Practice Paper 2 (Pure & Mechanics) (AQA A Level Maths: Mechanics)

Practice Paper Questions

1
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1 mark

Which one of these functions is decreasing for all real values of x?

Circle your answer.

straight h open parentheses x close parentheses equals negative 2 to the power of x minus 1 end exponent straight h open parentheses x close parentheses equals 2 to the power of x straight h open parentheses x close parentheses equals negative 2 to the power of negative x end exponent straight h open parentheses x close parentheses equals negative 2 to the power of 1 minus x end exponent

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2
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1 mark

Which one of the following equations has no real solutions?

Tick (✓) one box.

In open parentheses x minus 2 close parentheses equals 0 square
cosec space x equals 0 square
cot space x equals 0 square
open vertical bar 2 minus x close vertical bar equals 0 square

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3
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3 marks

In the expansion of left parenthesis 4 minus p x right parenthesis to the power of 6, the coefficient of the x to the power of 4 term is 19 440.
Given that p is a positive integer find the value of p.

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4a
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3 marks

Given that x is small, show that 

16 minus 7 sin x plus 2 tan 2 xnegative 18 cos xalmost equal to 9 x squared minus 3 x minus 2

4b
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3 marks

Hence, or otherwise, find approximate solution(s) to the equation

16 minus 7 sin x plus 2tan 2 x minus 18 cos x equals 0

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5
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5 marks

Use a suitable substitution to find

integral negative 15 sin space open parentheses 5 x minus 2 close parentheses space space straight d x

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6
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5 marks

The line with equation y equals x minus 1 intersects the circle with equation
open parentheses x minus 5 close parentheses squared plus open parentheses y minus 4 close parentheses squared equals 18 at two distinct points.
Find the coordinates of the two points of intersection.

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7
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3 marks

Determine if the circles with equations

open parentheses x plus 4 close parentheses squared plus y squared equals 9 space space and space open parentheses space x minus 2 close parentheses squared plus y squared equals 9

intersect once, twice or not at all.  Fully explain your answer.

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8
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3 marks

Prove that the square of an odd number is always odd.

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9
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4 marks

Prove by contradiction that there are an infinite number of even numbers.

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10
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4 marks
(i)
Find the Cartesian equation for the curve defined by the parametric equations

  space x equals space 4 cos space theta   space y equals 4 sin space theta

(ii)
Describe the shape of the curve.

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11a
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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.

11b
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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

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12a
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5 marks

A large block of ice used by sculptors is in the shape of a cuboid with dimensions x m by 2x m by 5x m.  The block melts uniformly with its surface area decreasing at a constant rate of k m2 s-1. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

  fraction numerator 15 k x over denominator 34 end fraction space straight m cubed space straight s to the power of negative 1 end exponent.

12b
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3 marks

In the case whenspace k equals 0.2, the block of ice remains solid enough to be sculpted as long as the rate of melting, by volume, does not exceed 0.05 space straight m cubed space straight s to the power of negative 1 end exponent.

Find the value of x for the largest block of ice that can be used for ice sculpting under such conditions, giving your answer as a fraction in its lowest terms.

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13
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1 mark

A tractor is driven at a constant speed of 8 ms-1 along a straight horizontal path.

Only one of the statements below is correct.

Identify the correct statement.

Tick (✓) one box.

The driving force of the tractor exceeds the total force resisting its motion. square
The resultant force acting on the tractor depends on its mass. square
The tractor is accelerating. square
The resultant force acting on the tractor is zero. square

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14
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1 mark

A number of forces act on a particle such that the resultant force is open parentheses table row cell negative 4 end cell row 10 end table close parentheses N.

One of the forces acting on the particle is open parentheses table row cell negative 2 end cell row 7 end table close parentheses N.

Calculate the total of the other forces acting on the particle.

Circle your answer.

open parentheses table row 2 row cell negative 3 end cell end table close parentheses space straight N open parentheses table row cell negative 6 end cell row 17 end table close parentheses space straight N open parentheses table row cell negative 2 end cell row 3 end table close parentheses space straight N open parentheses table row 6 row cell negative 17 end cell end table close parentheses space straight N

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15
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1 mark

A particle, P, is moving with constant velocity negative 3 bold i space plus 15 bold j.

A second particle, Q, is moving with constant velocity b bold i space minus space 20 bold j.

Q travels in a direction which is parallel to the motion of P.

Find b.

Circle your answer.

