Mathwise Academy | Grade 11 Mathematics Paper 1 - Preparatory Paper 1
M

MATHWISE ACADEMY

Grade 11 Mathematics: Preparatory Paper 1 • Paper 1

INSTRUCTIONS AND INFORMATION

  • This question paper consists of 5 questions totaling 150 marks.
  • Answer ALL questions.
  • Clearly show ALL calculations, diagrams, and graphs used in determining your answers. Full marks will not necessarily be awarded for answers only.
  • You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
  • Round off answers to TWO decimal places, unless stated otherwise.
  • Diagrams are NOT necessarily drawn to scale.

QUESTION 1 [45]

1.1 Solve for \( x \) in each of the following:

  • 1.1.1 \( x^2 - 5x - 14 = 0 \) (3)
  • 1.1.2 \( 3x^2 - 7x - 1 = 0 \) (correct to TWO decimal places) (4)
  • 1.1.3 \( \sqrt{x + 5} - x = -1 \) (5)
  • 1.1.4 \( x^2 - 3x \le 4 \) (4)
  • 1.1.5 \( x^{\frac{2}{3}} = 16 \) (3)
  • 1.2 Solve simultaneously for \( x \) and \( y \):
    \( y - 2x = -1 \) and \( x^2 + xy + y^2 = 7 \)     (6)

    • 1.3 The roots of a quadratic equation are given by \( x = \dfrac{-2 \pm \sqrt{4 - 20k}}{2} \).

    • 1.3.1 Determine the value(s) of \( k \) for which the equation will have equal roots. (2)
    • 1.3.2 For which value(s) of \( k \) will the roots be non-real? (2)
  • 1.4 Prove that the roots of the equation \( 3x^2 + px - 2p^2 = 0 \) are rational for all real values of \( p \). (3)

    • 1.5 Simplify the following expressions fully, without using a calculator:

    • 1.5.1 \( \dfrac{3^{2x+1} \cdot 15^{2x-3}}{27^{x-1} \cdot 5^{2x-4}} \) (5)
    • 1.5.2 \( \dfrac{\sqrt{50} + \sqrt{18}}{\sqrt{8}} \) (4)
  • 1.6 Solve for \( x \): \( 2^{x+1} + 2^x = 24 \) (4)

    • QUESTION 2 [25]

    2.1 Consider the quadratic number pattern: 3 ; 10 ; 21 ; 36 ; ...

    • 2.1.1 Write down the next TWO terms of the pattern. (2)
    • 2.1.2 Determine the general term of the pattern in the form \( T_n = an^2 + bn + c \). (4)
    • 2.1.3 Calculate the value of the \( 20^{th} \) term. (2)
    • 2.1.4 Which term in the pattern has a value of 406? (4)

    2.2 The first three terms of an arithmetic sequence are: \( 2x - 1 \) ; \( 3x + 2 \) ; \( 5x - 1 \).

    • 2.2.1 Determine the value of \( x \). (3)
    • 2.2.2 Determine the formula for the nth term, \( T_n \), of the sequence. (3)

    2.3 A different quadratic sequence has a first term of 5. The first differences between the terms of this quadratic sequence form an arithmetic sequence defined by \( d_n = 4n + 1 \).

    • 2.3.1 Write down the first three terms of the first difference sequence. (2)
    • 2.3.2 Hence, determine the second and third terms of the quadratic sequence. (2)
    • 2.3.3 Determine the general term of the quadratic sequence, \( T_n \). (3)

    QUESTION 3 [45]

    3.1 Given the functions \( f(x) = -x^2 - 4x + 5 \) and \( g(x) = x + 5 \).

    • 3.1.1 Determine the coordinates of the \( x \)- and \( y \)-intercepts of \( f \). (4)
    • 3.1.2 Determine the coordinates of the turning point of \( f \). (3)
    • 3.1.3 Sketch the graphs of \( f \) and \( g \) on the grid provided below. Clearly show all intercepts with the axes and the turning point. (4)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.1.4 Calculate the coordinates of the points of intersection of \( f \) and \( g \). (4)
    • 3.1.5 Use your graph to determine the values of \( x \) for which \( f(x) > g(x) \). (2)

