Edexcel A Level Maths Paper 1: Pure Mathematics 1 June 2019 question paper 9MA0/01

f(x) = 3x³ + 2ax² – 4x + 5a

Given that (x + 3) is a factor of f(x), find the value of the constant a.

Figure 1 shows a plot of part of the curve with equation y = cos x where x is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.

Use Diagram 1 to show why the equation

has only one real root, giving a reason for your answer.

Given that the root of the equation is , and that is small,

use the small angle approximation for cos x to estimate the value of to 3 decimal places.

Show that = where A and n are constants to be found.

Hence deduce the range of values for x for which 0

Find the first three terms, in ascending powers of x, of the binomial expansion of

giving each coefficient in its simplest form.

Possible values of x that could be substituted into this expansion are:

Without evaluating your expansion,

state, giving a reason, which of the three values of x should not be used

f(x) = 2x² + 4x + 9 x

Write f(x) in the form a(x + b)² + c, where a, b and c are integers to be found.

Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where

g(x) = 2(x – 2)² + 4x – 3 x

Find the range of the function

Solve, for –180° θ 180°, the equation

giving your answers, where necessary, to one decimal place.

[ Solutions based entirely on graphical or numerical methods are not acceptable. ]

Deduce the smallest positive solution to the equation

In a simple model, the value, £ V, of a car depends on its age, t, in years.

The following information is available for car A

its value when new is £20 000 its value after one year is £16 000

Use an exponential model to form, for car A, a possible equation linking V with t.

The value of car A is monitored over a 10-year period. Its value after 10 years is £2 000

Evaluate the reliability of your model in light of this information.

The following information is available for car B

it has the same value, when new, as car A its value depreciates more slowly than that of car A

Explain how you would adapt the equation found in (a) so that it could be used to model the value of car B.

Figure 2 shows a sketch of part of the curve with equation y = x (x + 2)(x – 4).

The region R₁ shown shaded in Figure 2 is bounded by the curve and the negative x-axis.

Show that the exact area of R₁ is

The region R₂ also shown shaded in Figure 2 is bounded by the curve, the positive x-axis and the line with equation x = b, where b is a positive constant and 0 b 4

Given that the area of R₁ is equal to the area of R₂

verify that b satisfies the equation

The roots of the equation 3b² – 20b + 20 = 0 are 1.225 and 5.442 to 3 decimal places. The value of b is therefore 1.225 to 3 decimal places.

Explain, with the aid of a diagram, the significance of the root 5.442

Given that a b 0 and that a and b satisfy the equation

log a – log b = log(a – b)

show that

Write down the full restriction on the value of b, explaining the reason for this restriction.

Prove that for all n , n² + 2 is not divisible by 4

“Given x , the value of 3x – 28 is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true.

A competitor is running a 20 kilometre race.

She runs each of the first 4 kilometres at a steady pace of 6 minutes per kilometre. After the first 4 kilometres, she begins to slow down.

In order to estimate her finishing time, the time that she will take to complete each subsequent kilometre is modelled to be 5% greater than the time that she took to complete the previous kilometre.

Using the model,

show that her time to run the first 6 kilometres is estimated to be 36 minutes 55 seconds,

estimate the total time, in minutes and seconds, that she will take to complete the race.

f(x) = 10e^–0.25x sin x, x 0

Show that the x coordinates of the turning points of the curve with equation y = f(x) satisfy the equation tan x = 4

Figure 3 shows a sketch of part of the curve with equation y = f(x).

Sketch the graph of H against t where

showing the long-term behaviour of this curve.

The function H(t) is used to model the height, in metres, of a ball above the ground t seconds after it has been kicked.

Using this model, find

the maximum height of the ball above the ground between the first and second bounce.

Explain why this model should not be used to predict the time of each bounce.

Explain why you can deduce that q = 4

Figure 4 shows a sketch of part of the curve C. The region R, shown shaded in Figure 4, is bounded by the curve C, the x-axis and the line with equation x = 3

Show that the exact value of the area of R is a ln 2 + b ln 3, where a and b are rational constants to be found.

The curve C, in the standard Cartesian plane, is defined by the equation

The curve C passes through the origin O

Find the value of at the origin.

Use the small angle approximation for sin 2 y to find an equation linking x and y for points close to the origin.

Explain the relationship between the answers to (a) and (b)(i).