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

Practice Paper Questions

1
Sme Calculator
3 marks

A triangular field is shown in the diagram below. Calculate the area of the field, give your answer to the nearest square metre.q7-5-1-basic-trigonometry-edexcel-a-level-pure-maths-easy

Did this page help you?

2
Sme Calculator
4 marks

A curve has equation y equals straight f open parentheses x close parentheses.

Describe the transformation of the curve given by the equations below:

(i)

y equals straight f left parenthesis x right parenthesis plus 2 comma

(ii)

straight y equals straight f left parenthesis x minus 2 right parenthesis comma

(iii)

straight y equals 3 straight f left parenthesis x right parenthesis comma

(iv)
y equals straight f left parenthesis 2 x right parenthesis.

Did this page help you?

3a
Sme Calculator
1 mark

The functions straight f left parenthesis x right parenthesis and g left parenthesis x right parenthesis are defined as follows

straight f left parenthesis x right parenthesis space equals space 3 x squared space plus space 2 space space space space space space space space space space space space space space space x element of straight real numbers
straight g left parenthesis x right parenthesis space equals space 1 space long dash space 3 x space space space space space space space space space space space space space space space x element of straight real numbers

Write down the range of straight f left parenthesis x right parenthesis.

3b
Sme Calculator
4 marks
Find
(i)
straight f g open parentheses x close parentheses
(ii)
g straight f open parentheses x close parentheses
3c
Sme Calculator
2 marks

Solve the equation straight f left parenthesis x right parenthesis space equals space straight g left parenthesis x right parenthesis space plus space 1

Did this page help you?

4
Sme Calculator
5 marks

Determine the number of points of inflection on the curve with equation

y equals x to the power of 4 plus 3 x cubed plus 2

and determine their coordinates.

Did this page help you?

5a
Sme Calculator
2 marks

The function straight f left parenthesis x right parenthesis is defined as

 space straight f open parentheses x close parentheses equals x squared minus ln space left parenthesis x plus 2 right parenthesis space   space x greater than 0

Use the sign change rule to show there is a root to the equationspace straight f open parentheses x close parentheses equals 0 space in the interval 1 less than x less than 1.2.

5b
Sme Calculator
2 marks

Find straight f apostrophe open parentheses x close parentheses.

5c
Sme Calculator
4 marks

Use the Newton-Raphson method with x subscript 0 equals 1 to find the root in the interval 1 less than x less than 1.2 correct to three decimal places.

Did this page help you?

6a
Sme Calculator
4 marks

The curvespace C space has parametric equations

x equals t squared minus 4   space y equals 3 t

Show that at the point (0 , 6),space t equals 2 spaceand find the value of  fraction numerator straight d y over denominator straight d x end fraction  at this point.

6b
Sme Calculator
3 marks

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

Did this page help you?

7a
Sme Calculator
2 marks

The graph defined by the parametric equations

x equals t to the power of 3 space end exponent    y equals 2 t squared minus 1

is shown below.

q2a-9-2-further-parametric-equations-medium-a-level-maths-pure

The point wherespace t equals t subscript 1 space end subscript has coordinates (1 , 1).

The point where t equals t subscript 2 has coordinates (8 , 7).

Find the values of t subscript 1 and t subscript 2.

7b
Sme Calculator
5 marks
(i)
Show that the shaded area can be found using the integral
integral subscript 1 superscript 2 open parentheses 6 t to the power of 4 minus 3 t squared close parentheses d t

(ii)
Hence find the shaded area.

Did this page help you?

8a
Sme Calculator
1 mark

Verify that the point A(1 , 1) lies on the curve with equation

ln open parentheses space x y close parentheses space plus x y squared equals 1.

8b
Sme Calculator
8 marks

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.
Find the area of the triangle OBC.

Did this page help you?

9a
Sme Calculator
2 marks

A train leaves station O from rest with constant acceleration bold a space equals space left parenthesis 0.3 bold i space plus space 0.7 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent.
80 seconds later it passes through (but does not stop at) station A at which point its acceleration changes to bold a space equals space left parenthesis 0.5 bold i space plus space 0.3 bold j right parenthesis space straight m space straight s to the power of negative 2 end exponent. 180 seconds later the train passes through station B.

 

Find the displacement of the train from station O when it passes through station A.

9b
Sme Calculator
2 marks

Find the velocity of the train as it passes through station A.

9c
Sme Calculator
2 marks

Find the displacement of the train between stations A and B.

