Mathematics

1. 1.

   A straight line passes through the points (1, 5) and (4, 17). Find the gradient of the line and write its equation in the form y = mx + c.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Uses gradient = (17 − 5)/(4 − 1). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains gradient = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Substitutes a point to find c, e.g. 5 = 4(1) + c. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Writes the equation y = 4x + 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check the equation using the second point as well; both points must satisfy it.

   Marking points

   • Uses gradient = (17 − 5)/(4 − 1).
   • Obtains gradient = 4.
   • Substitutes a point to find c, e.g. 5 = 4(1) + c.
   • Writes the equation y = 4x + 1.

   Examiner tip: Check the equation using the second point as well; both points must satisfy it.

2. 2.

   The curve y = x² − 2x − 8 crosses the x-axis at two points. Find the coordinates of both points by factorising.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Factorises to (x − 4)(x + 2) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Solves to obtain x = 4 and x = −2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: States both points as (4, 0) and (−2, 0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The x-axis crossing points require y = 0, and coordinates need both x and y values stated.

   Marking points

   • Factorises to (x − 4)(x + 2) = 0.
   • Solves to obtain x = 4 and x = −2.
   • States both points as (4, 0) and (−2, 0).

   Examiner tip: The x-axis crossing points require y = 0, and coordinates need both x and y values stated.

3. 3.

   Find the gradient of the line perpendicular to y = 2x + 3.

   [2 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: States that perpendicular gradients multiply to give −1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains the perpendicular gradient = −1/2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The perpendicular gradient is the negative reciprocal of the original gradient.

   Marking points

   • States that perpendicular gradients multiply to give −1.
   • Obtains the perpendicular gradient = −1/2.

   Examiner tip: The perpendicular gradient is the negative reciprocal of the original gradient.

4. 4.

   A ball is thrown and its height h metres after t seconds is given by h = 20t − 5t². Calculate the time at which the ball reaches its maximum height, and state the maximum height.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Recognises the maximum of a downward parabola occurs at the midpoint between its roots, or uses t = −b/(2a) with a = −5, b = 20. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains t = 2 seconds. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Substitutes t = 2 into h = 20(2) − 5(2)². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: States maximum height = 20 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For h = at² + bt + c, the maximum or minimum always occurs at t = −b/(2a) — a formula worth memorising for any quadratic model.

   Marking points

   • Recognises the maximum of a downward parabola occurs at the midpoint between its roots, or uses t = −b/(2a) with a = −5, b = 20.
   • Obtains t = 2 seconds.
   • Substitutes t = 2 into h = 20(2) − 5(2)².
   • States maximum height = 20 m.

   Examiner tip: For h = at² + bt + c, the maximum or minimum always occurs at t = −b/(2a) — a formula worth memorising for any quadratic model.

5. 5.

   A function is given by f(x) = 3x − 2. Find f⁻¹(x), the inverse function.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Sets y = 3x − 2 and rearranges to make x the subject: x = (y + 2)/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Swaps x and y (or renames appropriately). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: States f⁻¹(x) = (x + 2)/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Check your inverse by substituting a value into f(x) and then into f⁻¹(x) — you should get back your original value.

   Marking points

   • Sets y = 3x − 2 and rearranges to make x the subject: x = (y + 2)/3.
   • Swaps x and y (or renames appropriately).
   • States f⁻¹(x) = (x + 2)/3.

   Examiner tip: Check your inverse by substituting a value into f(x) and then into f⁻¹(x) — you should get back your original value.

6. 6.

   Given f(x) = x² + 1 and g(x) = 2x, find the composite function fg(x).

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Recognises that fg(x) means substituting g(x) into f. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Substitutes g(x) = 2x into f(x) = x² + 1 to get (2x)² + 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: States fg(x) = 4x² + 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: fg(x) means 'apply g first, then f' — always work from the inside outward, substituting the innermost function first.

   Marking points

   • Recognises that fg(x) means substituting g(x) into f.
   • Substitutes g(x) = 2x into f(x) = x² + 1 to get (2x)² + 1.
   • States fg(x) = 4x² + 1.

   Examiner tip: fg(x) means 'apply g first, then f' — always work from the inside outward, substituting the innermost function first.

7. 7.

   A straight line has equation y = −3x + 6. State its gradient and y-intercept, and find its x-intercept.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: States gradient = −3 and y-intercept = 6, read directly from the equation. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Sets y = 0 to find the x-intercept: 0 = −3x + 6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Solves to obtain x = 2, so the x-intercept is (2, 0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For y = mx + c, m and c can be read directly; the x-intercept always requires setting y = 0 and solving.

