June 2025 Paper 3 Q5

OCR MEICurrent spec10 marksLogs & ExponentialsNumerical Methods

5 The diagram shows the curve with equation \(y = x^4 - 2^x\), for values of \(x\) close to zero.

Curve y = x^4 - 2^x near the origin: crosses the x-axis once to the left of O and once to the right, with a minimum just right of the y-axis below the x-axis
(a) In this question you must show detailed reasoning.
Show that the equation \(x^4 - 2^x = 0\) has a root which lies between 1 and 2. [2]
(b) Show that the equation \(x^4 - 2^x = 0\) can be written in the form \(x = 2^{0.25x}\) for positive values of \(x\). [1]
(c) Use the iterative formula \(x_{n+1} = 2^{0.25x_n}\) with \(x_0 = 0.4\) to determine a root of \(x^4 - 2^x = 0\) to 2 decimal places. [2]
(d) The diagram in the Printed Answer Booklet shows the line with equation \(y = x\) and the curve with equation \(y = 2^{0.25x}\).
Sketch a cobweb or staircase diagram, on the diagram in the Printed Answer Booklet, for the iterative formula \(x_{n+1} = 2^{0.25x_n}\) starting with \(x_0 = 0.4\). Show at least two iterations. [2]
(e) Show that the equation \(x^4 - 2^x = 0\) can be written in the form \(x = 4\log_2 x\) for positive values of \(x\). [1]
(f) In this question you must show detailed reasoning.
Show that the iteration \(x_{n+1} = 4\log_2 x_n\), with \(x_0 = 1\), will not find a root of \(x^4 - 2^x = 0\). [2]
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