Mathwise Academy | Grade 9 Test 1 - Straight Line Graphs & Probability

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Straight Line Graphs & Probability

Time: 1 Hour Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
You may use an approved non-programmable scientific calculator.
Leave all probability answers as fractions in their simplest form unless specified otherwise.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: PROBABILITY BASICS [12]

1.1 A bag contains \( 5 \) red marbles, \( 4 \) blue marbles, and \( 3 \) yellow marbles. One marble is drawn from the bag at random.

1.1.1 What is the total number of possible outcomes? (1)

1.1.2 Calculate the probability of drawing a red marble. (2)

1.1.3 Calculate the probability of drawing a blue OR a yellow marble. (2)

1.1.4 Calculate the probability of drawing a green marble. (1)

1.2 A fair \( 8 \)-sided spinner is numbered from \( 1 \) to \( 8 \).

1.2.1 What is the probability of the spinner landing on a prime number? (3)

1.3 Theoretical vs Empirical Probability:
A student rolls a standard \( 6 \)-sided die \( 60 \) times. The die lands on the number '4' exactly \( 12 \) times.

1.3.1 Calculate the relative frequency (empirical probability) of rolling a '4' in this experiment. (2)

1.3.2 Does the relative frequency match the theoretical probability exactly? Briefly explain why or why not. (1)

QUESTION 2: COMPOUND EVENTS (TREE DIAGRAMS) [10]

2.1 A game involves two steps: first, flipping a fair coin (Heads or Tails), and second, rolling a fair \( 4 \)-sided die (numbered 1, 2, 3, 4).

2.1.1 Draw a complete tree diagram to illustrate all the possible outcomes of this game. Clearly show the probabilities on the branches. (4)

2.1.2 List the complete sample space for this game (e.g., H1, H2, etc.). (2)

2.1.3 Calculate the probability of getting a Tail AND an even number. (2)

2.1.4 Calculate the probability of getting a Head AND a number greater than \( 2 \). (2)

QUESTION 3: DRAWING AND INTERPRETING GRAPHS [14]

3.1 Consider the linear equation: \( y = 2x - 4 \)

3.1.1 Determine the \( y \)-intercept of the graph. (1)

3.1.2 Calculate the \( x \)-intercept of the graph. (2)

3.1.3 Use your intercepts to draw a neat sketch graph of \( y = 2x - 4 \) on a Cartesian plane. Clearly label the axes and the intercept coordinates. (4)

3.2 An equation of a straight line is given as: \( 3y + 6x = 12 \).

3.2.1 Rewrite the equation in the standard form \( y = mx + c \). (3)

3.2.2 State the gradient (\( m \)) of this line. (2)

3.2.3 Is the graph of this line increasing or decreasing? Give a reason for your answer. (2)

QUESTION 4: DETERMINING THE EQUATION OF A LINE [14]

4.1 Determine the equation of the straight line, in the form \( y = mx + c \), for each of the following scenarios:

4.1.1 A line passing through the \( y \)-intercept \( (0; 2) \) and the point \( B(2; 8) \). (4)

4.1.2 A line passing through the origin \( (0; 0) \) and the point \( C(4; -2) \). (3)

4.1.3 Line 1 has the equation \( y = 4x - 5 \). Line 2 is parallel to Line 1 and passes through the \( y \)-axis at \( -3 \). Write down the equation of Line 2. (3)

4.2 Study the graph of the straight line below.

4.2.1 Determine the equation of this straight line. Show your calculations for the gradient. (4)

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Mathwise Academy | Grade 9 Test 2 - Straight Line Graphs & Probability (Higher Order)

MATHWISE ACADEMY

Grade 9 Mathematics: Test 2 • Straight Line Graphs & Probability (Higher Order)

Time: 1 Hour Total Marks: 50

INSTRUCTIONS AND INFORMATION

Answer ALL the questions.
Read all questions carefully before you begin.
Show ALL working out where applicable. Full marks will not necessarily be awarded for answers only.
You may use an approved non-programmable scientific calculator.
Leave all probability answers as fractions in their simplest form unless specified otherwise.
To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: PROBABILITY APPLICATIONS [12]

