10 What actually comes up

Three years of real questions, part by part, with the setup and values from the paper and the grade band each part sat at according to that year's marking schedule. Start with the pattern table — it is the fastest way to decide what to revise.

Across 3 years

9 questions · 51 parts 24 Achieved 27 Merit 9 Excellence

The pattern — what to prepare

Topic	Appears in	Grade level	What to practise
Chain rule on a power / composite (√( ), ( )⁵)	Every year	Achieved	Derivative of the outside, times the derivative of the inside, un-simplified is fine.
Product / quotient rule with e^x, ln, trig	Every year	Achieved–Merit	Recognise structure: product or chain? (d/dx) ln f = (f')/f and (d/dx)e^f = f'e^f.
Gradient of the normal (negative reciprocal of tangent)	2023, 2025	Merit	Find dy/dx at the point, then m_normal = −1/(m_tangent).
Equation of a tangent / tangent x-intercept	Every year	Merit–Excellence	dy/dx at the point, then the point–gradient line y − y₁ = m(x − x₁); then find where it crosses the axis.
Stationary points & their nature (first or second derivative test)	Every year	Achieved–Excellence	Set dy/dx = 0; test with (d²y)/dx² or a sign table.
Points of inflection ((d²y)/dx² = 0)	Every year	Merit–Excellence	In parametric and implicit too; check concavity change, not just the zero.
Related rates (chain rule linking two rates)	Every year	Merit–Excellence	Write the geometric formula, differentiate, chain the given rate in from the given values.
Optimisation (max area / max volume)	Every year	Merit–Excellence	One-variable function from the constraint; set dA/dx = 0; justify max.
Parametric differentiation (dy/dx = (dy/dt)/(dx/dt))	Every year	Achieved–Excellence	Differentiate each with respect to t, then divide; second derivative needs the chain rule again, not (d²y/dt²)/(d²x/dt²).
Interpreting a graph (continuity, differentiability, limits)	Every year	Achieved	Read where the graph has a corner, a jump, or a vertical tangent; f'(x) = 0 at flat spots.

not tried got it to review 2025 an oven, a harbour, a melting ice ball 17 parts

The 2025 paper ↗ Marking schedule ↗

Question 1 — derivatives, normal, tangents

A chain-rule power, an oven cooling written in half-powers of t, a trig normal, a tangent to an exponential-plus-quadratic, and a parametric curve where inflection and stationary have to coincide.

Not tried yet: a Differentiate f(x) = (5x³ − 2x + 1)⁵. You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b The temperature of an oven is C = 60/√t + 5√t + 15, where 0 < t ≤ 12, C is the temperature in °C and t is the time in minutes after the oven has been switched off. Find the rate of change of the temperature of the oven 4 minutes after the oven was switched off.
- C = 60/√t + 5√t + 15
- 0 < t ≤ 12
- t = 4 min
Worked solution → Achieved
Not tried yet: c What is the gradient of the normal to the curve y = sin³ x cos x when x = π/4?Worked solution → Merit
Not tried yet: d The graph shows f(x) = x² + e^2x and the tangent to the curve when x = 1. Find the point P, the x-intercept of this tangent.Worked solution → Merit
Not tried yet: e A curve is defined by the parametric equations x = 3t² + 6t + 1 and y = 2t³. Find the coordinates of any points of inflection on this curve that are also stationary points. You may assume that any inflection points you find are inflection points.Worked solution → Merit to Excellence

Question 2 — logs, exponentials, tangents, radius of curvature

A reciprocal minus a log, a product that has to be differentiated to decide increasing or decreasing, two tangents of the same gradient to a rational curve, a parametric tangent in sec and tan, and a given curvature formula to apply.

