Mathwise Academy | Grade 9 Test 1 - Numeric & Geometric Patterns
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Numeric & Geometric Patterns

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 scientific calculator.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: LINEAR NUMBER PATTERNS [15]

1.1 Consider the following number sequence:

7   ;   11   ;   15   ;   19   ;   ........
  • 1.1.1 Write down the next two terms in the sequence. (2)
  • 1.1.2 Determine the general rule (or \(n^{\text{th}}\) term) for the sequence in the form \(T_n = ........\) (3)
  • 1.1.3 Calculate the value of the \(45^{\text{th}}\) term (\(T_{45}\)). (2)
  • 1.1.4 Which term in the sequence is equal to \(203\)? (3)

1.2 The general rule for a different number pattern is given as \( T_n = -3n + 12 \).

  • 1.2.1 Determine the first three terms of this sequence. (3)
  • 1.2.2 Which term in this sequence will have a value of \(-48\)? (2)

QUESTION 2: NON-LINEAR AND OTHER PATTERNS [12]

2.1 Consider the following geometric number pattern:

3   ;   -6   ;   12   ;   -24   ;   ........
  • 2.1.1 Write down the constant ratio for this sequence. (1)
  • 2.1.2 Determine the next two terms in the sequence. (2)

2.2 Consider the number sequence:

1   ;   8   ;   27   ;   64   ;   ........
  • 2.2.1 State the rule for this sequence in words. (2)
  • 2.2.2 Determine the value of the \(10^{\text{th}}\) term. (2)

2.3 A sequence with a constant difference is given as:

[5]
\(x\)   ;   12   ;   17   ;   \(y\)   ;   27

Calculate the values of \(x\) and \(y\). Show your reasoning.

QUESTION 3: GEOMETRIC (VISUAL) PATTERNS [13]

A pattern is built using matchsticks to form a chain of squares, as shown below:

Pattern 1
Pattern 2
Pattern 3
  • 3.1 Copy and complete the table below for the number of matchsticks required for each pattern number. (2)
    Pattern Number (\(n\)) 1 2 3 4 5
    Number of Matchsticks (\(M\)) 4 7 10
  • 3.2 Determine the general formula for the number of matchsticks, \(M\), in terms of the pattern number, \(n\). (3)
  • 3.3 Use your formula to calculate how many matchsticks are required to build Pattern 40. (2)
  • 3.4 If a certain pattern in the sequence requires exactly 106 matchsticks to build, which pattern number is it? (3)
  • 3.5 A student has a box containing exactly 500 matchsticks. What is the largest complete pattern number they can build? (3)

QUESTION 4: APPLICATION PATTERNS [10]

A local theatre has seats arranged in rows. The first row has 15 seats, the second row has 19 seats, the third row has 23 seats, and this pattern continues uniformly towards the back of the theatre.

  • 4.1 How many seats are there in the \(12^{\text{th}}\) row? (3)
  • 4.2 Which specific row contains exactly 87 seats? (3)
  • 4.3 The theatre has a total of 20 rows. The manager claims that the final row has fewer than 90 seats.
    Prove, with a calculation, whether the manager's statement is TRUE or FALSE.     (4)
--- END OF PAPER ---
Mathwise Academy | Grade 9 Test 1 - Numeric & Geometric Patterns
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Numeric & Geometric Patterns

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 scientific calculator.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: LINEAR NUMBER PATTERNS [15]

1.1 Consider the following number sequence:

7   ;   11   ;   15   ;   19   ;   ........
  • 1.1.1 Write down the next two terms in the sequence. (2)
  • 1.1.2 Determine the general rule (or \(n^{\text{th}}\) term) for the sequence in the form \(T_n = ........\) (3)
  • 1.1.3 Calculate the value of the \(45^{\text{th}}\) term (\(T_{45}\)). (2)
  • 1.1.4 Which term in the sequence is equal to \(203\)? (3)