Did this page help you?

10a
Sme Calculator
3 marks

In a cheese-rolling competition, a cylindrical block of cheese is rolled down a hill and its acceleration,a space m space s to the power of negative 2 end exponent , is modelled by the equation.

 a equals 1 plus 0.1 t                        0 less or equal than t less or equal than 20

where t is the time in seconds. The block of cheese reaches the bottom of the hill after 20 seconds.

 

Find the velocity of the block of cheese when it reaches the bottom of the hill.

10b
Sme Calculator
3 marks

Show that the distance down the hill, as travelled by the block of cheese, is 330 m to two significant figures.

Did this page help you?

11
Sme Calculator
13 marks

Two particles A and B, of identical mass, are connected by means of a light inextensible string. Particle A is held motionless on a rough fixed plane inclined at 30° to the horizontal, and that plane is connected at its top to another rough fixed plane inclined at 70° to the horizontal.  The string passes over a smooth light pulley fixed at the top of the two planes so that B is hanging downwards in contact with the second plane.   This situation is shown in the diagram below:

mech-3-3-h-q10

The parts of the string between A and the pulley and between B and the pulley each lie along a line of greatest slope of the respective planes. The coefficient of friction between the particles and the planes is 0.15 in both cases.

The system is released from rest with the string taut, and with particle B a vertical distance of 0.75 space straight m from the ground.  Particle B descends along the slope until it reaches the ground, at which point it 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 reached the ground.

Did this page help you?

12
Sme Calculator
8 marks

In the following diagram A B is a ladder of length 10 m and mass 34 kg.  End A of the ladder is resting against a smooth vertical wall, while end B rests on rough horizontal ground so that the ladder makes an angle of 60 degree with the ground as shown below:

4-1-h-qu7

A housepainter with a mass of 75 kg has decided to climb up the ladder without taking any additional precautions to prevent the bottom of the ladder from slipping.  The ladder may be modelled as a uniform rod lying in a vertical plane perpendicular to the wall, and the housepainter may be modelled as a particle.  The coefficient of friction between the ground and the ladder is 0.4.

Luckily, the housepainter’s partner convinces him not to climb up the ladder without providing some additional support at the bottom to prevent slipping.  If the housepainter had continued with his original plan, however, how far above the ground would he have been when the ladder began to slip?

Did this page help you?

13a
Sme Calculator
3 marks

A particle is projected from a point on a horizontal plane with initial velocity U space straight m space straight s to the power of negative 1 end exponent at an angle of alpha degree above the horizontal.  The particle moves freely under gravity. g space straight m space straight s to the power of negative 2 end exponent is the constant of acceleration due to gravity.

Show that the time of flight of the particle, T seconds, is given by

 straight T space equals space fraction numerator 2 U space sin space alpha over denominator g end fraction

13b
Sme Calculator
3 marks

Show that the range of the particle, R m, on the horizontal plane is given by

straight R space equals space fraction numerator U squared space sin space 2 alpha over denominator g end fraction 

Did this page help you?

14
Sme Calculator
5 marks

For a particle modelled as a projectile with initial velocity U space straight m space straight s to the power of negative 1 end exponent at an angle of α° above the horizontal, show that the equation of the trajectory of the particle is given by

y equals left parenthesis t a n space alpha right parenthesis space x minus fraction numerator g x squared over denominator 2 U squared c o s squared space alpha end fraction

Did this page help you?

15a
Sme Calculator
2 marks

The flight of a particle projected with an initial velocity of U space straight m space straight s to the power of negative 1 end exponent at an angle α above the horizontal is modelled as a projectile moving under gravity only. The particle is projected from the point  (x0, y0) with the upward direction being taken as positive, and with the coordinates being expressed in metres. g space straight m space straight s to the power of negative 2 end exponent is the constant of acceleration due to gravity.

Write down expressions for

(i)
the x-coordinate of the projectile at time t seconds
(ii)
the y-coordinate of the projectile at time t seconds.
15b
Sme Calculator
4 marks

For a particular projectile, space tan space alpha equals space 3 over 4, U space equals space 10 space straight m space straight s to the power of negative 1 end exponent and the particle is projected from the point left parenthesis 3 space comma space 8 right parenthesis.  Find an expression for the trajectory of the particle, giving your answer in the form

 y equals fraction numerator a x squared plus b x plus c over denominator 128 end fraction

where the constants a, b and c are expressed in terms of g.

Did this page help you?