Not tried yet: a Differentiate f(x) = 7/x − ln(3x³ − 2x² + 5). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b The depth of water, in metres, in a particular harbour is P = (30t − 5t²)e^−t, where t is measured in hours and 0 < t ≤ 4. Show whether the depth of the water in the harbour is increasing or decreasing after 2 hours.
- P = (30t − 5t²)e^−t
- 0 < t ≤ 4
- t = 2 hours
Worked solution → Achieved
Not tried yet: c The graph shows y = f(x) = x²/(x + 4). Two tangents to this curve have the equation y = −3x + c. Find the coordinates of the two points of contact of these tangents with the curve y = f(x).Worked solution → Merit
Not tried yet: d Find the equation of the tangent to the graph defined by the pair of equations x = 2 sec t and y = 5 tan t, at the point where t = π/6.Worked solution → Merit
Not tried yet: e The radius of curvature measures how much a curve is bending at a point, and is given by p = ((1 + (dy/dx)²)^3/2)/((d²y)/dx²). Find the radius of curvature for the curve y = cos²(2x) when x = π/3.
- p = ((1 + (dy/dx)²)^3/2)/((d²y)/dx²)
Worked solution → Merit to Excellence

Question 3 — the graph, related rates, rectangle optimisation

Reading differentiability, stationary points and a limit straight off a drawn graph, then a log quotient, a related-rates sphere, a parametric gradient, and a rectangle inscribed in a curve.

Not tried yet: a i The graph shows the function y = f(x). Find the value(s) of x where f(x) is not differentiable.Worked solution → Achieved
Not tried yet: a ii Find the value(s) of x where f'(x) = 0.Worked solution → Achieved
Not tried yet: a iii What is the value of lim_{x → −2} f(x)? State clearly if the value does not exist.Worked solution → Achieved
Not tried yet: b A curve has equation y = (ln x)/2x. Find the x-coordinate(s) of any stationary point(s) on the curve.Worked solution → Achieved
Not tried yet: c A solid spherical lump of ice is melting while maintaining its spherical shape. At the instant when the radius is 6 cm, the radius is decreasing by 0.05 cm s⁻¹. Find the rate at which the volume of the spherical ice ball is decreasing at the instant when the radius is 6 cm.
- r = 6 cm
- dr/dt = −0.05 cm s⁻¹
Worked solution → Merit
Not tried yet: d The equation of a curve is given by the parametric equations x = 5/e^2t and y = 5e^2t. Find the coordinates of the point(s) on the curve where the gradient is −2.Worked solution → Merit
Not tried yet: e The diagram shows part of the symmetrical graph y² = 16x − x². A rectangle ABCD is drawn inside the curve with vertices B and C on the curve. Find the length AD so that the rectangle has its maximum area. You can assume that the value found is a maximum.
- y² = 16x − x²
Worked solution → Merit to Excellence

2024 a baking flour cone, an exponential optimisation 17 parts

The 2024 paper ↗ Marking schedule ↗

Question 1 — derivatives & tangents

A chain-rule root, a product-rule tangent, a decreasing-interval question mixing a square with a log, stationary points of an exponential product, and a tangent at a point of inflection.

Not tried yet: a Differentiate f(x) = √(4 − 9x⁴). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b A curve is defined by the equation y = (x² + 3x + 2) sin x. Find the gradient of the tangent to this curve when x = 0.Worked solution → Achieved
Not tried yet: c For the function y = 3(2x − 7)² + 60 ln x + 12, where x > 0, find the range of values of x for which the function is decreasing.Worked solution → Merit
Not tried yet: d Find the x-value(s) of any stationary points on the graph of y = (2x − 1)e^−2x, and determine their nature.Worked solution → Merit
Not tried yet: e A curve is defined by y = (2x² − 1 − 2x ln x)/x, where x > 0. The curve has a point of inflection at the point P. Find the equation of the tangent to the curve at the point P.Worked solution → Merit to Excellence

Question 2 — parametric, velocity, solution & inflection

A parametric derivative, a velocity from a log displacement, a second-derivative verification, an inflection point on (ln x)/x, and a turning point that survives an unknown constant.

