Pearson Edexcel Level 3 GCE

Question 1

Given that \( \theta \) is small and is measured in radians, use the small angle approximations to find an approximate value of:

Question 2

A curve \( C \) has the equation:

(a) Find:

• (i) \( \frac{dy}{dx} \)
• (ii) \( \frac{d^2y}{dx^2} \)
(3 Marks)

(b) Verify that \( C \) has a stationary point when \( x = 4 \). (2 Marks)

(c) Determine the nature of this stationary point, giving a reason for your answer. (2 Marks)

Question 3

Figure 1 shows a sector AOB of a circle with center O and radius \( r \) cm. The angle AOB is \( \theta \) radians. The area of the sector AOB is \( 11 \, \text{cm}^2 \). Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of \( r \). (4 Marks)

Figure 1:

Question 4

The curve with the equation \( y = 2\ln(8 – x) \) meets the line \( y = x \) at a single point, \( x = \alpha \).

(a) Show that \( 3 < \alpha < 4 \). (2 Marks)

Figure 2 shows the graph of \( y = 2\ln(8 – x) \) and the graph of \( y = x \).

A student uses the iteration formula:

In an attempt to find an approximation for \( \alpha \),Using the graph and starting with \( x_1 = 4 \):

(b) Determine whether or not this iteration formula can be used to find an approximation for \( \alpha \), justifying your answer. (2 Marks)

Question 5

Given that:

Show that:

Where \( A \) is a rational constant to be found. (5 Marks)

Question 6

The circle \( C \) has center \( A \) with coordinates \( (7, 5) \).

The line \( l \), with equation \( y = 2x + 1 \), is the tangent to \( C \) at the point \( P \), as shown in Figure 3.

(a) Show that an equation of the line \( PA \) is \( 2y + x = 17 \). (3 Marks)

(b) Find an equation for \( C \). (4 Marks)

The line with equation \( y = 2x + k \), \( k \neq 1 \), is also a tangent to \( C \).

(c) Find the value of the constant \( k \). (3 Marks)

Question 7

Given that \( k \in \mathbb{Z}^+ \),

(a) Show that:

\[ \int_{k}^{3k} \frac{2}{3x – k} \, dx \] is independent of \( k \). (4 Marks)

(b) Show that:

\[ \int_{k}^{2k} \frac{2}{(2x – k)^2} \, dx \] is inversely proportional to \( k \). (3 Marks)

Question 8

The depth of water, \( D \) metres, in a harbour on a particular day is modelled by the formula:

\[ D = 5 + 2 \sin(30t)^\circ \quad \text{where} \, 0 \leq t < 24 \] where \( t \) is the number of hours after midnight.

A boat enters the harbour at 6:30 AM, and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.

(a) Find the depth of the water in the harbour when the boat enters the harbour. (1 Marks)

(b) Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.) (4 Marks)

Question 9

Figure 4 shows a sketch of the curve with the equation:

\[ x^2 – 2xy + 3y^2 = 50 \]

The curve is used to model the shape of a cycle track with both \( x \) and \( y \) measured in km. The points \( P \) and \( Q \) represent points that are furthest west and furthest east of the origin \( O \), as shown in Figure 4.

(a) Using the equation of the curve, (4 Marks)

(b) Find the exact coordinates of the point \( P \). (5 Marks)

(c) Explain briefly how to find the coordinates of the point that is furthest north of the origin \( O \). (You do not need to carry out this calculation.) (1 Marks)

Figure 4:

Question 10

The height above ground, \( H \) metres, of a passenger on a roller coaster can be modelled by the differential equation:

\[ \frac{dH}{dt} = \frac{H \cos(0.25t)}{40} \]

where \( t \) is the time, in seconds, from the start of the ride.

Given that the passenger is 5 m above the ground at the start of the ride:

(a) Show that:

\[ H = 5 e^{0.1 \sin(0.25t)} \] (5 Marks)

(b) State the maximum height of the passenger above the ground. (1 Marks)

The passenger reaches the maximum height, for the second time, \( T \) seconds after the start of the ride.

(c) Find the value of \( T \). (2 Marks)

Question 11

(a) Use binomial expansions to show that:

\[ \sqrt{\frac{1 + 4x}{1 – x}} \approx 1 + \frac{5}{2}x – \frac{5}{8}x^2 \]

A student substitutes \( x = \frac{1}{2} \) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \( \sqrt{6} \). (6 Marks)

(b) Give a reason why the student should not use \( x = \frac{1}{2} \). (1 Marks)

(c) Substitute \( x = \frac{1}{11} \) into: \[ \sqrt{\frac{1 + 4x}{1 – x}} \approx 1 + \frac{5}{2}x – \frac{5}{8}x^2 \]

to obtain an approximation to \( \sqrt{6} \). Give your answer as a fraction in its simplest form. (3 Marks)

Question 12

The value, £\( V \), of a vintage car \( t \) years after it was first valued on 1st January 2001, is modelled by the equation:

\[ V = A p^t \]

where \( A \) and \( p \) are constants. Given that the value of the car was £32,000 on 1st January 2005 and £50,000 on 1st January 2012:

(a) (i) Find \( p \) to 4 decimal places.

(ii) Show that \( A \) is approximately 24,800. (4 Marks)

(b) With reference to the model, interpret:

• (i) The value of the constant \( A \).
• (ii) The value of the constant \( p \).

(c) Using the model, find the year during which the value of the car first exceeds £100,000. (4 Marks)

Question 13

Show that:

\[ \int_{0}^{2} 2x \sqrt{x + 2} \, dx = \frac{32}{15} \left(2 + \sqrt{2}\right) \] (7 Marks)

Question 14

A curve \( C \) has parametric equations:

\[ x = 3 + 2 \sin t \]

\[ y = 4 + 2 \cos 2t \]

where \( 0 \leq t < 2\pi \).

(a) Show that all points on \( C \) satisfy \[ y = 6 – (x – 3)^2 \] (2 Marks)

(b) (i) Sketch the curve \( C \).

(ii) Explain briefly why \( C \) does not include all points of \[ y = 6 – (x – 3)^2 \], \( x \in \mathbb{R} \).

The line with equation \( x + y = k \), where \( k \) is a constant, intersects \( C \) at two distinct points. (3 Marks)

(c) State the range of values of \( k \), writing your answer in set notation. (5 Marks)