   Marking points

   • States gradient = −3 and y-intercept = 6, read directly from the equation.
   • Sets y = 0 to find the x-intercept: 0 = −3x + 6.
   • Solves to obtain x = 2, so the x-intercept is (2, 0).

   Examiner tip: For y = mx + c, m and c can be read directly; the x-intercept always requires setting y = 0 and solving.

8. 8.

   Sketch, in words, how the graph of y = x² is transformed to give the graph of y = (x − 3)² + 2.

   [2 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
   2. Work through this mathematical step: States that the graph is translated 3 units in the positive x-direction (to the right). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: States that the graph is also translated 2 units in the positive y-direction (upward). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For y = (x − a)² + b, the graph of y = x² is translated by vector (a, b) — a common source of sign confusion for the x-direction.

   Marking points

   • States that the graph is translated 3 units in the positive x-direction (to the right).
   • States that the graph is also translated 2 units in the positive y-direction (upward).

   Examiner tip: For y = (x − a)² + b, the graph of y = x² is translated by vector (a, b) — a common source of sign confusion for the x-direction.

9. 9.

   A car's distance-time graph shows it travelling at a constant 15 m/s for 20 seconds, then stationary for 10 seconds. Calculate the total distance travelled and the average speed over the whole 30 seconds.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Uses distance = speed × time for the moving phase: 15 × 20. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains distance = 300 m (no additional distance while stationary). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Uses average speed = total distance ÷ total time = 300 ÷ 30. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Obtains average speed = 10 m/s. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Average speed uses the total time of the journey, including any stationary periods, not just the moving time.

   Marking points

   • Uses distance = speed × time for the moving phase: 15 × 20.
   • Obtains distance = 300 m (no additional distance while stationary).
   • Uses average speed = total distance ÷ total time = 300 ÷ 30.
   • Obtains average speed = 10 m/s.

   Examiner tip: Average speed uses the total time of the journey, including any stationary periods, not just the moving time.

10. 10.

    A quadratic graph y = x² − 4x + k touches the x-axis at exactly one point. Calculate the value of k.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that touching the x-axis at exactly one point means the discriminant b² − 4ac = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes a = 1, b = −4, c = k: (−4)² − 4(1)(k) = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Solves 16 − 4k = 0 to obtain k = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A discriminant of zero always corresponds to a repeated root — the graph touching (not crossing) the x-axis at a single point.

    Marking points

    • States that touching the x-axis at exactly one point means the discriminant b² − 4ac = 0.
    • Substitutes a = 1, b = −4, c = k: (−4)² − 4(1)(k) = 0.
    • Solves 16 − 4k = 0 to obtain k = 4.

    Examiner tip: A discriminant of zero always corresponds to a repeated root — the graph touching (not crossing) the x-axis at a single point.

11. 11.

    Points A(2, 3) and B(8, 11) are given. Find the midpoint of AB and the length of AB.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates the midpoint as (5, 7). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the distance formula: √((x₂−x₁)² + (y₂−y₁)²). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Calculates the length AB = √(6² + 8²) = √100 = 10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The distance formula is Pythagoras' theorem applied to the horizontal and vertical differences between the two points.

    Marking points

    • States the midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2).
    • Calculates the midpoint as (5, 7).
    • States the distance formula: √((x₂−x₁)² + (y₂−y₁)²).
    • Calculates the length AB = √(6² + 8²) = √100 = 10.

    Examiner tip: The distance formula is Pythagoras' theorem applied to the horizontal and vertical differences between the two points.

12. 12.

    Find the equation of the line parallel to y = 2x − 5 that passes through the point (3, 4).

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States that a parallel line has the same gradient, so m = 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes the point into y = 2x + c: 4 = 2(3) + c. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Solves to find c = −2, giving the equation y = 2x − 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Parallel lines always share the same gradient — only the y-intercept (c) needs to be found using the given point.

    Marking points

    • States that a parallel line has the same gradient, so m = 2.
    • Substitutes the point into y = 2x + c: 4 = 2(3) + c.
    • Solves to find c = −2, giving the equation y = 2x − 2.

    Examiner tip: Parallel lines always share the same gradient — only the y-intercept (c) needs to be found using the given point.

13. 13.

    Express y = x² − 6x + 11 in the form (x − a)² + b, and hence state the coordinates of the turning point of the curve.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: Completes the square: x² − 6x + 11 = (x − 3)² − 9 + 11. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Simplifies to (x − 3)² + 2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the turning point as (3, 2), a minimum since the coefficient of x² is positive. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For y = (x − a)² + b, the turning point is always at (a, b) — read it directly once the square is completed.

    Marking points

    • Completes the square: x² − 6x + 11 = (x − 3)² − 9 + 11.
    • Simplifies to (x − 3)² + 2.
    • States the turning point as (3, 2), a minimum since the coefficient of x² is positive.