    3.2 Sketched conceptually is the graph of \( h(x) = \dfrac{4}{x - 2} + 1 \).

    • 3.2.1 Write down the equations of the asymptotes of \( h \). (2)
    • 3.2.2 Calculate the coordinates of the \( x \)- and \( y \)-intercepts of \( h \). (3)
    • 3.2.3 Sketch the graph of \( h \) on the grid below. (3)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.2.4 Determine the equation of the axis of symmetry of \( h \) that has a negative gradient. (4)

    3.3 Given the exponential function \( k(x) = 2^{x+1} - 4 \).

    • 3.3.1 Write down the equation of the horizontal asymptote of \( k \). (1)
    • 3.3.2 Determine the \( x \)- and \( y \)-intercepts of \( k \). (3)
    • 3.3.3 Sketch the graph of \( k \) on the grid below. (3)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.3.4 Write down the range of \( m(x) = k(x) + 2 \). (2)
    • 3.3.5 The graph of \( k \) is shifted 1 unit to the left and 3 units upwards to form the graph of \( p \). Write down the equation of \( p(x) \). (3)
    • 3.3.6 The graph of \( k \) is reflected in the \( x \)-axis to form the graph of \( q \). Write down the equation of \( q(x) \) in the form \( y = ... \) (4)

    QUESTION 4 [15]

    4.1 A printing press is purchased for R 85 000. After 5 years, its value depreciates to R 35 000.

    • 4.1.1 Calculate the annual rate of depreciation if it is calculated on the reducing balance method. (4)
    • 4.1.2 Calculate the annual rate of depreciation if it is calculated on the straight-line method. (3)
  • 4.2 Sihle deposits R 20 000 into a savings account. Two years later, he deposits a further R 15 000 into the same account. One year after that, he withdraws R 5 000 from the account. The interest rate for the first 3 years is 8,5% p.a. compounded monthly. For the remaining period, the interest rate changes to 9,2% p.a. compounded quarterly. Calculate how much money Sihle will have in his savings account at the end of 5 years from his initial deposit. (8)

    • QUESTION 5 [20]

    5.1 A survey of 80 Grade 11 learners was conducted to determine their participation in three sports: Athletics (A), Basketball (B), and Cricket (C). The results are as follows:

    • 35 learners play Athletics.
    • 30 learners play Basketball.
    • 28 learners play Cricket.
    • 10 learners play both Athletics and Basketball.
    • 8 learners play both Basketball and Cricket.
    • 12 learners play both Athletics and Cricket.
    • 5 learners play all three sports.

    • 5.1.1 Draw a Venn diagram in the space below to represent this information. (5)
    • 5.1.2 How many learners do not play any of the three sports? (2)
    • 5.1.3 Determine the probability that a learner chosen at random plays exactly one sport. (3)

    5.2 The contingency table below shows the results of a survey on whether left-handedness or right-handedness is related to gender in a sample of 120 people.

    Left-Handed Right-Handed Total
    Male 12 48 60
    Female 8 \( a \) \( b \)
    Total 20 100 120
    • 5.2.1 Determine the values of \( a \) and \( b \). (2)
    • 5.2.2 Use calculations to determine whether the events "being male" and "being left-handed" are independent. (4)

    5.3 Events A and B are mutually exclusive. It is given that \( P(A) = 0,4 \), \( P(B) = x \), and \( P(A \text{ or } B) = 0,7 \).

    • 5.3.1 Calculate the value of \( x \). (2)
    • 5.3.2 If events A and B were independent instead of mutually exclusive (with \( P(A) = 0,4 \) and \( P(B) = x \)), what would the new value of \( x \) be to maintain \( P(A \text{ or } B) = 0,7 \)? (2)
    --- END OF PAPER ---
    Mathwise Academy | Grade 11 Mathematics Paper 1 - Preparatory Paper 2
    M

    MATHWISE ACADEMY

    Grade 11 Mathematics: Preparatory Paper 2 • Paper 1

    INSTRUCTIONS AND INFORMATION

    • This question paper consists of 5 questions totaling 150 marks.
    • Answer ALL questions.
    • Clearly show ALL calculations, diagrams, and graphs used in determining your answers. Full marks will not necessarily be awarded for answers only.
    • You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
    • Round off answers to TWO decimal places, unless stated otherwise.
    • Diagrams are NOT necessarily drawn to scale.

