f(x) = 7cos x + sin x Given that f(x) = Rcos(x - α), where R > 0 and 0 < α < 90°, a) find the exact value of R and the value of α to one decimal place - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 8

Starting with the equation:

$7cos x + sin x = 5$

We rearrange it as follows:

$\sqrt{50}cos(x - 8.1°) = 5$

This simplifies to:

$cos(x - 8.1°) = \frac{5}{\sqrt{50}} = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}$

Thus:

$x - 8.1° = 45° \text{ or } 315°$

Solving for x gives:

$x = 53.1°\text{ or } 323.1°$

For the equation to have only one solution, the value of k must equal the maximum or minimum value of the function:

$k = \pm \sqrt{50}$

Thus, the values of k are:

$k = \sqrt{50} \text{ and } k = -\sqrt{50}$