Lemmara logoLemmaraWorked solution

0580/42 Feb/Mar 2021, Question 6: Reciprocal Functions

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 18 marks

اقرأ هذا الحل مشروحاً بالعربي ←

The question, in short

Use the curve y = a + bx² through (0, 4) and (1, 1) to find constants, tangents and intersections, and complete and draw a rational graph.

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

Part (ai) · 2 marks

Substitute the two given points into y = a + bx² and solve for a and b.

  1. 1At (0, 4): 4 = a + b × 0 so a = 4
  2. 2At (1, 1): 1 = 4 + b so b = −3
  3. 3a = 4, b = −3

Answera = 4, b = −3

Part (aii) · 1 mark

State the tangent line equation at the point (0, 4) on the curve.

  1. 1(0, 4) is the highest point, so the tangent is horizontal
  2. 2y = 4

Answery = 4

Part (aiii) · 2 marks

Given the tangent at x = −1 is y = 6x + 7, find the tangent at x = 1.

  1. 1The curve is symmetrical about the y-axis, so the gradient at x = 1 is −6
  2. 2Tangent passes through (1, 1): 1 = −6 × 1 + c, c = 7
  3. 3y = −6x + 7

Answery = −6x + 7

Part (bi) · 3 marks

Calculate the missing y values in the table for y = 1 + 5/(3 − x).

  1. 1x = −1: 1 + 5/4 = 2.25
  2. 2x = 0: 1 + 5/3 = 2.67
  3. 3x = 1: 1 + 5/2 = 3.5

Answer| x | −2 | −1.5 | −1 | −0.5 | 0 | 0.5 | 1 | 1.5 | | y | 2 | 2.11 | 2.25 | 2.43 | 2.67 | 3 | 3.5 | 4.33 |

Part (bii) · 4 marks

Plot the completed table points and draw the curve y = 1 + 5/(3 − x) from x = −2 to 1.5.

  1. 1Plot (−2, 2), (−1.5, 2.11), (−1, 2.25), (−0.5, 2.43), (0, 2.67), (0.5, 3), (1, 3.5), (1.5, 4.33)
  2. 2Join with a smooth increasing curve

Answercorrect curve

Part (ci) · 2 marks

Sketch y = 4 − 3x² on the grid and state the x values where it meets the other graph.

  1. 1Read the x-values where the two curves cross
  2. 2x ≈ −0.75 and x ≈ 0.56

Answer−0.78 to −0.72, 0.55 to 0.59

Part (cii) · 4 marks

Form a cubic equation with integer coefficients from the two intersection solutions.

  1. 1Part (c)(i) finds where y = 1 + 5/(3 − x) meets y = 4 − 3x²
  2. 2So 1 + 5/(3 − x) = 4 − 3x², giving 5/(3 − x) = 3 − 3x²
  3. 3Multiply by (3 − x): 5 = 9 − 3x − 9x² + 3x³
  4. 4Rearrange: 3x³ − 9x² − 3x + 4 = 0

Answer3x³ − 9x² − 3x + 4 = 0

Stuck on a step? Read the free lesson: Reciprocal Functions

Practise more reciprocal functions questions

Real 0580 past-paper questions, marked on your working like the exam, with a worked solution for every part.

Practise free on Lemmara

More past-paper questions on reciprocal functions

Also searched as: 0580/42/F/M/21 · 0580_m21_qp_42 · 0580 m21 42 · question 6.
Lemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education. The worked solutions are Lemmara’s own.