    QUESTION 1 [45]

    1.1 Solve for \( x \) in each of the following:

    • 1.1.1 \( 3x^2 - 2x - 6 = 0 \) (correct to TWO decimal places) (4)
    • 1.1.2 \( (x - 2)(x + 3) \le 6 \) (5)
    • 1.1.3 \( \sqrt{x + 2} + x = 4 \) (5)
    • 1.1.4 \( 3^x - 3^{x-2} = 24 \) (4)
  • 1.2 Solve simultaneously for \( x \) and \( y \):
    \( x + y = 5 \) and \( x^2 - xy + y^2 = 13 \)     (6)

    • 1.3 Simplify the following expressions fully, without the use of a calculator:

    • 1.3.1 \( \dfrac{2^{2026} - 2^{2024}}{2^{2024}} \) (3)
    • 1.3.2 \( \dfrac{\sqrt{45} - \sqrt{20}}{\sqrt{5}} \) (3)
    • 1.3.3 \( \dfrac{x^2 - 1}{x^2 + x - 2} \div \dfrac{x^2 - 2x - 3}{x^2 - 4} \) (5)

    1.4 Given the quadratic equation: \( kx^2 - 4x + k = 0 \).

    • 1.4.1 Determine the discriminant of the equation in terms of \( k \). (2)
    • 1.4.2 For which values of \( k \) will the equation have real roots? (4)
  • 1.5 Solve for \( x \) and \( y \) without using a calculator, if \( x, y \in \mathbb{R} \):
    \( (x - 3)^2 + (y + 2)^2 = 0 \)     (4)

    • QUESTION 2 [25]

    2.1 Consider the quadratic sequence: \( -2 \ ; \ -1 \ ; \ 2 \ ; \ 7 \ ; \ \dots \)

    • 2.1.1 Write down the next TWO terms of the sequence. (2)
    • 2.1.2 Determine the general term of the sequence in the form \( T_n = an^2 + bn + c \). (4)
    • 2.1.3 Calculate the value of the \( 50^{th} \) term. (2)
    • 2.1.4 Which term in the sequence has a value of 223? (4)

    2.2 The first three terms of an arithmetic sequence are: \( 2x - 5 \) ; \( 3x \) ; \( x + 20 \).

    • 2.2.1 Calculate the value of \( x \). (3)
    • 2.2.2 Calculate the numerical value of the first term of the sequence. (1)
    • 2.2.3 Determine the formula for the nth term of the sequence. (2)
  • 2.3 The first differences of a particular quadratic sequence form an arithmetic sequence defined by \( T_n = 6n - 2 \). If the first term of the quadratic sequence is 4, determine the general term of the quadratic sequence, \( Q_n \). (7)

    • QUESTION 3 [45]

    3.1 The equation of a hyperbola is given by \( f(x) = \dfrac{a}{x - p} + q \). The asymptotes of \( f \) are \( x = 2 \) and \( y = -1 \). The graph passes through the point \( (4 ; 0) \).

    • 3.1.1 Determine the values of \( a \), \( p \), and \( q \). (4)
    • 3.1.2 Calculate the coordinates of the \( y \)-intercept of \( f \). (3)
    • 3.1.3 Sketch the graph of \( f \) on the grid provided below. Clearly show all asymptotes and intercepts with the axes. (4)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.1.4 Determine the equation of the axis of symmetry of \( f \) that has a negative gradient. (3)

