify the lowest common multiple (LCM) from the lists. The smallest number that appears in both lists is $24$.

The correct option is B.

1.4 Step 1: Calculate the square root of $16$. $\sqrt{16} = 4$ Step 2: Calculate the square root of $9$. $\sqrt{9} = 3$ Step 3: Add the results. $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$

The correct option is B.

1.5 Step 1: Analyze the given sequence: $1; 4; 9; 16; ...$ Step 2: Identify the pattern. $1 = 1^2$ $4 = 2^2$ $9 = 3^2$ $16 = 4^2$ The sequence consists of perfect squares of consecutive natural numbers. Step 3: Determine the next term in the sequence. The next number in the sequence of bases is $5$. The next term is $5^2 = 25$.

The correct option is B.

1.6 Step 1: Recognize the form of the expression $x^2 - 4$. This is a difference of two squares, which follows the formula $a^2 - b^2 = (a-b)(a+b)$. Step 2: Identify $a$ and $b$ in the given expression. Here, $a^2 = x^2$, so $a = x$. And $b^2 = 4$, so $b = \sqrt{4} = 2$. Step 3: Apply the difference of squares formula. $x^2 - 4 = (x-2)(x+2)$

The correct option is C.

1.7 Step 1: Write the given equation. $2x - 1 = 5$ Step 2: Add $1$ to both sides of the equation to isolate the term with $x$. $2x - 1 + 1 = 5 + 1$ $2x = 6$ Step 3: Divide both sides by $2$ to solve for $x$. $\frac{2x}{2} = \frac{6}{2}$ $x = 3$

The correct option is B.

1.8 Step 1: Substitute the given value of $x = -1$ into the expression $x^2 - 2x + 1$. $(-1)^2 - 2(-1) + 1$ Step 2: Evaluate each term. $(-1)^2 = (-1) \times (-1) = 1$ $-2(-1) = 2$ Step 3: Add the results. $1 + 2 + 1 = 4$

The correct option is C.

1.9 Step 1: Recall the slope-intercept form of a linear equation, which is $y = mx + c$. In this form, $m$ represents the gradient (slope) of the line, and $c$ represents the y-intercept. Step 2: Compare the given equation $y = 2x - 3$ with the slope-intercept form. By comparison, the coefficient of $x$ is $2$. Therefore, the gradient $m = 2$.

The correct option is A.

1.10 Step 1: Understand that a line passing through the origin $(0; 0)$ has a y-intercept of $0$. Its equation is of the form $y = mx$. Step 2: Use the given point $(1; 2)$ to find the gradient $m$. Substitute $x=1$ and $y=2$ into the equation $y = mx$. $2 = m(1)$ $m = 2$ Step 3: Write the equation of the line using the calculated gradient. $y = 2x$

The correct option is B.

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