Questions and Worked Solutions for Edexcel Pure Maths Paper 1 June 2018.

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2. A curve C has equation

y = x^{2} - 2x - 24 √x, x > 0

(a) Find (i) dx/dy

(ii) d^{2}x/dy^{2}

(b) Verify that C has a stationary point when x = 4

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

3. Figure 1 shows a sector AOB of a circle with centre O and radius r cm.

The angle AOB is θ radians.

The area of the sector AOB is 11cm^{2}

Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r.

4. The curve with equation y = 2ln(8 - x) meets the line y = x at a single point, x = α.

(a) Show that 3 < α < 4

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

A student uses the iteration formula

x_{n+1} = 2ln(8 - x_{n}), n ∈ ℕ
in an attempt to find an approximation for α.

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 α, justifying your answer.

6. The circle C has centre 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

(b) Find an equation for C.

The line with equation y = 2x + k, k ≠ 1 is also a tangent to C.

(c) Find the value of the constant k.

8. The depth of water, D metres, in a harbour on a particular day is modelled by the formula

D = 5 + 2 sin (30t)° 0 ≤ t < 24

where t is the number of hours after midnight.

A boat enters the harbour at 6:30am and it takes 2 hours to load its cargo.

The boat requires the depth of water to be at least 3.8metres before it can leave the harbour.

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

(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.)

9. Figure 4 shows a sketch of the curve with equation x^{2} - 2xy + 3y^{2} = 50

(a) Show that

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.

Using part (a),

(b) find the exact coordinates of the point P.

(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).

10. The height above ground, H metres, of a passenger on a roller coaster can be modelled by the differential equation

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

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

(a) show that H = 5e^{0.1sin (0.25t)}

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

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

(c) Find the value of T

11. (a) Use binomial expansions to show that

A student substitutes x = 1/2 into both sides of the approximation shown in part (a) in an attempt to find an approximation to √6

(b) Give a reason why the student should not use x = 1/2

(c) Substitute x = 1/11 into

to obtain an approximation to √6 . Give your answer as a fraction in its simplest form.

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 = Ap^{t}

where A and p are constants

Given that the value of the car was £32000 on 1st January 2005 and £50000 on 1st January 2012

(a) (i) find p to 4 decimal places,

(ii) show that A is approximately 24 800

(b) With reference to the model, interpret

(i) the value of the constant A,

(ii) the value of the constant p.

Using the model,

(c) find the year during which the value of the car first exceeds £100000

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Edexcel Pure Maths Paper 1 June 2018 Past Paper

1. Given that θ is small and is measured in radians, use the small angle approximations to find an approximate value of2. A curve C has equation

y = x

(a) Find (i) dx/dy

(ii) d

(b) Verify that C has a stationary point when x = 4

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

3. Figure 1 shows a sector AOB of a circle with centre O and radius r cm.

The angle AOB is θ radians.

The area of the sector AOB is 11cm

Given that the perimeter of the sector is 4 times the length of the arc AB, find the exact value of r.

4. The curve with equation y = 2ln(8 - x) meets the line y = x at a single point, x = α.

(a) Show that 3 < α < 4

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

A student uses the iteration formula

x

Using the graph and starting with x

(b) determine whether or not this iteration formula can be used to find an approximation for α, justifying your answer.

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

(b) Find an equation for C.

The line with equation y = 2x + k, k ≠ 1 is also a tangent to C.

(c) Find the value of the constant k.

8. The depth of water, D metres, in a harbour on a particular day is modelled by the formula

D = 5 + 2 sin (30t)° 0 ≤ t < 24

where t is the number of hours after midnight.

A boat enters the harbour at 6:30am and it takes 2 hours to load its cargo.

The boat requires the depth of water to be at least 3.8metres before it can leave the harbour.

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

(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.)

9. Figure 4 shows a sketch of the curve with equation x

(a) Show that

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.

Using part (a),

(b) find the exact coordinates of the point P.

(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).

10. The height above ground, H metres, of a passenger on a roller coaster can be modelled by the differential equation

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

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

(a) show that H = 5e

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

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

(c) Find the value of T

11. (a) Use binomial expansions to show that

A student substitutes x = 1/2 into both sides of the approximation shown in part (a) in an attempt to find an approximation to √6

(b) Give a reason why the student should not use x = 1/2

(c) Substitute x = 1/11 into

to obtain an approximation to √6 . Give your answer as a fraction in its simplest form.

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 = Ap

where A and p are constants

Given that the value of the car was £32000 on 1st January 2005 and £50000 on 1st January 2012

(a) (i) find p to 4 decimal places,

(ii) show that A is approximately 24 800

(b) With reference to the model, interpret

(i) the value of the constant A,

(ii) the value of the constant p.

Using the model,

(c) find the year during which the value of the car first exceeds £100000

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