    Examiner tip: For y = (x − a)² + b, the turning point is always at (a, b) — read it directly once the square is completed.

14. 14.

    Find the coordinates of the points where the line y = 2x − 1 intersects the curve y = x² − 2x − 1.

    [4 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Sets the two expressions for y equal: 2x − 1 = x² − 2x − 1. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Rearranges into x² − 4x = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Factorises: x(x − 4) = 0, giving x = 0 or x = 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Substitutes back to find the points (0, −1) and (4, 7). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Substitute each x-value back into the simpler (linear) equation to find its matching y-coordinate.

    Marking points

    • Sets the two expressions for y equal: 2x − 1 = x² − 2x − 1.
    • Rearranges into x² − 4x = 0.
    • Factorises: x(x − 4) = 0, giving x = 0 or x = 4.
    • Substitutes back to find the points (0, −1) and (4, 7).

    Examiner tip: Substitute each x-value back into the simpler (linear) equation to find its matching y-coordinate.

15. 15.

    The value of a car, $V, after t years is modelled by V = 18000 × (0.85)ᵗ. Calculate the value of the car after 5 years, to the nearest dollar.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Substitutes t = 5 into V = 18000 × (0.85)⁵. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates (0.85)⁵ ≈ 0.4437. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States V ≈ $7987 to the nearest dollar. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: In an exponential decay model like this, the base (here 0.85) must be less than 1, since the value decreases every year.

    Marking points

    • Substitutes t = 5 into V = 18000 × (0.85)⁵.
    • Calculates (0.85)⁵ ≈ 0.4437.
    • States V ≈ $7987 to the nearest dollar.

    Examiner tip: In an exponential decay model like this, the base (here 0.85) must be less than 1, since the value decreases every year.

16. 16.

    State the equations of the asymptotes of the graph y = 3/x + 2.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States the vertical asymptote as x = 0, where the function is undefined. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the horizontal asymptote as y = 2, the value the graph approaches as x becomes very large or very negative. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For y = k/x + c, the vertical asymptote is always x = 0 and the horizontal asymptote is always y = c.

    Marking points

    • States the vertical asymptote as x = 0, where the function is undefined.
    • States the horizontal asymptote as y = 2, the value the graph approaches as x becomes very large or very negative.

    Examiner tip: For y = k/x + c, the vertical asymptote is always x = 0 and the horizontal asymptote is always y = c.

17. 17.

    A speed-time graph shows a car accelerating uniformly from 0 to 20 m/s over 8 seconds, then travelling at a constant 20 m/s for a further 12 seconds. Calculate the total distance travelled.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the distance during acceleration as the area of a triangle: (1/2) × 8 × 20 = 80 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates the distance during constant speed as 20 × 12 = 240 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Adds both distances to obtain a total of 320 m. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: On a speed-time graph, the area between the graph and the time axis always represents distance travelled, whatever the shape of that region.

    Marking points

    • Calculates the distance during acceleration as the area of a triangle: (1/2) × 8 × 20 = 80 m.
    • Calculates the distance during constant speed as 20 × 12 = 240 m.
    • Adds both distances to obtain a total of 320 m.

    Examiner tip: On a speed-time graph, the area between the graph and the time axis always represents distance travelled, whatever the shape of that region.

18. 18.

    A speed-time graph is a straight line showing speed increasing from 4 m/s to 22 m/s over 6 seconds. Calculate the acceleration.

    [2 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses acceleration = (change in speed)/(time taken) = (22 − 4)/6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the acceleration as 3 m/s². Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The gradient of a speed-time graph always represents acceleration, in the same way the gradient of a distance-time graph represents speed.

    Marking points

    • Uses acceleration = (change in speed)/(time taken) = (22 − 4)/6.
    • States the acceleration as 3 m/s².

    Examiner tip: The gradient of a speed-time graph always represents acceleration, in the same way the gradient of a distance-time graph represents speed.

19. 19.

    The curve y = x² − 5x + 6 crosses the x-axis at two points. Find the x-coordinates of these points, and state the equation of the curve's line of symmetry.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Factorises x² − 5x + 6 = (x − 2)(x − 3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the x-axis crossing points at x = 2 and x = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the line of symmetry as x = 2.5, the midpoint of the two roots. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The line of symmetry of any quadratic always passes exactly halfway between its two x-axis crossing points, when they exist.

    Marking points

    • Factorises x² − 5x + 6 = (x − 2)(x − 3).
    • States the x-axis crossing points at x = 2 and x = 3.
    • States the line of symmetry as x = 2.5, the midpoint of the two roots.

