Higher June 2018 Paper 6 Q9
9 The graph of \(y = x^3 - 7x - 12\) is shown below.
The root of the equation \(x^3 - 7x - 12 = 0\) is \(p\).
(a) Calculate \(y\) when \(x = 3\). [1]
(b) Show that \(3 \lt p \lt 4\). [2]
(c) Find a smaller interval that contains the value of \(p\).
You must show calculations to support your answer. [3]
You must show calculations to support your answer. [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| −6 | 1 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| [\(x = 4\),] \(y = 24\) Change of sign, so \(p\) lies between 3 and 4 oe | 2 | B1 for 24 seen If using \(3.27 \lt x \lt 4\) rather than 4: SC2 evaluate \(y\) correctly (see table in (c)), state change of sign oe and that because \(3 \lt p \lt\) their \(x\)-value, then so \(3 \lt p \lt 4\). 0 for just evaluating y. | After \(x = 4\), \(y = 24\) scored: Examples just sufficient for second mark include: change of sign \(-6 \lt 0 \lt 24\) \(x = 3\) gives an answer < 0 and \(x = 4\) gives an \(\gt 0\) Examples insufficient for second mark: so \(p\) lies between 3 and 4 |
| Answer | Marks | Part marks and guidance | |||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Examples: when \(x = 3.5\), \(y = 6.4\), so \(3 \lt p \lt 3.5\) when \(x = 3.1\), \(y = -3.9\), so \(3.1 \lt p \lt 4\) when \(x = 3.1\), \(y = -3.9\) and when \(x = 3.5\), \(y = 6.4\), so \(3.1 \lt p \lt 3.5\) | 3 | M2 for one further value of \(y\) evaluated correctly, possibly rot to 2 or more sf, for a value of \(x\) such that \(3 \lt x \lt 4\) OR M1 for working shown to calculate one further value of \(y\) for a value of \(x\) such that \(3 \lt x \lt 4\) Note after SC considered in (b): if SC2 was awarded then they must use a value of \(x\) that produces a smaller interval than \(3 \lt x \lt\) their x-value in (b); if SC2 was not awarded then \(3 \lt x \lt 4\) applies If 0 scored, award SC1 or SC2 if evidence for M1 or M2 has not yet been credited in (b) | Solution is approx. 3.2670 Common values:
eg M2 only for when \(x = 3.1\), \(y = -3.9\) so \(3.1 \lt p \lt 3.5\) (as 3.5 has not been correctly justified) Calculations in support of \(x = 3\) or \(x = 4\) need not be repeated from parts (a) or (b). | ||||||||||||||||||||||||||||