AQA A Level Maths Paper 1 June 2018 question paper 7357/1

Circle your answer.

The graph of y 5^x is transformed by a stretch in the y-direction, scale factor 5

State the equation of the transformed graph.

Circle your answer.

State the period of this sequence.

Circle your answer.

The function f is defined by f (x) e^x4, x

Find f ¹(x) and state its domain.

A curve is defined by the parametric equations

Find the Cartesian equation of the curve in the form xy ax by c, where a, b and c are integers.

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

Hence, find the first three terms of the binomial expansion of

to seven decimal places.

Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used.

Explain clearly whether Edward’s approximation will be an overestimate, an underestimate, or if it is impossible to tell.

Edward goes on to use the expansion from part (b) to find an approximation

Explain why Edward’s approximation is invalid.

Three points A, B and C have coordinates A (8, 17), B (15, 10) and C (2, 7)

Show that angle ABC is a right angle.

A, B and C lie on a circle.

Explain why AC is a diameter of the circle.

Determine whether the point D (8, 2) lies inside the circle, on the circle or outside the circle.

Fully justify your answer.

The diagram shows a sector of a circle OAB.

C is the midpoint of OB.

Angle AOB is radians.

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that 2 sin

Use the Newton-Raphson method with ₁ , to find ₃ as an approximation for . Give your answer correct to five decimal places.

Given that 1.89549 to five decimal places, find an estimate for the percentage error in the approximation found in part (b).

An arithmetic sequence has first term a and common difference d.

The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms.

Show that 4a 70d 4a² 20ad 25d²

Given that the sixth term of the sequence is 25, find the smallest possible value of a.

A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using

where m₀ milligrams is the initial mass of caffeine in the body and m milligrams is the mass of caffeine in the body after t hours.

On average, it takes 5.7 hours for the mass of caffeine in the body to halve.

One cup of strong coffee contains 200 mg of caffeine.

The scientist drinks two strong cups of coffee at 8 am. Use the model to estimate the mass of caffeine in the scientist’s body at midday.

The scientist wants the mass of caffeine in her body to stay below 480 mg

Use the model to find the earliest time that she could drink another cup of strong coffee.

Give your answer to the nearest minute.

State a reason why the mass of caffeine remaining in the scientist’s body predicted by the model may not be accurate.

The daily world production of oil can be modelled using

where V is volume of oil in millions of barrels, and t is time in years since 1 January 1980.

The model is used to predict the time, T, when oil production will fall to zero.

Show that T satisfies the equation

values of T, T, and T, giving your answers to three decimal places. 1 2 3

Explain the relevance of using T₀ 38

From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as

Use the models to show that the country’s use of oil and the world production of oil will be equal during the year 2029.

p(x) 30x³ 7x² 7x 2

Prove that (2x 1) is a factor of p(x)

Factorise p (x) completely.

Prove that there are no real solutions to the equation

A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible.

The company models the logo on an x - y plane as shown in the diagram.

Use calculus to find the maximum area of the rectangle.

Fully justify your answer.

Some students are trying to prove an identity for sin (A B).

They start by drawing two right-angled triangles ODE and OEF, as shown.

The students’ incomplete proof continues,

Let angle DOE A and angle EOF B.

In triangle OFR,

Explain why in Line 4 leads to cos A sin B in Line 5

Complete Line 4 and Line 5 to prove the identity

Explain why the argument used in part (a) only proves the identity when A and B are acute angles.

Another student claims that by replacing B with B in the identity for sin (A B) it is possible to find an identity for sin (A B).

Assuming the identity for sin (A B) is correct for all values of A and B, prove a similar result for sin (A B).

A curve has equation y x³ 48x

The point A on the curve has x coordinate 4

The point B on the curve has x coordinate 4 h

Show that the gradient of the line AB is h² 12h

Explain how the result of part (a) can be used to show that A is a stationary point on the curve.