A1 MATHEMATICS HOLIDAY HOME WORK

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WORKSHEET 1- PURE MATHEMATICS QUADRATIC EQUATIONS AND INEQUALITIES

1) Factorise completely 3 24 3x x x . [3]

2) a) Express 2 6 16x x in the form 2( )x p q , where p and q are integers. [2]

b) A curve has equation y = 2 6 16x x .

Using your answer from part (a), or otherwise:

(i) find the coordinates of the vertex (minimum point) of the curve; [2]

(ii) sketch the curve, indicating the value where the curve crosses the y-axis. [2]

(iii) state the equation of the line of symmetry of the curve. [1]

3) The quadratic equation 2 ( 4) (4 1) 0x m x m , where m is a constant, has equal roots.

a) Show that 2 8 12 0m m . [3]

b) Hence find the possible values of m. [2]

4) Given that f(x) = 2( 6 )( 2) 3x x x x ,

a) Express f(x) in the form 2( )x ax bx c , where a, b and c are constants. [3]

b) Hence factorise f(x) completely. [4]

c) Sketch the graph of y = f(x), showing the coordinates of each point at which the graph

meets the axes. [3]

5) Given that 2f ( ) 6 18, 0,x x x x

a) Express f(x) in the form 2( )x a b , where a and b are integers. [3]

The curve C with equation y = f(x), x ≥ 0, meets the y-axis at P and has a minimum point

at Q.

b) Sketch the graph of C, showing the coordinates of P and Q. [4]

The line y = 41 meets C at the point R.

c) Find the x-coordinate of R, giving your answer in the form 2p q , where p and q are

integers. [5]

6) Solve the equation 4 25 6 0x x . [5]

7) Find the coordinates of the point where the line with equation x – 4y = 3 intersects the line

with equation 3x + 5y = 26. [3]

8) The line L has equation y + 2x = 12 and the curve C has equation 2 4 9y x x .

(i) Show that the x-coordinates of the points of intersection of L and C satisfy the equation

2 2 3 0x x [1]


(ii) Hence find the coordinates of the points of intersection of L and C. [4]

9) Solve the simultaneous equations

x + y = 2

x2 + 2y = 12 [6]

10) Solve each of the following inequalities:

a) 3(𝑥 – 1)

5(𝑥 + 6)+ 1 >

4

5(𝑥 + 6) [3]

b) 2 6 0x x [4]

11) Find the set of values of x for which

a) 3(2x + 1) > 5 – 2x, [2]

b) 2x2 – 7x + 3 > 0, [4]

c) both 3(2x + 1) > 5 – 2x and 2x2 – 7x + 3 > 0. [2]

12) Find the set of values for x for which

(a) 6x – 7 < 2x + 3, [2]

(b) 2x2 – 11x + 5 < 0. [4]

13) a) Using the method of completing the square, show that 4x2 – 4x + 3 is always positive

for all real values of x. [3]

b) Solve the inequality 32𝑥 − 243

𝑥2+7𝑥−60 > 4. [6]

14) a) Show that using the substitution y = 2x, the equation 2x + 2 + 23 – x = 33 can be written

as 4y2 – 33y + 8 = 0. [2]

b) Hence solve the equation 2x + 2 + 23 – x = 33. [4]

15) a) Given that y = 3x, find expression in terms of y for

i) 3x + 1

ii) 32x – 1 [2]

b) Hence, or otherwise, solve the equation 3x + 1 - 32x – 1 = 6, giving non exact answers to

2 decimal places. [4]

“That feeling you get when all of your homework is done”

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WORKSHEET 2- PURE MATHEMATICS COORDINATE GEOMETRY

1) The point A has coordinates (1, 7) and the point B has coordinates (5, 1).

a) (i) Find the gradient of the line AB. [2]

(ii) Hence, or otherwise, show that the line AB has equation 3x + 2y = 17. [2]

b) The line AB intersects the line with equation x – 4y = 8 at the point C. Find the

coordinates of C. [3]

c) Find an equation of the line through A which is perpendicular to AB. [3]

2) The line l1 passes through the point (9, -4) and has gradient 13 .

a) Find an equation for l1 in the form ax + by + c = 0, where a, b and c are integers. [3]

The line l2 passes through the origin O and has gradient -2. The lines l1 and l2 intersect

at the point P.

b) Calculate the coordinates of P. [4]

Given that l1 crosses the y-axis at the point C,

c) Calculate the exact area of triangle OCP. [3]

3)

The points A(1, 7), B(20, 7) and C(p, q) form the vertices of a triangle ABC, as shown in

the diagram. The point D(8, 2) is the mid-point of AC.

a) (i) Find the value of p and the value of q. [2]

(ii) Calculate the distance between points A and C. [2]

The line l, which passes through D and is perpendicular to AC, intersects AB at E.

b) Find an equation for l, in the form ax + by + c = 0, where a, b and c are integers. [4]

c) Find the exact x-coordinate of E. [2]

4) The points A(3, 2) and B(8 , 4) are at the ends of a diameter of the circle.