1.2 The general rule for a different number pattern is given as \( T_n = -3n + 12 \).

  • 1.2.1 Determine the first three terms of this sequence. (3)
  • 1.2.2 Which term in this sequence will have a value of \(-48\)? (2)

QUESTION 2: NON-LINEAR AND OTHER PATTERNS [12]

2.1 Consider the following geometric number pattern:

3   ;   -6   ;   12   ;   -24   ;   ........
  • 2.1.1 Write down the constant ratio for this sequence. (1)
  • 2.1.2 Determine the next two terms in the sequence. (2)

2.2 Consider the number sequence:

1   ;   8   ;   27   ;   64   ;   ........
  • 2.2.1 State the rule for this sequence in words. (2)
  • 2.2.2 Determine the value of the \(10^{\text{th}}\) term. (2)

2.3 A sequence with a constant difference is given as:

[5]
\(x\)   ;   12   ;   17   ;   \(y\)   ;   27

Calculate the values of \(x\) and \(y\). Show your reasoning.

QUESTION 3: GEOMETRIC (VISUAL) PATTERNS [13]

A pattern is built using matchsticks to form a chain of squares, as shown below:

Pattern 1
Pattern 2
Pattern 3
  • 3.1 Copy and complete the table below for the number of matchsticks required for each pattern number. (2)
    Pattern Number (\(n\)) 1 2 3 4 5
    Number of Matchsticks (\(M\)) 4 7 10
  • 3.2 Determine the general formula for the number of matchsticks, \(M\), in terms of the pattern number, \(n\). (3)
  • 3.3 Use your formula to calculate how many matchsticks are required to build Pattern 40. (2)
  • 3.4 If a certain pattern in the sequence requires exactly 106 matchsticks to build, which pattern number is it? (3)
  • 3.5 A student has a box containing exactly 500 matchsticks. What is the largest complete pattern number they can build? (3)

QUESTION 4: APPLICATION PATTERNS [10]

A local theatre has seats arranged in rows. The first row has 15 seats, the second row has 19 seats, the third row has 23 seats, and this pattern continues uniformly towards the back of the theatre.

  • 4.1 How many seats are there in the \(12^{\text{th}}\) row? (3)
  • 4.2 Which specific row contains exactly 87 seats? (3)
  • 4.3 The theatre has a total of 20 rows. The manager claims that the final row has fewer than 90 seats.
    Prove, with a calculation, whether the manager's statement is TRUE or FALSE.     (4)
--- END OF PAPER ---
Mathwise Academy | Grade 9 Test 1 - Algebraic Expressions
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Algebraic Expressions

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 scientific calculator.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: ALGEBRAIC TERMINOLOGY AND SUBSTITUTION [10]

1.1 Consider the following algebraic expression: \(\quad 5x^3 - 4x^2y + 7x - 12\)

  • 1.1.1 How many terms are in this algebraic expression? (1)
  • 1.1.2 Write down the constant term. (1)
  • 1.1.3 Write down the coefficient of \(x^2y\). (1)
  • 1.1.4 What is the degree of this expression with respect to \(x\)? (1)

1.2 Evaluate the following algebraic expression if \( a = -2 \), \( b = 3 \), and \( c = 0 \):

[3]
\[ a^2 - 3ab + c \]

1.3 Write down an algebraic expression for the following statement:

[3]

"The square of a number \(x\), decreased by twice another number \(y\)."

QUESTION 2: ADDITION AND SUBTRACTION OF POLYNOMIALS [8]

2.1 Simplify the following expressions by grouping and adding/subtracting like terms:

  • 2.1.1 \( 4ab^2 - 3a^2b + 2ab^2 + 5a^2b \) (4)
  • 2.1.2 \( (3x^2 - 5x + 2) - (x^2 - 3x + 7) \) (4)

QUESTION 3: MULTIPLICATION OF ALGEBRAIC EXPRESSIONS [18]

