Practice Paper 3 (Pure & Mechanics) (OCR A Level Maths: Pure)

Practice Paper Questions

1
Sme Calculator
4 marks

Calculate the values of x and y in the triangle below. Giving your answers to 3 significant figures.

q2-5-1-basic-trigonometry-edexcel-a-level-pure-maths-medium

Did this page help you?

2a
Sme Calculator
2 marks

Show that x squared plus y squared plus 5 x minus 2 y minus 5 equals 0 can be written in the form open parentheses x minus a close parentheses squared plus open parentheses y minus b close parentheses squared equals r squared, where a comma space b and r are constants to be found.

2b
Sme Calculator
2 marks

Hence write down the centre and radius of the circle with equation x squared plus y squared plus 5 x minus 2 y minus 5 equals 0.

Did this page help you?

3
Sme Calculator
4 marks

The line space y plus 2 x equals 11 spacemeets the circle with equation x squared plus y squared plus 6 x minus 14 y equals negative 38.

(i)
Show that the line and circle meet at one point only.

(ii)
Find the coordinates of the point of intersection.

Did this page help you?

4
Sme Calculator
3 marks

Solve the equation open vertical bar 4 x space plus space 2 close vertical bar equals 5.

Did this page help you?

5
Sme Calculator
4 marks

Describe a sequence of transformations that map the graph of y space equals space In space x onto the graph of y space equals space 5 space plus space In open parentheses 1 half x space plus space 4 close parentheses.

Did this page help you?

6a
Sme Calculator
3 marks

Given  y equals x squared plus ln space x space minus 2 x,  where x greater than 0,  show that

fraction numerator d squared y over denominator d x squared end fraction equals 2 minus 1 over x squared

6b
Sme Calculator
3 marks

Solve the inequality

fraction numerator d squared y over denominator d x squared end fraction greater than 0

and hence determine the set of values for which the graph of space y equals x squared plus ln space x minus 2 x space  is convex.

Did this page help you?

7a
Sme Calculator
2 marks

The function space straight f left parenthesis x right parenthesis spaceis defined as

 space straight f open parentheses x close parentheses equals sin space 3 x minus ln space 2 x     x greater than 0, where x is in radians.

Find straight f apostrophe left parenthesis x right parenthesis.

7b
Sme Calculator
4 marks

Use the Newton-Raphson method with x subscript 0 equals 0.8 to find a root, α, of the equation straight f open parentheses x close parentheses equals 0, correct to four decimal places.

7c
Sme Calculator
1 mark

The graph of y equals straight f left parenthesis x right parenthesis has a local maximum point at x equals straight beta. Briefly explain why the Newton-Raphson method would fail if the exact value of β was used for x subscript 0.

Did this page help you?

8
Sme Calculator
6 marks

The diagram below shows part of the graph with equation y equals 5 minus 3 e to the power of negative x end exponent.

The trapezium rule is to be used to estimate the shaded area of the graph which is given by the integral

integral subscript 1 superscript 2 5 minus 3 e to the power of negative x end exponent space space straight d x

(i)
Given that 4 strips are to be used, calculate the width of each strip, h.

(ii)
Complete the table of values below, giving each entry correct to three significant figures.

x 

1

1.25

1.5

1.75

2

y 

3.90

   

4.48

 

(iii)
Use the trapezium rule with the values from the table in part (ii) to find an estimate of the shaded area.

Did this page help you?

9
Sme Calculator
5 marks

Prove the identity

fraction numerator 1 minus tan squared space x over denominator cos space 2 x end fraction identical to sec squared space x space space space space space space space space space space space space x not equal to fraction numerator 2 straight k plus 1 over denominator 4 end fraction straight pi

Did this page help you?

10a
Sme Calculator
4 marks

Express

fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction


as partial fractions.

10b
Sme Calculator
3 marks

Hence, or otherwise, find

integral fraction numerator x squared minus 4 x plus 7 over denominator open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction space straight d x

Did this page help you?

11
Sme Calculator
4 marks

The diagram below shows the velocity-time graph for a train travelling between two stations, starting at station P and finishing at station Q. The graph indicates velocity in kilometres per hour and time in minutes.

edexcel-al-maths-mechanics-topic-2-1-h---q3

(i)
Find the distance between station P and station Q.

(ii)
Find the deceleration of the train in the last 20 minutes.

Did this page help you?