    3.2 Given the functions \( g(x) = -x^2 - 2x + 8 \) and \( h(x) = -x + 2 \).

    • 3.2.1 Determine the \( x \)- and \( y \)-intercepts of \( g \). (4)
    • 3.2.2 Determine the coordinates of the turning point of \( g \). (3)
    • 3.2.3 Calculate the coordinates of the points of intersection of \( g \) and \( h \). (5)
    • 3.2.4 Sketch the graphs of \( g \) and \( h \) on the grid below. (5)
    x y -6 -4 -2 2 4 6 10 8 6 2 0 -2
    • 3.2.5 Use your graphs to determine the values of \( x \) for which \( g(x) \ge h(x) \). (2)
    • 3.2.6 Write down the range of \( m(x) = -g(x) \). (2)

    3.3 Given the exponential function \( k(x) = 2^{x+1} - 4 \).

    • 3.3.1 Write down the equation of the horizontal asymptote of \( k \). (1)
    • 3.3.2 Calculate the \( y \)-intercept of \( k \). (2)
    • 3.3.3 Calculate the \( x \)-intercept of \( k \). (2)
    • 3.3.4 The graph of \( k \) is translated 2 units downwards and 1 unit to the right to form the graph of \( p \). Determine the equation of \( p(x) \) in the form \( y = \dots \) (3)
    • 3.3.5 Sketch the graph of \( k(x) \) on the grid below. (2)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6

    QUESTION 4 [15]

    • 4.1 A transport company purchases a delivery vehicle for R 250 000. After 6 years, the vehicle is traded in for its book value of R 120 000. Calculate the annual rate of depreciation if it is calculated on the reducing balance method. (4)
    • 4.2 Convert a nominal interest rate of 12% p.a. compounded monthly to an effective annual interest rate. (3)
    • 4.3 Samuel invests R 30 000 in a savings account. For the first 2 years, the interest rate is 9% p.a. compounded monthly. For the next 3 years, the interest rate changes to 10,5% p.a. compounded quarterly. At the end of the second year, Samuel withdraws R 10 000 from the account. Calculate the total amount Samuel will have in the account at the end of the 5-year period. (8)

    QUESTION 5 [20]

    5.1 For two events A and B, it is given that \( P(A) = 0,4 \) and \( P(B) = x \).

    • 5.1.1 If events A and B are mutually exclusive, and \( P(A \text{ or } B) = 0,65 \), determine the value of \( x \). (2)
    • 5.1.2 If events A and B are independent, and \( P(A \text{ or } B) = 0,65 \), determine the value of \( x \). (4)

    5.2 A survey of 100 Grade 11 learners was conducted to find out which elective subjects they take. The subjects are Mathematics (M), Physical Sciences (S), and Accounting (A).

    • 40 learners take Mathematics.
    • 50 learners take Physical Sciences.
    • 25 learners take Accounting.
    • 15 learners take Mathematics and Physical Sciences.
    • 10 learners take Physical Sciences and Accounting.
    • 8 learners take Mathematics and Accounting.
    • 5 learners take all three subjects.

    • 5.2.1 Draw a Venn diagram in the space below to represent this information. (5)
    • 5.2.2 Calculate the probability that a learner chosen at random takes exactly ONE of these three subjects. (3)
    • 5.2.3 How many learners do not take any of these three subjects? (2)

    5.3 A bag contains 6 red balls and 4 blue balls. Two balls are drawn from the bag, one after the other, without replacement.

    • 5.3.1 Draw a tree diagram to represent all possible outcomes of this experiment. Clearly indicate the probabilities on the branches. (4)
    • 5.3.2 Calculate the probability of drawing one red ball and one blue ball in any order. (2)
    --- END OF PAPER ---
    Mathwise Academy | Grade 11 Mathematics Paper 1 - Preparatory Paper 3
    M

    MATHWISE ACADEMY

    Grade 11 Mathematics: Preparatory Paper 3 • Paper 1

    INSTRUCTIONS AND INFORMATION

    • This question paper consists of 5 questions totaling 150 marks.
    • Answer ALL questions.
    • Clearly show ALL calculations, diagrams, and graphs used in determining your answers. Full marks will not necessarily be awarded for answers only.
    • You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
    • Round off answers to TWO decimal places, unless stated otherwise.
    • Diagrams are NOT necessarily drawn to scale.