1.1 A biased spinner is divided into four colours. The probabilities of landing on three of the colours are given in the table below:

Colour	Red	Blue	Green	Yellow
Probability	\( 0,15 \)	\( 0,45 \)	\( 0,2 \)	\( x \)

1.1.1 Calculate the value of \( x \), the probability of landing on Yellow. (2)

1.2 Algebraic Probability:
A jar contains some red marbles, \( 5 \) blue marbles, and \( 3 \) green marbles. The probability of randomly drawing a blue marble is exactly \( \dfrac{1}{4} \).

1.2.1 Let the number of red marbles be \( r \). Set up an equation and solve for \( r \) to determine the number of red marbles in the jar. (4)

1.3 Relative Frequency (Empirical Probability):
A factory produces lightbulbs. The quality control manager tests \( 200 \) lightbulbs at random from the production line and finds that \( 12 \) of them are defective (broken).

1.3.1 Calculate the relative frequency of a lightbulb being defective. (2)

1.3.2 Calculate the relative frequency of a lightbulb being in good working condition. (2)

1.3.3 If the factory produces \( 5\,000 \) lightbulbs in a week, predict how many of them will be defective based on this sample. (2)

QUESTION 2: COMPOUND EVENTS & TWO-WAY TABLES [12]

2.1 A group of \( 80 \) Grade 9 learners were asked to choose their favourite extramural activity. They could only choose one: Sport, Music, or Drama. The results are recorded in the two-way contingency table below. Some information is missing.

	Sport	Music	Drama	TOTAL
Boys	22	a	5	35
Girls	16	14	b	c
TOTAL	d	22	20	80

2.1.1 Determine the missing values \( a \), \( b \), \( c \), and \( d \). (4)

2.1.2 If a learner is chosen at random from the whole group, what is the probability that the learner is a Girl? (2)

2.1.3 What is the probability that a learner chosen at random is a Boy who chose Sport? (2)

2.2 Tree Diagrams:
Bag A contains \( 2 \) Red tokens and \( 3 \) Blue tokens. Bag B contains \( 4 \) Red tokens and \( 1 \) Blue token. A student first draws one token at random from Bag A, and then draws one token at random from Bag B.

2.2.1 Draw a complete tree diagram to illustrate all the possible outcomes of this two-step experiment. Clearly show the probabilities on the branches. (4)

QUESTION 3: DRAWING AND INTERPRETING GRAPHS [13]

3.1 Consider the linear equation given in standard form: \(\quad 3x + 2y = 12\)

3.1.1 Rewrite the equation into the gradient-intercept form (\( y = mx + c \)). (3)

3.1.2 Write down the gradient (\( m \)) and the \( y \)-intercept (\( c \)) of the line. (2)

3.1.3 Calculate the \( x \)-intercept of the graph. (2)

3.1.4 Draw a neat sketch graph of the line on a Cartesian plane. Clearly indicate the coordinates of the \( x \)-intercept and the \( y \)-intercept. (3)

3.1.5 Is the graph increasing or decreasing? (1)

3.1.6 Prove algebraically whether the point \( (2; 3) \) lies on this straight line. (2)

QUESTION 4: DETERMINING THE EQUATION OF A LINE [13]

4.1 Determine the equation of the straight line, in the form \( y = mx + c \), for each of the following scenarios. Show all your calculations:

4.1.1 A line that passes through the coordinates \( A(-2; 4) \) and \( B(1; -5) \). (5)

4.1.2 Line 1 has the equation \( y = -2x + 1 \). Line 2 is parallel to Line 1 and passes through the point \( (0; 6) \). Determine the equation of Line 2. (3)

4.2 Study the graph of the straight line below.

4.2.1 Determine the equation of this straight line. Show your working to calculate the gradient. (5)

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