12a
Sme Calculator
3 marks

A diver jumps from the edge of a diving board that is 10 m above the surface of the water.  The diver leaves the board at an angle of 80° above the horizontal with a speed of 4 space straight m space straight s to the power of negative 1 end exponent.  The diver is then modelled as a projectile until they splash into the swimming pool below.

Find the time of the dive, giving your answer in seconds to three significant figures.

12b
Sme Calculator
3 marks

Find the maximum height above the water achieved by the diver, giving your answer to the nearest tenth of a metre.

Did this page help you?

13a
Sme Calculator
7 marks

Two particles A and B, of masses 4 kg and 3 kg respectively, are connected by means of a light inextensible string.  Particle A is held motionless on a rough fixed plane inclined at 35 degree to the horizontal.  The string passes over a smooth light pulley fixed at the top of the plane so that B is hanging vertically downwards as shown in the diagram below:

3-3-q-10-m-a-level-maths-mechanics

The string between A and the pulley lies along a line of greatest slope of the plane, and B hangs freely from the pulley.  The coefficient of friction between particle A and the plane is 0.15.

The system is released from rest with the string taut.

Calculate the acceleration of the two objects and the tension in the string as B descends.

13b
Sme Calculator
6 marks

After descending for 3.2 seconds, particle B strikes the ground and immediately comes to rest.  Particle A continues to move up the slope until the forces of gravity and friction cause it to come momentarily to rest.

Find the total distance travelled by particle A between the time that the system is first released from rest and the time that particle A comes momentarily to rest again after B has struck the ground.

Did this page help you?

14a
Sme Calculator
4 marks

A stone is thrown from the edge of a deep cave such that it will fall into the cave and has velocity

bold v equals left parenthesis left parenthesis 0.4 t right parenthesis bold i plus left parenthesis 2 minus 0.3 t squared right parenthesis bold j right parenthesis space straight m space straight s to the power of negative 1 end exponent

at time t seconds after it is thrown. 

Find the position of the stone at the time it is about to fall into the cave (rather than being in the air above the cave).

14b
Sme Calculator
3 marks

Find the maximum height above the cave the stone reaches.

14c
Sme Calculator
2 marks

The deepest known cave in the world has a depth of 2212 m. (The Veryovkina Cave in Georgia.) The model above suggests the stone would take around 28 space seconds to reach this depth. Consider the average speed and the acceleration of the stone to highlight a problem with this model.

Did this page help you?

15a
Sme Calculator
3 marks

An ice skater moves across a straight section of a frozen river such that their position, at time t seconds relative to an origin is given by

 bold r equals open parentheses table row cell 2 t to the power of 1 half end exponent plus t end cell row cell t to the power of 1 half end exponent end cell end table close parentheses space straight m space space space space space space space space space space space space space space space space 0 less or equal than t less or equal than 225

(i)        Find the distance the skater is from the origin after 25 seconds.


(ii)       As they skate forwards, the skater slowly crosses the width of the river.
            It takes 225 seconds for the skater to cross the river.
            How wide is the river?

15b
Sme Calculator
4 marks

Show that the magnitude of acceleration of the skater at time t seconds is given by a equals 0.25 square root of 5 t to the power of negative 3 end exponent end root space straight m space straight s to the power of negative 2 end exponent.

Did this page help you?

16a
Sme Calculator
9 marks

In the following diagram AB is a ladder of length 2a and mass ml.  End A of the ladder is resting against a rough vertical wall, while end B rests on rough horizontal ground so that the ladder makes an angle of θ with the ground as shown below:

 4-1-vh-qu8 

A person with mass mp is standing on the ladder a distance d from end B. The ladder may be modelled as a uniform rod lying in a vertical plane which is perpendicular to the wall, and the person may be modelled as a particle. The coefficient of friction between the wall and the ladder is μA, and the coefficient of friction between the ground and the ladder is μB. It may be assumed that 0 less than theta less than 90 degree.

 

Given that the ladder is at rest in limiting equilibrium, show that

 R subscript B equals fraction numerator a m subscript l plus d m subscript p over denominator 2 a mu subscript B left parenthesis mu subscript A plus t a n space theta right parenthesis end fraction space g 

where RB is the normal reaction force exerted by the ground on the ladder at point B and where g is the constant of acceleration due to gravity.

16b
Sme Calculator
2 marks

Hence find an equivalent expression for RA, the normal reaction force exerted by the wall on the ladder at point A when the ladder is at rest in limiting equilibrium.

Did this page help you?