    QUESTION 1 [45]

    1.1 Solve for \( x \) in each of the following:

    • 1.1.1 \( 2x^2 - 5x + 2 = 0 \) (3)
    • 1.1.2 \( x^2 - 3x - 5 = 0 \) (correct to TWO decimal places) (4)
    • 1.1.3 \( x - \sqrt{x + 2} = 4 \) (5)
    • 1.1.4 \( x^2 + 5x \le -6 \) (4)
    • 1.1.5 \( 2^{x+1} - 2^{x-1} = 24 \) (4)
  • 1.2 Solve simultaneously for \( x \) and \( y \):
    \( y - 2x = 3 \) and \( x^2 + y^2 - xy = 7 \)     (6)
  • 1.3 Given the quadratic equation: \( kx^2 - 2x + k = 0 \). Determine the values of \( k \) for which the roots of the equation will be real. (4)

    • 1.4 Simplify the following expressions fully, without the use of a calculator:

    • 1.4.1 \( \dfrac{3^{2027} - 3^{2025}}{3^{2025}} \) (3)
    • 1.4.2 \( \dfrac{\sqrt{27} + \sqrt{12}}{\sqrt{75}} \) (4)
    • 1.4.3 \( \dfrac{2x^2 - x - 1}{x^3 - x} \div \dfrac{2x + 1}{x^2 + x} \) (4)
  • 1.5 Solve for \( x \) : \( x^{\frac{3}{2}} = 64 \) (4)

    • QUESTION 2 [25]

    2.1 Consider the quadratic sequence: \( 4 \ ; \ 7 \ ; \ 14 \ ; \ 25 \ ; \ \dots \)

    • 2.1.1 Write down the next TWO terms of the sequence. (2)
    • 2.1.2 Determine the general term of the sequence in the form \( T_n = an^2 + bn + c \). (4)
    • 2.1.3 Calculate the value of the \( 20^{th} \) term. (2)
    • 2.1.4 Which term in the sequence has a value of 253? (4)

    2.2 The first differences of a quadratic sequence form an arithmetic sequence defined by \( T_n = 4n + 2 \).

    • 2.2.1 If the first term of the quadratic sequence is 1, determine the second and third terms of the quadratic sequence. (3)
    • 2.2.2 Determine the general term of the quadratic sequence, \( Q_n \). (4)

    2.3 The first three terms of a linear sequence are: \( 3p - 2 \) ; \( p + 4 \) ; \( 2p + 1 \).

    • 2.3.1 Calculate the value of \( p \). (3)
    • 2.3.2 Determine the value of the \( 50^{th} \) term in the sequence. (3)

    QUESTION 3 [45]

    3.1 The graphs of the functions \( f(x) = -x^2 + 2x + 8 \) and \( g(x) = -x + 2 \) are given.

    • 3.1.1 Determine the coordinates of the \( x \)- and \( y \)-intercepts of \( f \). (4)
    • 3.1.2 Determine the coordinates of the turning point of \( f \). (3)
    • 3.1.3 Calculate the coordinates of the points of intersection of \( f \) and \( g \). (5)
    • 3.1.4 Sketch the graphs of \( f \) and \( g \) on the grid provided below. Clearly show all intercepts, the turning point, and points of intersection. (5)
    x y -6 -4 -2 2 4 6 10 8 6 2 0 -2
    • 3.1.5 Use your graphs to determine the values of \( x \) for which \( f(x) > g(x) \). (2)
    • 3.1.6 Determine the maximum vertical distance between \( f \) and \( g \) for \( x \in [-2 ; 4] \). (4)

    3.2 Sketched conceptually is the graph of a hyperbola \( h(x) = \dfrac{k}{x - 1} + 2 \). The graph passes through the point \( (2 ; 5) \).