3.1 Expand and simplify the following expressions completely:

  • 3.1.1 \( 2x(3x^2 - 4x + 1) \) (3)
  • 3.1.2 \( (x - 5)(x + 3) \) (3)
  • 3.1.3 \( (2x - 3)(3x - 4) \) (4)
  • 3.1.4 \( (x - 4)^2 \) (4)
  • 3.1.5 \( (3x + 2y)(3x - 2y) \) (4)

QUESTION 4: DIVISION AND MIXED OPERATIONS [14]

4.1 Simplify the following algebraic fractions:

  • 4.1.1 \( \dfrac{15x^3 - 10x^2 + 5x}{5x} \) (4)
  • 4.1.2 \( \dfrac{12a^2b^3 - 8a^3b^2}{-4a^2b} \) (4)

4.2 Simplify the following expression combining multiplication and subtraction:

[6]
\[ (x + 2)^2 - x(x - 3) \]
--- END OF PAPER ---
Mathwise Academy | Grade 9 Test 2 - Algebraic Expressions
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 2 • Algebraic Expressions

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 scientific calculator.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: ALGEBRAIC TERMINOLOGY AND SUBSTITUTION [12]

1.1 Consider the following algebraic expression: \(\quad -7p^4 + 3p^2q - 5pq + 11\)

  • 1.1.1 What is the degree of this expression with respect to the variable \(p\)? (1)
  • 1.1.2 Write down the coefficient of \(p^2q\). (1)
  • 1.1.3 Write down the constant term. (1)
  • 1.1.4 Calculate the value of the expression if \( p = 0 \) and \( q = -2 \). (1)

1.2 Evaluate the following algebraic expression if \( x = -3 \), \( y = -1 \), and \( z = 4 \):

[4]
\[ 2x^2 - 3yz + x \]

1.3 Write down an algebraic expression for each of the following statements:

  • 1.3.1 The sum of the squares of two numbers, \(a\) and \(b\). (2)
  • 1.3.2 A number \(y\) is multiplied by \(5\), and the result is subtracted from \(20\). (2)

QUESTION 2: ADDITION AND SUBTRACTION OF POLYNOMIALS [8]

2.1 Simplify the following algebraic expressions:

  • 2.1.1 \( 7x^2y - 4xy^2 - 3x^2y + 8xy^2 - xy^2 \) (3)
  • 2.1.2 Subtract \( (2a^2 - 5ab + 3b^2) \) from \( (5a^2 + 2ab - b^2) \) (5)

QUESTION 3: MULTIPLICATION OF ALGEBRAIC EXPRESSIONS [16]

3.1 Expand and simplify the following expressions completely:

  • 3.1.1 \( -3a^2(4a^2 - 5ab + 2b) \) (3)
  • 3.1.2 \( (2x - 5)(3x + 1) \) (3)
  • 3.1.3 \( (4x - 3)^2 \) (3)
  • 3.1.4 \( (5x + 2y)(5x - 2y) \) (3)

3.2 Simplify the following compound expression:

[4]
\[ 2(x - 1)(x + 3) - x(2x - 5) \]

QUESTION 4: DIVISION AND MIXED OPERATIONS [14]

4.1 Divide the following polynomial by the monomial:

[4]
\[ \dfrac{18a^4b^3 - 12a^3b^2 + 6a^2b}{-6a^2b} \]

4.2 Simplify the following expression combining fractions:

[5]
\[ \dfrac{4x^2 - 8x}{2x} + \dfrac{15x^2 + 5x}{5x} \]

4.3 Simplify the following higher-order expression:

[5]
\[ (2x - 3)^2 - (2x + 3)(2x - 3) \]
--- END OF PAPER ---
Mathwise Academy | Grade 9 Test 1 - Algebraic Equations
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 1 • Algebraic Equations

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.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: SOLVING LINEAR EQUATIONS [12]

1.1 Solve for \( x \) in the following linear equations. Show all your steps clearly:

  • 1.1.1 \( 3x - 5 = x + 7 \) (3)
  • 1.1.2 \( 2(x - 3) = 3(2x + 2) \) (4)
  • 1.1.3 \( 5 - 3(x + 2) = 2x - 11 \) (5)