4 8 –4 –8

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16
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4 marks

A B is a uniform rod of length 1.7 m and weight 50 N. A B rests horizontally on supports placed at points C and D, with A C space equals space 0.4 m, C D space equals space 1 m and D B space equals space 0.3 m as shown in the diagram below:

q3-4-1-medium-aqa-a-level-maths-mechanics

Calculate the magnitude of the reaction force at each of the support points.

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17a
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2 marks

The position vector of a particle at time t seconds is given by

 bold r bold space equals space left parenthesis 12 e to the power of negative 0.1 t end exponent right parenthesis bold i bold space plus space left parenthesis 24 e to the power of negative 0.2 t end exponent right parenthesis bold j space space straight m

(i)
Write down the initial position of the particle.

(ii)
Briefly explain why the particle gets closer to the origin but never actually meets it.
17b
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2 marks

Find an expression for the velocity of the particle at time t seconds.

17c
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3 marks
(i)
Find an expression for the acceleration of the particle at time t seconds.

(ii)
Find the magnitude of acceleration of the particle at time t space equals space 4 seconds, giving your answer to three significant figures.

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18a
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3 marks

A TV camera runs along a horizontal track next to the pitch at a football match.

The pitch is 100 m long and the camera is initially positioned at the halfway line.
The camera moves from rest at a constant velocity and after 5 seconds has displacement 35 m.

It remains in this location for 3 seconds before moving back to the halfway line at a constant speed of 10 space straight m space straight s to the power of negative 1 end exponent.

For the next 4 seconds the camera moves with constant velocity negative 2.5 space straight m space straight s to the power of negative 1 end exponent.


Sketch a displacement-time graph to illustrate the motion of the camera.

18b
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2 marks
(i)
Write down the final displacement of the camera and the time when it reaches this position.

(ii)
How far has the camera travelled by the time it reaches this position?

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19
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5 marks

A ball is projected upwards from ground level. 1.6 seconds after reaching its maximum height the ball hits the ground. Find the maximum height the ball reaches and the velocity with which it was projected. Give your answers to an appropriate degree of accuracy.

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20a
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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 open parentheses space 0 space comma space h space close parentheses 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.

Find, in terms of U comma space alpha comma space h comma space g space and time t as appropriate, expressions for 

(i)
the x - coordinate of the projectile at time t seconds,
(ii)
the y - coordinate of the projectile at time t.
20b
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4 marks

For a particular projectile, sin space alpha space equals space 8 over 17U space equals space 51 space straight m space straight s to the power of negative 1 end exponent and the particle is projected from the point open parentheses space 0 space comma space 6 space close parentheses.  Find an expression for the trajectory of the particle, giving your answer in the form y space equals space a x space plus space b g x squared space plus space c where a comma b space and space c are rational constants.

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21
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5 marks

A particle of mass m space kg is suspended by a light inextensible string, with the other end of the string attached to a fixed point P.  With the string at an angle of 40 degree to the vertical, equilibrium is maintained by a horizontal force of 12 N which acts on the particle as shown in the diagram below:

rhJZokPx_3-3-q-3-e-a-level-maths-mechanics

Find

(i)
the tension in the string
(ii)
the value of m.

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22
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8 marks

Two particles A and B, of masses 2.7 kg and 2.2 kg respectively, are connected by means of a light inextensible string.  Particle A is held motionless on a rough fixed plane inclined at 25° 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:

mech-3-3-h-q9

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 μ.

 

The system is released from rest with the string taut.  Given that particle B descends 1.82 m in the first 3 seconds after it is released, find the value of μ.

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23a
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4 marks

A remote-controlled car is driven around a large playground with velocity,  bold v subscript bold c bold space straight m space straight s to the power of negative 1 end exponent, at time t seconds, given by

bold v subscript bold c bold space equals space open parentheses 0.45 straight t squared space plus space 2 straight t space minus space 16 close parentheses bold i space plus space open parentheses 0.75 straight t squared space minus 1 close parentheses bold j

(i)
The remote-controlled car is initially set in motion from position open parentheses negative 6 space comma space 15 close parentheses. Find the position vector bold r subscript bold c of the car at time t seconds.

(ii)
Find the distance of the remote-controlled car from the origin after 15 seconds.
23b
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3 marks

At the same time as the remote-controlled car is started, a remote-controlled truck is also set into motion. The truck has position vector, rT m, at time t space seconds given by

 bold r subscript bold T equals left parenthesis 0.15 t cubed minus 6 right parenthesis bold i plus left parenthesis 0.25 t cubed minus 1 right parenthesis bold j

Determine the time(s) at which the car and the truck will collide, if at all.

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