    • 3.2.1 Determine the value of \( k \). (2)
    • 3.2.2 Calculate the coordinates of the \( x \)- and \( y \)-intercepts of \( h \). (3)
    • 3.2.3 Sketch the graph of \( h \) on the grid below, clearly showing the asymptotes and intercepts. (4)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.2.4 Determine the equation of the axis of symmetry of \( h \) that has a positive gradient. (3)

    3.3 Given the exponential function \( p(x) = 2^{-x} - 4 \).

    • 3.3.1 Write down the equation of the horizontal asymptote of \( p \). (1)
    • 3.3.2 Determine the coordinates of the \( x \)- and \( y \)-intercepts of \( p \). (4)
    • 3.3.3 Sketch the graph of \( p \) on the grid below. (3)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6
    • 3.3.4 The graph of \( p \) is transformed to become the graph of \( q(x) = 2^x \). Describe the transformations required to map \( p \) onto \( q \) in words. (2)

    QUESTION 4 [15]

    • 4.1 A manufacturing machine depreciates at a rate of 15% p.a. on the reducing balance method. After 4 years, the book value of the machine is R 45 000. Calculate the original cost of the machine. (4)
    • 4.2 Sarah invests R 25 000 in an account. For the first 2 years, the interest rate is 10% p.a. compounded monthly. For the remaining 3 years, the interest rate changes to 12% p.a. compounded quarterly. Calculate the final amount Sarah will have in her account after the 5-year period. (5)
    • 4.3 Convert a nominal interest rate of 14% p.a. compounded monthly to an effective annual interest rate. (3)
    • 4.4 A new laptop costs R 18 000. If the average rate of inflation is 6% p.a., calculate the expected cost of the same laptop in 3 years' time. (3)

    QUESTION 5 [20]

    • 5.1 Events A and B are mutually exclusive. It is given that \( P(A) = 0,3 \) and \( P(B) = 0,5 \). Calculate \( P(A \text{ or } B) \). (2)
    • 5.2 Events C and D are independent. It is given that \( P(C) = 0,4 \) and \( P(C \text{ or } D) = 0,7 \). Calculate \( P(D) \). (4)

    5.3 A survey of 50 Grade 11 learners was conducted regarding their participation in three sports: Rugby (R), Cricket (C), and Hockey (H). The following information was gathered:

    • 20 learners play Rugby.
    • 25 learners play Cricket.
    • 15 learners play Hockey.
    • 10 learners play both Rugby and Cricket in total.
    • 8 learners play both Cricket and Hockey in total.
    • 7 learners play both Rugby and Hockey in total.
    • 3 learners play all three sports.

    • 5.3.1 Draw a Venn diagram in the space below to represent this information. (5)
      • Draw Venn Diagram Here
    • 5.3.2 Calculate the probability that a learner chosen at random plays exactly TWO of the three sports. (2)
    • 5.3.3 Calculate the probability that a learner chosen at random plays none of these three sports. (1)

    5.4 A bag contains 5 green marbles and 7 yellow marbles. Two marbles are drawn consecutively from the bag without replacement.

    • 5.4.1 Draw a tree diagram to represent all possible outcomes. Clearly indicate the probabilities on the branches. (4)
      • Draw Tree Diagram Here
    • 5.4.2 Calculate the probability of drawing two marbles of different colours. (2)
    --- END OF PAPER ---
    Mathwise Academy | Grade 11 Mathematics Paper 1 - Preparatory Paper 4
    M

    MATHWISE ACADEMY

    Grade 11 Mathematics: Preparatory Paper 4 • Paper 1

    INSTRUCTIONS AND INFORMATION

    • This question paper consists of 5 questions totaling 150 marks.
    • Answer ALL questions.
    • Clearly show ALL calculations, diagrams, and graphs used in determining your answers. Full marks will not necessarily be awarded for answers only.
    • You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.
    • Round off answers to TWO decimal places, unless stated otherwise.
    • Diagrams are NOT necessarily drawn to scale.

    QUESTION 1 [45]

    1.1 Solve for \( x \) in each of the following:

    • 1.1.1 \( 2x^2 + 7x - 4 = 0 \) (3)
    • 1.1.2 \( x^2 - 4x - 8 = 0 \) (correct to TWO decimal places) (4)
    • 1.1.3 \( -2x^2 + 5x \ge -3 \) (4)
    • 1.1.4 \( \sqrt{3x + 1} - x = -1 \) (5)
  • 1.2 Solve simultaneously for \( x \) and \( y \):
    \( 2y + x = 3 \) and \( y^2 + 2xy = -4x \)     (6)

    • 1.3 Consider the equation: \( 2x^2 - 4x + (k - 1) = 0 \).

