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Preview Higher Mathematics practice

Higher Mathematics

Higher Mathematics diagnostic starter

10-20 minutes | Projectable starter, mini-whiteboards or printed half-sheet

  1. Find the remainder when f(x) = 2x² + 6x + 1 is divided by x − 2.
  2. Find the gradient of the line through (-4, 0) and (0, −4).
  3. Find the area under y = 3x + 6 from x = 0 to x = 5.
  4. Find the discriminant of x² + 8x + 6 = 0 and state the nature of the roots.
  5. In triangle ABC, a = 13, A = 60° and B = 30°. Use the sine rule to express b exactly
  6. State the multiplier for growth of 8% per time period.
  7. A geometric sequence has u₁ = 8 and ratio 0.5. State uₙ.
  8. Find a unit vector in the direction of (5, 12).

Answer key and teacher notes

  1. 21

    polynomials quadratics: By the remainder theorem, the remainder is f(2) = 21

    Common misconception: Dividing the polynomial when substitution is enough.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  2. -1

    straight line circles: Gradient = change in y/change in x = -4 − 00 − −4 = −1

    Common misconception: Dividing change in x by change in y.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  3. 1352

    integration: Integrate to 3/2x² + 6x, then evaluate from 0 to 5

    Common misconception: Using only rectangle area or only triangle area.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  4. 40; two real roots

    polynomials quadratics: Use b² − 4ac = 8² − 4(1)(6) = 40

    Common misconception: Using b² + 4ac

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  5. 13/√3

    trigonometry: Use b/sin B = a/sin A, then substitute the exact sine values

    Common misconception: Pairing the wrong side and angle.

    Next step: Ask pupils to write a full exam-style method line and check restrictions.

  6. 1.08

    exponentials logarithms: Add 8% from 100%, then write as a decimal multiplier.

    Common misconception: Using the percentage itself as the multiplier.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  7. uₙ = 8(0.5)ⁿ⁻¹

    recurrence sequences: Use uₙ = arⁿ⁻¹

    Common misconception: Using arⁿ.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.

  8. (513, 1213)

    vectors: Divide each component by the magnitude 13.

    Common misconception: Dividing by one component instead of the magnitude.

    Next step: Generate a fresh attempt on the matching Higher Mathematics topic page.