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0580/41 Oct/Nov 2021, Question 8: Equation of the Tangent Line

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 16 marks

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The question, in short

For f(x) = 3 − 5x and g(x) = 18 − 3x − x², find values, inverse, completed-square form, sketch with turning point, and tangent at x = 4.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 2 marks

Find the value of x for which f(x) = −5.

  1. 13 − 5x = −5
  2. 25x = 8, x = 1.6

Answer1.6

Part (aii) · 2 marks

Find the inverse function f⁻¹(x).

  1. 1x = 3 − 5y
  2. 25y = 3 − x
  3. 3f⁻¹(x) = (3 − x)/5

Answer(3 − x)/5

Part (bi) · 3 marks

Rewrite g(x) as b − (a + x)².

  1. 1x² + 3x = (x + 1.5)² − 2.25
  2. 2g(x) = 18 − (x + 1.5)² + 2.25
  3. 3g(x) = 20.25 − (1.5 + x)²

Answer20.25 − (1.5 + x)²

Part (bii) · 3 marks

Sketch y = g(x) on axes, marking the turning point coordinates.

  1. 1∩-shaped parabola with maximum at (−1.5, 20.25)
  2. 2Crosses the y-axis at 18 and the x-axis at x = −6 and x = 3
  3. 3Mark the turning point (−1.5, 20.25)

AnswerCorrect sketch with max in correct quadrant at (−1.5, 20.25)

Part (biii) · 6 marks

Find the tangent to y = 18 − 3x − x² at x = 4, in y = mx + c form.

  1. 1dy/dx = −3 − 2x
  2. 2At x = 4: gradient = −3 − 8 = −11
  3. 3y = 18 − 12 − 16 = −10
  4. 4−10 = −11 × 4 + c, so c = 34
  5. 5y = 34 − 11x

Answery = 34 − 11x

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Also searched as: 0580/41/O/N/21 · 0580_w21_qp_41 · 0580 w21 41 · question 8.
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