    • 1.3.1 Determine the value(s) of \( k \) for which the equation will have non-real roots. (4)
    • 1.3.2 Prove that if \( k = 1 \), the roots of the equation will be rational. (2)

    1.4 Simplify the following expressions fully, without the use of a calculator:

    • 1.4.1 \( \dfrac{5^{x+1} \cdot 10^{x-2}}{2^{x-3} \cdot 25^{x-1}} \) (4)
    • 1.4.2 \( \dfrac{\sqrt{48} - \sqrt{12}}{\sqrt{27}} \) (3)

    1.5 Solve for \( x \) without using a calculator:

    • 1.5.1 \( x^{\frac{3}{4}} = 8 \) (3)
    • 1.5.2 \( 3^x + 3^{x+1} - 3^{x-1} = \dfrac{11}{27} \) (5)
  • 1.6 Given: \( P = \dfrac{\sqrt{x - 2}}{x^2 - 9} \). Determine the values of \( x \) for which \( P \) is a real number. (2)

    • QUESTION 2 [25]

    2.1 A quadratic number pattern has a second term of 1, a third term of -6, and a fifth term of -26.
    (Note: Minor error corrected from original paper where the 5th term was incorrectly given as -14)

    • 2.1.1 Show that the first term of the pattern is 6. (4)
    • 2.1.2 Determine the general term of the sequence in the form \( T_n = an^2 + bn + c \). (4)
    • 2.1.3 Calculate the value of the \( 30^{th} \) term. (2)
    • 2.1.4 Determine the maximum value of this quadratic pattern. (4)

    2.2 The first differences of a quadratic sequence form an arithmetic sequence. The first three terms of this arithmetic sequence are: \( 2p - 4 \) ; \( p + 1 \) ; \( 3p - 3 \).

    • 2.2.1 Determine the value of \( p \). (3)
    • 2.2.2 Write down the general term of the arithmetic sequence, \( d_n \). (2)
    • 2.2.3 If the first term of the original quadratic sequence is 5, calculate the fourth term of the quadratic sequence. (3)
  • 2.3 The sequence \( 4 \ ; \ t \ ; \ 32 \ ; \ \dots \) can be considered as either an arithmetic sequence or a geometric sequence. Determine the value(s) of \( t \) for both possibilities. (3)

    • QUESTION 3 [45]

    3.1 The graphs of \( f(x) = ax^2 + bx + c \) and \( g(x) = mx + k \) are sketched below. The turning point of \( f \) is \( T(2 ; 9) \). The graph of \( f \) intersects the \( x \)-axis at \( A(-1 ; 0) \) and \( B \). The straight line \( g \) passes through point \( A \) and the \( y \)-intercept of \( f \), which is point \( C \).

    x y O A(-1; 0) B T(2; 9) C g f
    • 3.1.1 Show that the equation of \( f \) can be written as \( f(x) = -x^2 + 4x + 5 \). (4)
    • 3.1.2 Determine the equation of \( g(x) \). (3)
    • 3.1.3 Calculate the coordinates of point \( B \). (2)
    • 3.1.4 Calculate the maximum vertical distance between \( f \) and \( g \) for \( x \in [-1 ; 0] \). (4)
    • 3.1.5 Use the graphs to determine the values of \( x \) for which \( f(x) \cdot g(x) < 0 \). (3)
    • 3.1.6 The graph of \( f \) is reflected in the \( y \)-axis to form the graph of \( h \). Write down the equation of \( h(x) \) in the form \( y = \dots \) (3)

    3.2 Sketched below is the graph of a hyperbola defined by \( p(x) = \dfrac{a}{x - h} + k \). The asymptotes of the graph intersect at the point \( (1 ; -2) \). The graph passes through the \( x \)-axis at \( (2 ; 0) \).