QUESTION 2: EQUATIONS INVOLVING FRACTIONS [13]

2.1 Solve for \( x \) in the following equations involving fractions:

  • 2.1.1 \( \dfrac{x}{3} + 2 = 5 \) (3)
  • 2.1.2 \( \dfrac{2x - 1}{3} = \dfrac{x + 3}{2} \) (5)
  • 2.1.3 \( \dfrac{x}{2} - \dfrac{x - 1}{5} = 2 \) (5)

QUESTION 3: EXPONENTIAL EQUATIONS [10]

3.1 Solve for \( x \) in the following exponential equations:

  • 3.1.1 \( 2^x = 32 \) (2)
  • 3.1.2 \( 3^{x-2} = 27 \) (3)
  • 3.1.3 \( 5 \cdot 2^x = 40 \) (2)
  • 3.1.4 \( 3^x = \dfrac{1}{9} \) (3)

QUESTION 4: ALGEBRAIC WORD PROBLEMS [15]

4.1 Use algebraic equations to solve the following word problems.
You must set up an equation for each problem.

  • 4.1.1 The sum of three consecutive integers is \( 84 \). Determine the three integers. (5)
  • 4.1.2 A father is currently \( 4 \) times as old as his son. In \( 5 \) years' time, the father will be \( 3 \) times as old as his son will be then. Calculate the current ages of both the father and the son. (5)
  • 4.1.3 The length of a rectangle is \( 3 \text{ cm} \) more than twice its breadth. If the perimeter of the rectangle is \( 42 \text{ cm} \), calculate the actual length and breadth of the rectangle. (5)
--- END OF PAPER ---
Mathwise Academy | Grade 9 Test 2 - Algebraic Equations
M

MATHWISE ACADEMY

Grade 9 Mathematics: Test 2 • Algebraic Equations

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.
  • To save a printable worksheet, tap the green "Download / Print PDF" button above and select "Save as PDF".

QUESTION 1: SOLVING LINEAR EQUATIONS [12]

1.1 Solve for \( x \) in the following linear equations. Show all your steps clearly:

  • 1.1.1 \( 4x - 3 = 2x + 9 \) (3)
  • 1.1.2 \( 3(x - 2) - 2(x + 1) = 4 \) (4)
  • 1.1.3 \( 2 - (x - 3) = 3x - 7 \) (5)

QUESTION 2: EQUATIONS INVOLVING FRACTIONS [13]

2.1 Solve for \( x \) in the following equations involving fractions:

  • 2.1.1 \( \dfrac{x}{4} - 1 = 3 \) (3)
  • 2.1.2 \( \dfrac{3x - 1}{2} = \dfrac{x + 2}{3} \) (5)
  • 2.1.3 \( \dfrac{x + 1}{3} - \dfrac{x - 2}{4} = 2 \) (5)

QUESTION 3: EXPONENTIAL EQUATIONS [10]

3.1 Solve for \( x \) in the following exponential equations:

  • 3.1.1 \( 3^x = 81 \) (2)
  • 3.1.2 \( 5^{x+1} = 125 \) (3)
  • 3.1.3 \( 2 \cdot 4^x = 128 \) (3)
  • 3.1.4 \( 2^x = \dfrac{1}{8} \) (2)

QUESTION 4: ALGEBRAIC WORD PROBLEMS [15]

4.1 Use algebraic equations to solve the following word problems.
You must set up an equation for each problem.

  • 4.1.1 The sum of three consecutive even numbers is \( 72 \). Determine the three numbers. (5)
  • 4.1.2 Sipho is \( 5 \) years older than his brother, Thabo. If the sum of their current ages is \( 31 \), calculate the age of each brother. (5)
  • 4.1.3 The length of a rectangle is \( 2 \text{ cm} \) more than three times its breadth. If the perimeter of the rectangle is \( 44 \text{ cm} \), calculate the actual length and breadth of the rectangle. (5)
--- END OF PAPER ---