(a) Find the coordinates of the centre of the circle. [2]


(b) Find an equation of the diameter AB, giving your answer in the form

ax + by + c = 0, where a, b and c are integers. [4]

(c) Find an equation of tangent to the circle at B. [3]

The line l passes through A and the origin.

(d) Find the coordinates of the point at which l intersects the tangent to the circle at B,

giving your answer as exact fractions. [4]

5) The points A and B have coordinates (4, 6) and (12, 2) respectively. The straight line l1

passes through A and B.

(a) Find an equation for l1 in the form ax + by = c, where a, b and c are integers. [4]

The straight line l2 passes through the origin and has gradient –4.

(b) Write down an equation for l2. [1]

The lines l1 and l2 intercept at the point C.

(c) Find the exact coordinates of the mid-point of AC. [5]

6) A function f is given by f(x) = 9 – (x – 2)2

(a) Write down the maximum value of f(x). [1]

(b) Sketch the graph of y = f(x), showing the coordinates of the points at which the graph

meets the coordinate axes. [5]

7) The circle C has centre (-3, 2) and passes through the point (2, 1).

(a) Show that the point with the coordinates (-4, 7) lies on C. [3]

(b) Find a equation for the tangents to C at the point (-4, 7). Give your answer in the

form ax + by + c = 0, where a, b and c are integers. [5]

8) The points A and B on the graph of y = f(x) have coordinates (–2, –7) and (3, 8)

respectively.

(a) Find, in the form y = mx + c, an equation of the straight line through A and B. [4]

(b) Find the coordinates of the point at which the line AB crosses the x-axis. [2]

The mid-point of AB lies on the line with equation y = kx, where k is a constant.

(c) Find the value of k. [2]

9) The points A(1, 2), B (7, 2) and C(k, 4), where k is a constant, are the vertices of ABC.

Angle ABC is a right angle.

(a) Find the gradient of AB. [2]

(b) Calculate the value of k. [2]

(c) Show that the length of AB may be written in the form p5, where p is an integer to

be found. [3]

(d) Find the exact value of the area of ABC. [3]


(e) Find an equation for the straight line l passing through B and C. Give your answer in

the form ax + by + c = 0, where a, b and c are integers. [2]

10)

Figure shows the curve with equation y2 = 4(x – 2) and the line with equation

2x – 3y = 12.

The curve crosses the x-axis at the point A, and the line intersects the curve at the points

P and Q.

(a) Write down the coordinates of A. [1]

(b) Find, using algebra, the coordinates of P and Q. [6]

(c) Show that PAQ is a right angle. [4]

11)

The points A (3, 0) and B (0, 4) are two vertices of the rectangle ABCD, as shown in

Fig.


(a) Write down the gradient of AB and hence the gradient of BC. [3]

The point C has coordinates (8, k), where k is a positive constant.

(b) Find the length of BC in terms of k. [2]

Given that the length of BC is 10 and using your answer to part (b),

(c) find the value of k, [4]

(d) find the coordinates of D. [2]

12) A function maps a point P(x, y) to a point P/(2y – x, 2x – y)

a) Find the images of A/ and B/ for the points A(-3, 2), B(2, -1) under this function.

b) Find the gradient of the line joining B and B/.

c) Show that the midpoint of A/B/ is the image of the midpoint of AB under this

function.

d) Prove that the line joining the midpoints of AB and A/B/ is parallel to BB/.

13) A line is given by the equation 3x – y - 1 = 0.

a) Find the distance between the point (1, 3) and (2, 1) and the line.

b) Determine whether the points are on the same or the opposite side of the line.

c) Find the coordinates of two points which lie on the line 3x – y – 1 = 0 and are at a

distance of 2 ½ units from the line 6x – 8y + 3 = 0

14) a) Find the tangents of the angles between two lines 3x – y = 1 and x + y = 2.

b) Hence, find the angle between the two lines.

15) Three vertices of a quadrilateral are A(3, 26), B(-15, 2) and C(0, 2). Equation of CD is 4x

– 3y + 6 = 0 and AD had gradient - 24

7.

a) Prove that AB DC.

