Practice Paper 1 (Pure) (AQA A Level Maths: Mechanics)

Given that  find .

Circle your answer.

A geometric sequence has a sum to infinity of .

A second sequence is formed by multiplying each term of the original sequence by .

What is the sum to infinity of the new sequence?

Circle your answer.

5

–5

the sum to infinity does not exist

Milo is attempting to use proof by contradiction to show that the result of multiplying a prime number by a multiple of 3 is never a prime number.

Select the assumption he should make to start his proof.

Tick () one box.

Every prime multiplied by a multiple of 3 is never prime.
Every prime multiplied by a multiple of 3 is prime.
There exists a prime and a multiple of 3 whose product is prime.
There exists a prime and a multiple of 3 whose product is not prime.

The line l passes through the points (3, 4) and (9, 2).

Find the equation of the line l, giving your answer in the form .

Write down the gradient of a line perpendicular to l.

The line segment AB is the diameter of a circle.

A has coordinates (-7,-9) and B has coordinates (9, 3).

Find the coordinates of the centre of the circle and the length of the diameter.

An arithmetic series is defined by

Find an expression for  in terms of  and 

For a particular value of ,   and .

Find the value of .

The diagram below shows the graphs of and .

The iterative formula

is to be used to find an estimate for a root, , of the function.

Write down an expression for .

Using an initial estimate, , show, by adding to the diagram above, which of the two points (S or T) the sequence of estimates will converge to.
Hence deduce whether   is the -coordinate of point S or point T.

Find the estimates and , giving each to three decimal places.

Confirm that α = 4.146 correct to three decimal places.

Show that

sin 

Show that the equation  cosec² cosec   can be written as

Hence, or otherwise, solve the equation

cosec² cosec 

Carbon-14 is a radioactive isotope of the element carbon.
Carbon-14 decays exponentially – as it decays it loses mass.
Carbon-14 is used in carbon dating to estimate the age of objects.

The time it takes the mass of carbon-14 to halve (called its half-life) is approximately 5700 years.

 A model for the mass of carbon-14, m g, in an object of age  years is

where and  are constants.

For an object initially containing 100g of carbon-14, write down the value of .

Briefly explain why, if ,  will equal g  when  years.

Using the values from part (b), show that the value of is to three significant figures.

A different object currently contains 60g of carbon-14.
In 2000 years’ time how much carbon-14 will remain in the object?

An exponential model of the form    is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream,  hours after the drug was administered by injection.   and  are constants.

The graph below shows values of   plotted against  with a line of best fit drawn.

(i)        Use the graph and line of best fit to estimate at time .

(ii)       Work out the gradient of the line of best fit.

Use your answers to part (a) to write down an equation for the line of best fit in the form ,  where and  are constants.

Show that  can be rearranged to give 

Hence find estimates for the constants and .

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

Differentiate  with respect to x.

Use the substitution to find

Show that the general solution to the differential equation

is

where A is a constant.

On the same set of axes sketch the graphs of the solution for the instances where

In each case be sure to state where the graph intercepts the y-axis.

Show that the derivative function of the curve given by 

is given by

Find the equation of the normal to the curve given in part (a) at the point where , giving your answer in the form where and c are integers to be found.

Show that where  and  are constants to be found.

Hence factorise  completely.

Write down all the real roots of the equation .

The diagram below shows a sketch of the curve defined by the parametric equations

The four tangents from part (a) create a rectangle around the curve as shown below.

Find the percentage of the area of the rectangle enclosed by the curve

(the shaded area on the diagram).

The circle sector  is shown in the diagram below.

The angle at the centre is  radians , and the radii  and  are each equal to cm

Additionally, is parallel to , so that   and .

In the case when cm, show that the area of the shaded shape is given by sin .

.

Show that for small values of , the area of is approximately .

Given that  is small, write an approximation in terms of  for