Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The point P(k, 8) lies on C, where k is a constant. (a) Find the exact value of k. (b) The finite region R, shown shaded in Figure 4, is bounded by the curve C, the y-axis, the x-axis and the line with equation x = k. (c) Show that the area of R can be expressed in the form $$ \lambda \int_{0}^{\beta} \left( \theta \sec^2 \theta + \tan \theta \sec^2 \theta \right) d\theta $$ where \lambda, \alpha and \beta are constants to be determined. (c) Hence use integration to find the exact value of the area of R.

A-Level Edexcel Maths Pure Equation of a Straight Line: Figure 4 shows a sketch of part

Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The point P(k, 8) lies on C, where k is a constant - Edexcel - A-Level Maths Pure - Question 1 - 2017 - Paper 5

Question 1

$Figure-4-shows-a-sketch-of-part-of-the-curve-C-with-parametric-equations--$$--x-=-30-\,--ext{sin}-\theta,-\,-y-=-\sec-\theta,-\,-0-<-\theta-<-\frac{\pi}{2}-$$--The-point-P(k,-8)-lies-on-C,-where-k-is-a-constant-Edexcel-A-Level Maths Pure-Question 1-2017-Paper 5.png$

Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The p... show full transcript

Worked Solution & Example Answer:Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The point P(k, 8) lies on C, where k is a constant - Edexcel - A-Level Maths Pure - Question 1 - 2017 - Paper 5

Step 1

Find the exact value of k.

96%

114 rated

Answer

To find the value of k, we start from the parametric equation for y:

y = \sec \theta = 8

This implies that:

\sec \theta = 8\Rightarrow \cos \theta = \frac{1}{8}

Using the identity, we find \theta:

\theta = \cos^{-1}\left(\frac{1}{8}\right)

Next, substituting \theta back into the equation for x:

x = 30 \sin \theta = 30 \sqrt{1 - \cos^2 \theta} = 30 \sqrt{1 - \left(\frac{1}{8}\right)^2} = 30 \sqrt{\frac{63}{64}} = \frac{30 \sqrt{63}}{8}.

Thus, the exact value of k is:

k = \frac{30 \sqrt{63}}{8}.

Step 2

Show that the area of R can be expressed in the form $ \lambda \int_{0}^{\beta} (\theta \sec^2 \theta + \tan \theta \sec^2 \theta) d\theta $.

99%

104 rated

Answer

To find the area R, we can use the formula for the area in parametric form:

A = \int_{a}^{b} y \frac{dx}{d\theta} d\theta.

Here, we have:

y = \sec \theta, \quad \frac{dx}{d\theta} = 30 \cos \theta.

Thus, the area becomes:

A = \int_{0}^{\beta} \sec \theta (30 \cos \theta) d\theta = 30 \int_{0}^{\beta} \sec \theta d\theta.

Next, using the identity ( \sec \theta = \tan \theta + 1 ) allows us to express the integral as:

\int_{0}^{\beta} (\tan \theta + 1) d\theta.

Therefore, we confirm that the area can indeed be expressed in the required form as:

\lambda \int_{0}^{\beta} (\theta \sec^2 \theta + \tan \theta \sec^2 \theta) d\theta.

Step 3

Hence use integration to find the exact value of the area of R.

96%

101 rated

Answer

To compute the area R:

From previous parts:

We know:
$A = 30 \int_{0}^{\beta} \sec^2 \theta d\theta.$
Integrate: Using the fact that the integral of ( \sec^2 \theta ) is ( \tan \theta ):
$A = 30 [\tan \theta]_{0}^{\beta} = 30 (\tan \beta - \tan 0) = 30 \tan \beta.$
Evaluate at the limits: If we take ( \beta = \frac{\pi}{4} ):
- Then:( \tan \frac{\pi}{4} = 1), so:
$A = 30 \times 1 = 30.$

Thus, the exact area of region R is given by:

A = 30.

Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The point P(k, 8) lies on C, where k is a constant - Edexcel - A-Level Maths Pure - Question 1 - 2017 - Paper 5

Worked Solution & Example Answer:Figure 4 shows a sketch of part of the curve C with parametric equations $$ x = 30 \, ext{sin} \theta, \, y = \sec \theta, \, 0 < \theta < \frac{\pi}{2} $$ The point P(k, 8) lies on C, where k is a constant - Edexcel - A-Level Maths Pure - Question 1 - 2017 - Paper 5

Find the exact value of k.

Show that the area of R can be expressed in the form \( \lambda \int_{0}^{\beta} (\theta \sec^2 \theta + \tan \theta \sec^2 \theta) d\theta \).

Hence use integration to find the exact value of the area of R.

Join the A-Level students using SimpleStudy...