Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9). The line l is a tangent to C at P. The line l cuts the x-axis at the point Q. (a) Find the exact value of the x coordinate of Q. (4) The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line l. This region R is rotated through 360° about the x-axis. (b) Use integration to find the exact value of the volume of the solid generated. Give your answer in the form $ \frac{p}{q} $ where p and q are exact constants. [You may assume the formula $ V = \frac{1}{3} \pi r^2 h $ for the volume of a cone.]

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Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9) - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 4

Question 1

Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9). The line l is a tangent to C at P. The line... show full transcript

Worked Solution & Example Answer:Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9) - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 4

Step 1

Find the exact value of the x coordinate of Q.

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Answer

To find the x-coordinate of point Q, we first need to determine the equation of the tangent line l at point P (2, 9). The derivative of the curve ( y = 3^x ) with respect to x is:

$\frac{dy}{dx} = 3^x \ln(3)$

Evaluating this derivative at x = 2 gives:

$\frac{dy}{dx}\bigg|_{x=2} = 3^2 \ln(3) = 9 \ln(3)$

The equation of the tangent line at point P can be formulated using the point-slope form:

$y - 9 = 9 \ln(3)(x - 2)$

To find where this line intersects the x-axis (i.e., where y = 0), substitute y = 0:

$0 - 9 = 9 \ln(3)(x - 2)$

Solving for x:

$-9 = 9 \ln(3)(x - 2)$

$x - 2 = \frac{-9}{9 \ln(3)}$

$x = 2 - \frac{1}{\ln(3)}$

Thus, the exact value of the x-coordinate of Q is:

$x_Q = 2 - \frac{1}{\ln(3)}.$

Step 2

Use integration to find the exact value of the volume of the solid generated.

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Answer

The volume V of the solid formed by rotating region R about the x-axis can be calculated using the disk method:

$V = \pi \int_{0}^{x_Q} (3^x)^2 \, dx$

This simplifies to:

$V = \pi \int_{0}^{x_Q} 9^{x} \, dx$

The integral of ( 9^x ) is:

$\int 9^x \, dx = \frac{9^x}{\ln(9)} + C$

Thus, evaluating the definite integral:

$V = \pi \left[ \frac{9^x}{\ln(9)} \right]_{0}^{x_Q}$

Substituting the limits:

$V = \pi \left( \frac{9^{x_Q}}{\ln(9)} - \frac{9^{0}}{\ln(9)} \right)$

Simplifying further:

$V = \pi \left( \frac{9^{2 - \frac{1}{\ln(3)}}}{\ln(9)} - \frac{1}{\ln(9)} \right) = \frac{\pi}{\ln(9)} \left( 9^{2 - \frac{1}{\ln(3)}} - 1 \right)$

Thus, the volume of the solid generated is:

$V = \frac{p}{q}$

where p and q can be determined based on the expression derived.

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Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9) - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 4

Worked Solution & Example Answer:Figure 3 shows a sketch of part of the curve C with equation y = 3^x The point P lies on C and has coordinates (2, 9) - Edexcel - A-Level Maths Pure - Question 1 - 2014 - Paper 4

Find the exact value of the x coordinate of Q.

Use integration to find the exact value of the volume of the solid generated.

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