Higher Mathematics

Higher Maths Readiness Check

A pupil-facing snapshot of National 5 foundations and early Higher Mathematics skills. Equivalent exact answers are accepted; the result is intended to identify strengths, gaps and useful next steps.

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Readiness snapshot

Check key skills

Answer each question, then check it. When all questions are checked, your readiness result and suggested next practice will appear.

Score: 0/10

Question 1

The area of a design is modelled by A(x) = −x² + 10x + 9, where x > 0. Determine the maximum area and the value of x that gives it.

Differentiation
The maximum occurs at the stationary point of the model.
Maths tools
Hint

For a maximum or minimum from a model, differentiate and set the derivative equal to zero.

Question 2

A population model is u₀ = 81, uₙ₊₁ = 0.75uₙ + 5. Find u₁.

Recurrence and sequences
Choose an answer
Hint

Use the previous term once to get the next term.

Question 3

Find the angle between a = (1, 0) and b = (0, 1).

Vectors
The scalar product connects vector components with the angle between vectors.
Maths tools
Hint

Start with the scalar product formula.

Question 4

Find the discriminant of 3x² + 5x − 1 = 0 and state the nature of the roots.

Polynomials and quadratics
Choose an answer
Hint

Identify a, b and c before substituting.

Question 5

Solve 5ˣ = 125.

Exponentials and logarithms
Choose an answer
Hint

Same base means same exponent.

Question 6

A line has gradient 3. What is the gradient of a perpendicular line?

Straight line and circles
Choose an answer
Hint

Flip the fraction and change the sign.

Question 7

For y = 2cos θ − 1, state the midline of the graph.

Trigonometry
The midline is shifted vertically by the constant term.
Choose an answer
Hint

The multiplier changes amplitude, not the midline.

Question 8

For y = 1x³ − 2x, find the gradient at x = 1.

Differentiation
Choose an answer
Hint

Differentiate before substituting.

Question 9

Given dy/dx = 6x² − 8x and the curve passes through (1, 0), find the equation of the curve. Show your working.

Integration
The integrated cubic passes through the given point.
Maths tools
Hint

Integrate first, then use the point to find the constant of integration.

Question 10

Find f⁻¹(x) for f(x) = 5x − 7.

Algebra and functions
Maths tools
Hint

Reverse the operations after swapping x and y.