If AD and BC, when produced meet at P,

b) Find the coordinates of P.

c) Show that the ratio of the area of PDC to the area of PAB is 4 : 25.

_____________________________________________________________________________




WORKSHEET 1 - STATISTICS REPRESENTING DATA

1) A senior officer at a fire station carried out an investigation into the time taken for a fire

engine to reach the scene of the emergency. He recorded the time, t minutes, between

receiving a call for assistance and the arrival of the fire engine at the scene of the

emergency. The results are given in the table below:

(i) On graph paper draw a cumulative frequency graph to represent the data. [4]

(ii) From your graph estimate

a) the median time, [1]

b) the upper quartile of the times, [1]

c) the percentage of calls for which the time was above 13 minutes. [2]

2) Two chess players, A and B, play in a tournament. Each player plays against 11 different

opponents. Each game concludes when either the player has won, his opponent has

won, or the game has ended in a draw. The time, correct to the nearest minute, for each

game to conclude was noted and the results are given below.

For player A’s times, find

a) the median, [1]

b) the upper quartile, [1]

c) the lower quartile. [1]

(ii) On graph paper, using the same axis, draw two box-and-whisker plots to

represent the times played by A and B. [6]

(iii) Use the diagram to make two statements comparing the two sets of data. [2]

(iv) Can you state from the diagram or from the data which player was the more

successful? Give a reason for your answer. [1]


3) A doctor conducted a survey about the ages of patients who visited his surgery. She

chose a day at random and recorded the age, in completed years, of each patient visiting

the surgery on that particular day. The results are given in the table below.

(i) Draw a histogram to represent the data. [5]

(ii) Later the doctor decided that picking just one day at random was a bad idea, so

she conducted a second survey. This time patients were chosen at random from

all the patients who had visited the surgery in the last six months. The ages, in

completed years, of the selected patients are recorded below.

Draw a stem-and-leaf diagram to show the data, and state one advantage which the

stem-and-leaf diagram has when compared with a histogram. [4]

4) A student carried out a statistics survey in a supermarket: At a checkout she recorded

how many items each customer had in their shopping basket. The table below gives her

results.

(i) Calculate the mean and standard deviation of the number of items the customers

had in their shopping baskets. [5]

(ii) At the checkout each customer was given two free items. State the mean and the

standard deviation of the total number of items the customers now had. [2]

5) A class of 20 students takes a test. The score, x, for each student was recorded by the

teacher. The results are summarized by

2( 10) 208 and ( 10) 2716x x

(i) Calculate the standard deviation of the 20 scores. [3]

(ii) Show that 2 8876x . [3]

(iii) Two other students took the test later. Their scores were 18 and 16. Find the mean

and standard deviation of all 22 scores. [4]


6) For a sample of divorced women, the table below gives the age, a years, at which each

woman was divorced.

Calculate estimates of the mean and standard deviation of the ages at which women in

the sample were divorced. [5]

7) For each of 50 plants, the height, h cm, was measured and the value of (h – 100) was

recorded. The mean and standard deviation of (h -100) were found to be 24.5 and 4.8

respectively.

(i) Write down the mean and standard deviation of h. [2]

The mean and standard deviation of the heights of another 100 plants were found to be

123.0 cm and 5.1 cm respectively.

(ii) Describe briefly how the heights of the second group of plants compare with the first.

[2]

(iii) Calculate the mean height of all 150 plants. [2]

8) A mathematics student has been asked to calculate some properties of the volumes,

x cm³, of a sample of 8 solid objects. To make the calculations easier he decides to

subtract 200 cm³ from each volume. His results can be summarised as:

2( 200) 17000, ( 200) 86000000.x x

(i) Find the mean and standard deviation of the volumes of the 8 solid objects. [5]

(ii) Calculate the value of ( 300)x . [2]

9) For the data set: x -4, x + 4, x – 6, x + 6, x – 6, x + 12, find

(a) the median [3]

(b) the mean [2]

(c) the variance [4]


10) Each Monday morning, all the 60 GCSE students in a school take a short numeracy test,

marked out of 5. In one particular week, the head of mathematics disguises the results

as follows

She gives these figures to one of her sixth form mathematics classes and asks them to

work out;

(a) The value of x, a positive integer, [4]

(b) The mean and the standard deviation of the marks. [5]

What answers should the class have produced?