    Examiner tip: The line of symmetry of any quadratic always passes exactly halfway between its two x-axis crossing points, when they exist.

20. 20.

    A conversion graph is a straight line through the origin, converting miles (x) to kilometres (y). The line passes through the point (5, 8). Find the equation connecting y and x, and use it to convert 20 miles to kilometres.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the gradient through the origin: 8/5 = 1.6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the equation as y = 1.6x. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Substitutes x = 20 to obtain y = 32 km. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A conversion graph through the origin always has the simple form y = kx, since zero of one unit must always convert to zero of the other.

    Marking points

    • Calculates the gradient through the origin: 8/5 = 1.6.
    • States the equation as y = 1.6x.
    • Substitutes x = 20 to obtain y = 32 km.

    Examiner tip: A conversion graph through the origin always has the simple form y = kx, since zero of one unit must always convert to zero of the other.

21. 21.

    Solve the inequality x² − x − 6 > 0.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Factorises x² − x − 6 = (x − 3)(x + 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies the critical values as x = −2 and x = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the solution as x < −2 or x > 3, since the upward parabola is positive outside its roots. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Sketching the upward parabola and its two roots makes it visually clear which region is above the x-axis (positive) and which is below.

    Marking points

    • Factorises x² − x − 6 = (x − 3)(x + 2).
    • Identifies the critical values as x = −2 and x = 3.
    • States the solution as x < −2 or x > 3, since the upward parabola is positive outside its roots.

    Examiner tip: Sketching the upward parabola and its two roots makes it visually clear which region is above the x-axis (positive) and which is below.

22. 22.

    A straight-line graph has x-intercept (6, 0) and y-intercept (0, −4). Find the equation of the line.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the gradient between the two intercepts: (0 − (−4))/(6 − 0) = 4/6 = 2/3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Uses the y-intercept directly as c = −4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the equation as y = (2/3)x − 4. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The y-intercept of a line, (0, c), always gives the value of c directly — no substitution is needed once it is known.

    Marking points

    • Calculates the gradient between the two intercepts: (0 − (−4))/(6 − 0) = 4/6 = 2/3.
    • Uses the y-intercept directly as c = −4.
    • States the equation as y = (2/3)x − 4.

    Examiner tip: The y-intercept of a line, (0, c), always gives the value of c directly — no substitution is needed once it is known.

23. 23.

    The graph of y = (x − 1)(x + 2)(x − 3) crosses the x-axis at three points. State the coordinates of these three points.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: Sets each factor equal to zero: x − 1 = 0, x + 2 = 0, x − 3 = 0. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Solves to obtain x = 1, x = −2 and x = 3. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the three points as (1, 0), (−2, 0) and (3, 0). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a graph already given in fully factorised form, each factor set to zero gives one x-axis crossing point directly.

    Marking points

    • Sets each factor equal to zero: x − 1 = 0, x + 2 = 0, x − 3 = 0.
    • Solves to obtain x = 1, x = −2 and x = 3.
    • States the three points as (1, 0), (−2, 0) and (3, 0).

    Examiner tip: For a graph already given in fully factorised form, each factor set to zero gives one x-axis crossing point directly.

24. 24.

    A distance-time graph for a runner is curved and becomes steeper as time passes. Describe what this shape indicates about the runner's speed.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States that the gradient of a distance-time graph represents speed. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that since the graph becomes steeper over time, the runner's speed is increasing (the runner is accelerating). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A curved distance-time graph means changing speed: increasing steepness means speeding up, decreasing steepness means slowing down.

    Marking points

    • States that the gradient of a distance-time graph represents speed.
    • States that since the graph becomes steeper over time, the runner's speed is increasing (the runner is accelerating).

    Examiner tip: A curved distance-time graph means changing speed: increasing steepness means speeding up, decreasing steepness means slowing down.

25. 25.

    A table shows a plant's height (cm) over time: at week 0 it is 5 cm, and at week 4 it is 17 cm. Assuming the growth is linear, find the weekly growth rate and predict the plant's height at week 10.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses gradient = (17 − 5)/(4 − 0) = 3 cm per week as the growth rate. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the model as height = 3 × week + 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Substitutes week = 10 to predict a height of 35 cm. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Two data points are enough to find a linear model's gradient and intercept, but predicting far beyond the given weeks assumes the linear pattern continues, which may not hold in reality for plant growth.

    Marking points

    • Uses gradient = (17 − 5)/(4 − 0) = 3 cm per week as the growth rate.
    • States the model as height = 3 × week + 5.
    • Substitutes week = 10 to predict a height of 35 cm.

    Examiner tip: Two data points are enough to find a linear model's gradient and intercept, but predicting far beyond the given weeks assumes the linear pattern continues, which may not hold in reality for plant growth.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.