Not tried yet: a A function is defined parametrically by x = 3t² + 1 and y = cos t. Find an expression for dy/dx.Worked solution → Achieved
Not tried yet: b An object is travelling in a straight line. Its displacement, in metres, is s(t) = ln(3t² + 5t + 2), where t > 0 and t is time in seconds. Find the velocity of this object when t = 1 second.
- s(t) = ln(3t² + 5t + 2)
- t = 1 s
Worked solution → Achieved
Not tried yet: c Show that y = sin(x²) − cos(x) is a solution to the equation (d²y)/dx² + 4x²y = 2 cos(x²) + (1 − 4x²) cos x.Worked solution → Merit
Not tried yet: d Consider the function f(x) = (ln x)/x, x > 0. Find the coordinates of the point of inflection on the graph of the function. You can assume that the point you find is a point of inflection.Worked solution → Merit
Not tried yet: e The graph of y = xe^3x/(2x + k), where k is a non-zero constant, has a single turning point at Q. Find the x-coordinate of the point Q.Worked solution → Merit to Excellence

Question 3 — sec, graph, cone, triangle area

A product with sec, three readings off a drawn graph (continuity, stationary points, a limit), stationary points of a rational function, a related-rates cone, and an optimisation that has to be proved to a given exact value.

Not tried yet: a Differentiate y = √x sec(6x). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b i The graph shows the function y = f(x). For the function above, find the value(s) of x where f(x) is continuous but not differentiable.Worked solution → Achieved
Not tried yet: b ii For the function above, find the value(s) of x where f'(x) = 0.Worked solution → Achieved
Not tried yet: b iii What is the value of lim_{x → −1} f(x)? State clearly if the value does not exist.Worked solution → Achieved
Not tried yet: c Find the x-value(s) of any stationary point(s) on the graph of f(x) = (x² − 5x + 4)/(x² + 5x + 4). You do not need to determine the nature of any stationary point(s).Worked solution → Merit
Not tried yet: d Jamie is pouring flour to form a conical pile. The height of the pile is always the same as the diameter of the base of the cone. Flour is being added at a constant rate of 3 cm³ per second. Volume of a cone = (1/3)π r²h. At what rate is the height increasing when the pile is 4 cm in height?
- h = 2r
- dV/dt = 3 cm³ s⁻¹
- h = 4 cm
Worked solution → Merit
Not tried yet: e The diagram shows part of the graph of f(x) = e^−x², where x ≥ 0. The point P lies on the curve and the point Q lies on the x-axis so that OP = PQ, where O is the origin. Prove that the largest possible area of the triangle OPQ is 1/√2e. You do not need to show that the area you have found is a maximum.
- f(x) = e^−x², x ≥ 0
- OP = PQ
Worked solution → Merit to Excellence

2023 a police helicopter, a triangle under a cubic 17 parts

The 2023 paper ↗ Marking schedule ↗

Question 1 — derivatives, tangents, a proof

A chain-rule root, a product rate of change, a second tangent parallel to a given one, a parametric circle tangent to be proved in general form, and an optimisation whose answer is a fraction of a given area.

Not tried yet: a Differentiate y = √(3x − 2). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b Find the rate of change of the function f(t) = t²e^2t when t = 1.5.Worked solution → Achieved
Not tried yet: c The graph shows the curve y = 2/(x + 1)³ along with the tangent to the curve drawn at x = 1. A second tangent to this curve is drawn which is parallel to the first. Find the x-coordinate of the point where this second tangent touches the curve.Worked solution → Merit
Not tried yet: d The diagram shows a tangent passing through the point P(p, q), which lies on the circle with parametric equations x = 4 cosθ and y = 4 sinθ. Show that the equation of the tangent line is px + qy = p² + q².
- x = 4 cosθ
- y = 4 sinθ
- P(p, q) on the circle
Worked solution → Merit
Not tried yet: e The graph of y = x(x − 2m)², where m > 0, is shown. The total shaded area between the curve and the x-axis from x = 0 to x = 2m is A = 4m⁴/3. A right-angled triangle is constructed with one vertex at (0, 0) and another on the curve. Show that the maximum area of such a triangle is 3/8 of the total shaded area. You do not have to prove that the area you have found is a maximum.
- y = x(x − 2m)², m > 0
- total shaded area A = 4m⁴/3
Worked solution → Merit to Excellence