11) Four A level classes of 15 pupils each sit a trial statistics examinations. Their marks out

of a total of 99 are shown below:

(a) Draw up a grouped frequency distribution using classes 0-9, 10-19, ……., 90-99. [3]

(b) Calculate the mean and the standard deviation for those marks. [5]

(c) The marks are to be scaled using the relation Y = 5 + 2X, where X is the original mark.

Find the mean and the standard deviation of the scaled marks. [4]

(d) Write down the maximum and minimum values of Y. [2]

(e) If the scaling notation is written as Y = pX + q, and the mean and the standard deviation

of Y are chosen as 100 and 20 respectively, what to three significant figures are the

values of p and q? [4]

12) The average income, in £, of households in 11 regions of the United Kingdom are given

below:

a) Find the median and the upper and lower quartiles. [4]

b) On the graph paper, draw a whisker plot to represent these data. [4]


c) Describe the skewness of this distribution and identify a possible outlier. [3]

d) Calculate the mean and standard deviation of these data. [5]

Further investigation suggested that the £362.3 value could be incorrect and should in

fact be £326.3

e) Without carrying any further calculations indicate what effect this change would

have on

i) the standard Deviation [2]

ii) the interquartile range [2]

13) Summarised below is the data relating to the number of minutes, to the nearest minute,

that a random sample of 65 trains from Darlingborough were late arriving at a main

station.

a) Write down the values needed to complete the stem and leaf diagram. [2]

b) Find the Median and Quartiles of these times [5]

c) Find the 67th Percentile. [2]

d) On the graph paper construct a box plot for these data, showing clearly your scale.

[5]

e) Comment on the skewness of the distribution. [2]

14) The figures below give the lives, in months to the nearest whole number of 30 car

batteries selected at random from the output of a manufacturer.


a) Draw a stem and leaf diagram to illustrate these figures stating the (equal) class

intervals used. [4]

If the last reading had been 30 instead of 24,

b) How would you have divided up the figures so that the class intervals remain equal?

[3]

15) A class of 25 pupils takes tests in mathematics and physics (out of 100), with the following

results

a) Draw a back to back stem and leaf diagrams, suing class intervals 20 – 29, 30 – 39

etc. [5]

b) Calculate the mean and the standard deviation of each subject. [7]

c) Compare and contrast the data. [2]




WORKSHEET 1 – MECHANICS MIXED QUESTIONS

1. Three forces, in newtons, act at the point O as shown in the figure below. Find, the

magnitude and direction in which the particle starts to move. [4]

2. A horizontal force, of magnitude TN, acts on a body of mass 0·8 kg on a rough plane

inclined at an angle α to the horizontal, where sinα = 3/5. The coefficient of friction between

the body and the plane is 0·4. Given that the body is on the point of moving up the plane,

calculate the value of T. [5]

3. The driver of a car travelling at 20 m/s sees a second car 120 m in front, travelling in the

same direction at a uniform speed of 8 m/s.

(i) What is the least retardation that must be applied to the faster car so as to avoid a

collision? [4]

If the actual retardation is 1 m/s2, calculate

(ii) the time interval, in seconds, for the faster car to reach a point 66 m behind the slower

car. [3]

(iii) the shortest distance between the cars. [3]

4. A particle P of mass 4 kg is moving up a fixed rough plane at a constant speed of 16m/s

under the action of a force of magnitude 36 N. The plane is inclined at 300 to the horizontal.

The force acts in the vertical plane containing the line of greatest slope of the plane through

P, and acts at 300 to the inclined plane. The coefficient of friction between P and the plane

is . Find

a) the magnitude of the normal reaction between P and the plane, [2]

b) the value of . [2]

The force of magnitude 36 N is removed.


c) Find the distance that P travels between the instant when the force is removed and the

instant when it comes to rest. [5]

5. A ring is moving on a straight wire. Its velocity is v m/s at time t seconds after passing a

point Q. Model A for the motion of the ring gives the velocity time graph for 0 ≤ t ≤ 6 as

shown figure.

Use Model A to calculate

i) Acceleration of the ring when t = 0.5 sec. [2]

ii) The displacement of the ring when t = 2 sec and when t = 6 sec. [4]

5. A particle P of mass 4 kg is moving up a fixed rough plane at a constant speed of 16m/s

under the action of a force of magnitude 36 N. The plane is inclined at 300 to the horizontal.

The force acts in the vertical plane containing the line of greatest slope of the plane through

P, and acts at 300 to the inclined plane. The coefficient of friction between P and the plane

is . Find

a) the magnitude of the normal reaction between P and the plane, [2]

b) the value of . [2]

The force of magnitude 36 N is removed.

c) Find the distance that P travels between the instant when the force is removed and

the instant when it comes to rest. [5]


6.

