7 (a) By sketching the graphs of $y = \frac{1}{x}$ and $y = \sec 2x$ on the axes below, show that the equation \[ \frac{1}{x} = \sec 2x \] has exactly one solution for $x > 0$ - AQA - A-Level Maths Pure - Question 7 - 2019 - Paper 1

Step 1: Define the Function

Let us define the function: [ f(x) = \frac{1}{x} - \sec 2x ]

Step 2: Evaluate at 0.4 and 0.6

Next, we evaluate $f(x)$ at two points within the range [0.4, 0.6]:

• For $x = 0.4$: [ f(0.4) = \frac{1}{0.4} - \sec(0.8) \approx 2.5 - 1.151 = 1.349 > 0 ]

• For $x = 0.6$: [ f(0.6) = \frac{1}{0.6} - \sec(1.2) \approx 1.666 - 1.442 = 0.224 > 0 ]

Step 3: Values in the Interval

We can check at intermediate values:

• For $x = 0.5$: [ f(0.5) = \frac{1}{0.5} - \sec(1) \approx 2 - 1.850 > 0 ]

By observing that $f(x)$ changes sign between $0.4$ and $0.6$ and that it is continuous, we conclude that the solution lies between those two values.

Step 1: Starting from the Original Equation

We begin with the original equation: [ \frac{1}{x} = \sec(2x) ]

Step 2: Rearrangement

Using the identity (\sec \theta = \frac{1}{\cos \theta}), we can rewrite the equation as: [ \frac{1}{x} = \frac{1}{\cos(2x)} ]

Multiplying both sides by $x \cos(2x)$ gives: [ \cos(2x) = x ]

Next, applying the cosine inverse function, we have: [ 2x = \cos^{-1}(x) ]

Finally, dividing both sides by 2 results in: [ x = \frac{1}{2} \cos^{-1}(x) ]

This confirms the rearrangement.

Step 1: Initial Value

Start with the initial value $x_1 = 0.4$.

Step 2: Applying the Iterative Formula

Using the formula: [ x_{n+1} = \frac{1}{2} \cos^{-1}(x_n) ]

• For $n = 1$: [ x_2 = \frac{1}{2} \cos^{-1}(0.4) \approx 0.5796 ] (rounded to four decimal places)
• For $n = 2$:
  [ x_3 = \frac{1}{2} \cos^{-1}(0.5796) \approx 0.4763 ] (rounded to four decimal places)
• For $n = 3$:
  [ x_4 = \frac{1}{2} \cos^{-1}(0.4763) \approx 0.5372 ] (rounded to four decimal places)

Thus, we find:

• $x_2 \approx 0.5796$
• $x_3 \approx 0.4763$
• $x_4 \approx 0.5372$.

Step 1: Set Up the Graph

To illustrate the cobweb or staircase diagram, we will plot the function $y = \frac{1}{2} \cos^{-1}(x)$ and the line $y = x$.

Step 2: Positions

Indicate the positions of $x_2, x_3$, and $x_4$:

1. Start at $x_1 = 0.4$, move vertically to the curve.
2. Draw horizontal until it meets the line $y=x$ at $x_2$.
3. Similarly plot $x_3$ and $x_4$ with continued vertical and horizontal lines illustrating convergence until reaching close to a fixed point.

Note: Ensure to mark the points clearly on the graph.