Question 2 — trig, quotients, the helicopter

A quotient with cos, a cot gradient, a stationary point on an exponential quotient, inflection on x² ln x, and a related-rates chase where the rate given is an angle.

Not tried yet: a Differentiate f(x) = x²/(cos x). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b Find the gradient of the tangent to the curve y = cot(2x) at the point where x = π/12.Worked solution → Achieved
Not tried yet: c A curve is defined by f(x) = e^x/(x² + 2x). Find the x-value(s) of any point(s) on the curve where the tangent is parallel to the x-axis.Worked solution → Merit
Not tried yet: d Find the x-value(s) of any points of inflection on the graph of f(x) = 3x² ln(x). You can assume that your point(s) found are actually point(s) of inflection.Worked solution → Merit
Not tried yet: e A police helicopter flies above a straight horizontal motorway chasing a speeding car, at a constant 72 m s⁻¹ and a constant height of 400 metres. When the direct distance from the helicopter to the car is 2500 metres, the angle of depression θ between the horizontal and the line of sight is increasing at 0.002 rad s⁻¹. Calculate the speed of the car at this instant.
- helicopter 72 m s⁻¹
- height 400 m
- line of sight 2500 m
- dθ/dt = 0.002 rad s⁻¹
Worked solution → Merit to Excellence

Question 3 — gradient of normal, turning points, a proof

A log chain rule, three readings off a drawn graph, a parametric normal on a Ferris wheel, stationary points of a reciprocal pair, and a catenary to verify by substitution.

Not tried yet: a Differentiate y = ln(x² − x⁴ + 1). You do not need to simplify your answer.Worked solution → Achieved
Not tried yet: b i The graph shows the function y = f(x). Find the value(s) of x where f(x) is continuous but not differentiable.Worked solution → Achieved
Not tried yet: b ii Find the value(s) of x where f'(x) = 0 and f''(x) < 0 are both true.Worked solution → Achieved
Not tried yet: b iii What is the value of lim_{x → 6} f(x)? State clearly if the value does not exist.Worked solution → Achieved
Not tried yet: c Char rides a Ferris wheel. Her position is x = 5√2 sin((π t)/5) and y = 10 − 5√2 cos((π t)/5), where t is time in seconds from the start of the ride. Find the gradient of the normal to this curve at the point when t = 6.25 seconds after the start of the ride.
- x = 5√2 sin((π t)/5)
- y = 10 − 5√2 cos((π t)/5)
- t = 6.25 s
Worked solution → Merit
Not tried yet: d Find the co-ordinates of any stationary points on the graph of f(x) = 1/x − 2/x³, identifying their nature.Worked solution → Merit
Not tried yet: e A power line hangs between two poles. The curve y = f(x) that models its shape solves the differential equation a(d²y)/dx² = √(1 + (dy/dx)²), where a is a positive constant. Use differentiation to verify that the function y = (a/2)(e^x/a + e^−x/a) satisfies the above differential equation.
- a(d²y)/dx² = √(1 + (dy/dx)²)
Worked solution → Merit to Excellence

Read the last column as a to-do list. In all three papers every question climbs the same ladder: (a) and (b) are Achieved, (c) and (d) are Merit, and (e) is Merit to Excellence. Excellence is only ever on the (e)s, and each of those is a full problem (a proof, an optimisation, a related-rates chase or a parametric inflection) built from the same rules as the (a)s.

Practice for this standard covers the whole topic — flashcards and quiz.