By sketching a suitable pair of graphs, show that the equation ln x = 3x − x² has one real root.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. B1 Sketch a relevant graph, e.g. y = ln x ln(x) : sketch should imply y-axis is an asymptote. Through (1, 0) if marked. Correct shape.
2. B1 Sketch a second relevant graph, e.g. y=3x-x², and justify the given statement by marking the root on the sketch or by use of a suitable comment 3x - x²: Symmetrical. Through (0, 0) and (3, 0) if marked. If ln(x) correct accept parabola for +ve y only. If ln(x) incorrect then need parabola in 3 quadrants.
3. M1 Calculate the values of a relevant expression or pair of expressions at x = 2 and x = 2.8 Allow for a smaller interval. At least one value correct if comparing with 0. If using pairs then the pairing must be clear.
4. A1 Complete the argument correctly with correct calculated values e.g. 0.693 < 2 and 1.03 > 0.56 or 1.307 > 0, -0.47 < 0 using √3x-ln x 0.304>0,-0.085 < 0. Need to have calculated values to at least 2 sf.
5. M1 Use the iterative process correctly at least once
6. A1 Obtain final answer 2.63
7. A1 Show sufficient iterations to at least 4 dp to justify 2.63 to 2 dp or show there is a sign change in the interval (2.625, 2.635) SC Allow M1 A1 A0 to a candidate who starts at a point in the interval and reaches a premature conclusion