Higher June 2019 Paper 5 Q19
19 The graph of \(y = x^3 - x^2 - 2\) is drawn on the grid.
(a) Use the graph to solve \(x^3 - x^2 - 2 = 0\).
Give your answer correct to 1 decimal place. [1]
Give your answer correct to 1 decimal place. [1]
(b) The equation \(x^3 - x^2 + 5x - 6 = 0\) can be solved by finding the intersection of the graph of \(y = x^3 - x^2 - 2\) and the line \(y = ax + b\).
(i) Find the value of \(a\) and the value of \(b\). [2]
(ii) Hence, use the graph to solve the equation \(x^3 - x^2 + 5x - 6 = 0\).
Give your answer correct to 1 decimal place. [3]
Give your answer correct to 1 decimal place. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 1.7 cao | 1 | Mark at most accurate i.e. do not allow 1.65 = 1.7 | |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) [\(a\) =] –5 [\(b\) =] 4 | 2 | B1 for each or for \(-5x + 4\) seen | |
| (ii) Correct line ruled on grid | M2 | Strict FT \(y =\) their \(ax\) + their \(b\) M1 for correct \(y\)–intercept (FT their \(b\)) or correct gradient (FT their \(a\)) or for freehand or broken ‘correct’ line FT \(y =\) their \(ax\) + their \(b\) or for at least 3 correct plots and no incorrect plots FT \(y =\) their \(ax\) + their \(b\) | For M2 line must cross curve For M2 or M1, accuracy ± 1 small square at \(y\) – intercept (extend line if necessary provided it fits on the grid) and gradient ± 1 small square vertically for a run of 1 unit horizontally Do not follow through if \(a = 0\) and/or \(b = 0\) |
| 1.1 or 1.2 | A1 | Only award if M2 scored previously | |