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Higher Mathematics homework set

Scottish Higher Mathematics | 25-40 minutes | Mixed practice

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Pupil instructions

Answer each question in your jotter or on the printed sheet. Show enough working for your teacher to see your method.

  1. For y = 3x² + 12x, state where the function is increasing.

    Support hint: Increasing means dy/dx > 0

  2. For y = 4x³ − x, find the gradient at x = 0.

    Support hint: Differentiate before substituting.

  3. A geometric sequence has u₁ = 8 and ratio 0.5. State uₙ.

    Support hint: The exponent is n − 1 because u₁ is the first term.

  4. In triangle ABC, a = 13, A = 60° and B = 30°. Use the sine rule to express b exactly

    Support hint: Match each side with its opposite angle.

  5. Solve sin θ = −√32 for 0° ≤ θ < 360° .

    Support hint: Use CAST or the graph to locate the signs.

  6. Find the discriminant of 2x² + 5x + 2 = 0 and state the nature of the roots.

    Support hint: Identify a, b and c before substituting.

  7. State the multiplier for growth of 8% per time period.

    Support hint: Growth multipliers are above 1; decay multipliers are below 1.

  8. Find the area under y = 4x + 2 from x = 0 to x = 2.

    Support hint: The line is above the axis, so the integral gives the area.

  9. Find a unit vector in the direction of (5, 12).

    Support hint: A unit vector has length 1.

  10. f(x) = 5x + 6 and g(x) = x² + 5. Find f(g(x))

    Support hint: Put the whole expression for g(x) inside f.

  11. Use the factor theorem to test whether x + 1 is a factor of f(x) = x² + 2x + 1.

    Support hint: For x − a, substitute x = a.

  12. State the centre and radius of (x − 4)² + (y − 4)² = 25.

    Support hint: The bracket signs are opposite to the centre coordinates.

Answer key

  1. x > −2

    differentiation: dy/dx = 6x + 12. This is positive when x > −2

  2. -1

    differentiation: dy/dx = 12x² − 1. At x = 0, the gradient is −1

  3. uₙ = 8(0.5)ⁿ⁻¹

    recurrence sequences: Use uₙ = arⁿ⁻¹

  4. 13/√3

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

  5. 240°, 300°

    trigonometry: The related acute angle is 60°. Use the quadrants where sin is negative.

  6. 9; two real roots

    polynomials quadratics: Use b² − 4ac = 5² − 4(2)(2) = 9

  7. 1.08

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

  8. 12

    integration: Integrate to 2x² + 2x, then evaluate from 0 to 2

  9. (513, 1213)

    vectors: Divide each component by the magnitude 13.

  10. 5x² + 31

    algebra functions: Apply g first: f(g(x)) = 5(x² + 5) + 6 = 5x² + 31

  11. Yes, because f(a) = 0

    polynomials quadratics: Substitute x = −1. The value is 0, so x + 1 is a factor

  12. Centre (4, 4), radius 5

    straight line circles: Compare with (x − h)² + (y − k)² = r²; watch the signs inside brackets.