    x y O x = 1 y = -2 (1; -2) p (2; 0)
    • 3.2.1 Determine the values of \( a \), \( h \), and \( k \), and hence write down the equation of \( p(x) \). (4)
    • 3.2.2 Calculate the coordinates of the \( y \)-intercept of \( p \). (2)
    • 3.2.3 Determine the equation of the axis of symmetry of \( p \) that has a negative gradient. (3)
    • 3.2.4 Write down the domain of \( p \). (1)

    3.3 The graph of an exponential function \( m(x) = a \cdot b^{x+p} + q \) is sketched below. The horizontal asymptote is given by \( y = -3 \). The graph passes through the origin \( (0 ; 0) \) and the point \( (-1 ; -2) \).

    x y O y = -3 m (0; 0) (-1; -2)
    • 3.3.1 Use the given information to deduce the equation of \( m(x) \). (5)
    • 3.3.2 Use your equation to find the value of \( x \) for which \( m(x) = 24 \). (3)
    • 3.3.3 The graph of \( m \) is shifted 2 units downwards. Write down the new range of the graph. (2)
    • 3.4 A function \( w(x) \) has roots at \( x = -2 \) and \( x = 5 \). It has a local maximum turning point at \( (1,5 ; 12) \). Determine the coordinates of the turning point of \( k(x) \) if \( k(x) = w(x - 3) + 4 \). (3)
    • 3.5 Sketch the graph of \( y = -\dfrac{4}{x} \) for \( x \in [-4 ; 4] \) on the grid provided. (3)
    x y -6 -4 -2 2 4 6 6 4 2 -2 -4 -6

    QUESTION 4 [15]

    4.1 A company buys new industrial machinery for R 650 000. They have the option of calculating depreciation using either the straight-line method at 12% p.a. or the reducing balance method at 15% p.a.

    • 4.1.1 Calculate the book value of the machinery after 4 years using the straight-line method. (2)
    • 4.1.2 Calculate the book value of the machinery after 4 years using the reducing balance method. (2)
    • 4.1.3 Which method provides a higher book value after 4 years? (1)
    • 4.2 A bank offers an interest rate of 9,6% p.a. compounded monthly. Calculate the equivalent effective annual interest rate. (3)
    • 4.3 Samuel invests R 40 000 into a savings account to save for a deposit on a house.
      • For the first 2 years, the interest rate is 8,5% p.a. compounded quarterly.
      • After exactly 2 years, he adds another R 15 000 to the account.
      • For the remaining 3 years, the interest rate changes to 9,2% p.a. compounded monthly.
      Calculate the total amount in Samuel's account at the end of the 5-year period.       (7)

    QUESTION 5 [20]

    5.1 Two events, A and B, are such that \( P(A) = 0,45 \) and \( P(B) = 0,3 \).

    • 5.1.1 Calculate \( P(A \text{ or } B) \) if A and B are mutually exclusive. (2)
    • 5.1.2 Calculate \( P(A \text{ or } B) \) if A and B are independent. (4)

    5.2 The contingency table below shows the distribution of 150 Grade 11 learners based on their gender and whether they passed or failed a recent Mathematics test.

    Passed Failed Total
    Male 45 30 75
    Female \( x \) \( y \) 75
    Total 95 55 150
    • 5.2.1 Determine the values of \( x \) and \( y \). (2)
    • 5.2.2 If a learner is selected at random, calculate the probability that the learner is female AND failed the test. (2)
    • 5.2.3 Are the events "Being Male" and "Failing the test" independent? Support your answer with relevant calculations. (4)

    5.3 A school offers three extracurricular clubs: Chess (C), Drama (D), and Robotics (R). A survey of 80 learners yielded the following information:

    • 30 learners joined Chess.
    • 35 learners joined Drama.
    • 25 learners joined Robotics.
    • 12 learners joined both Chess and Drama.
    • 8 learners joined both Drama and Robotics.
    • 10 learners joined both Chess and Robotics.
    • 4 learners joined all three clubs.

    • 5.3.1 Draw a Venn diagram in the space below to represent the data. (5)
      • Draw Venn Diagram Here
    • 5.3.2 Calculate the probability that a randomly selected learner joined exactly one club. (1)
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