The velocity time graph in the figure represents the journey of a train P travelling along a

straight track between two stations which are 1.5 km apart. The train P leaves the first station,

accelerating uniformly from rest for 300 m until it reaches a speed of 30m/s. The train then

maintains this speed for T seconds before decelerating uniformly at 1.25m/s2, coming to rest

a) Find the acceleration of the train during the first 300 m of its journey and calculate

the value of T. [5]

b) Find an expression for the distance travelled the train P at any time t > T in terms

of t. [3]

A second train Q completes the same journey in the same total time. The train leaves the

first station, accelerating uniformly from rest untill it reaches a speed of V m/s and then

immediately decelerates uniformly untill it comes to rest at the next station.

c) Sketch, a velocity time graph which represents the journey of train Q. [2]

d) Find the value of V. [2]

7. Three horizontal forces, acting at a single point, have magnitudes 12N, 14n and 5 N and

act along bearings 0000, 0900 and 2700 respectively. Find the magnitude and bearing of

their resultant. [4]

8)

A particle P of mass 0.25 kg moves upwards with constant speed u m/s along a line of

greatest slope on a smooth plane inclined at 300 to the horizontal. The pulling force acting

on P has magnitude T N and acts at an angle of 200 to the line of greatest slope. Calculate


i) The value of T, [3]

ii) The magnitude of the contact force exerted on P by the plane. [3]

The pulling force T N acting on P suddenly removed, and P comes to instantaneous rest

0.4 s later.

iii) Calculate u. [4]

9)

In Fig. AOC = 90 and BOC = . A particle at O is in equilibrium under the action of

three coplanar forces. The three forces have magnitude 8 N, 12 N and X N and act along

OA, OB and OC respectively. Calculate

(a) the value, to one decimal place, of , [3]

(b) the value, to 2 decimal places, of X. [3]

10)

A box of mass 1.5 kg is placed on a plane which is inclined at an angle of 30 to the

horizontal. The coefficient of friction between the box and plane is 31 . The box is kept in

equilibrium by a light string which lies in a vertical plane containing a line of greatest


slope of the plane. The string makes an angle of 20 with the plane, as shown in Fig.

The box is in limiting equilibrium and is about to move up the plane. The tension in the

string is T newtons. The box is modelled as a particle.

Find the value of T. [10]

11) A ball is projected vertically upwards with a speed u m s1 from a point A which is 1.5 m

above the ground. The ball moves freely under gravity until it reaches the ground. The

greatest height attained by the ball is 25.6 m above A.

(a)Show that u = 22.4. [3]

The ball reaches the ground T seconds after it has been projected from A.

(b) Find, to 2 decimal places, the value of T. [4]

The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The

mass of the ball is 0.6 kg. The ground is assumed to exert a constant resistive force of

magnitude F newtons.

(c) Find, to 3 significant figures, the value of F. [6]

(d) State one physical factor which could be taken into account to make the model used

in this question more realistic. [1]

12) A woman travels in a lift. The mass of the woman is 50 kg and the mass of the lift is

950 kg. The lift is being raised vertically by a vertical cable which is attached to the top

of the lift. The lift is moving upwards and has constant deceleration of 2 m s–2. By

modelling the cable as being light and inextensible, find

(a) the tension in the cable, [3]

(b) the magnitude of the force exerted on the woman by the floor of the lift. [3]

13)

A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The

rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The

rope is inclined to the plane at an angle, where tan = ¾, and the plane is at an angle of


30° to the horizontal, as shown in Figure. The coefficient of friction between the box and

the inclined plane is 1/3 and the box is on the point of slipping up the plane. By modelling

the box as a particle and the rope as a light inextensible string, find the tension in the

rope. [8]

14) A lorry is moving along a straight horizontal road with constant acceleration. The lorry

passes a point A with speed u m s–1, (u < 34), and 10 seconds later passes a point B

with speed 34 m s–1. Given that AB = 240 m, find

(a) the value of u, [3]

(b) the time taken for the lorry to move from A to the mid-point of AB. [6]

15) A car is travelling along a straight horizontal road. The car takes 120 s to travel between

two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set

of traffic lights and moves with constant acceleration for 30 s until its speed is 22 m s–1.

The car maintains this speed for T seconds. The car then moves with constant

deceleration, coming to rest at the second set of traffic lights.

(a) Sketch, in the space below, a speed-time graph for the motion of the car between the

two sets of traffic lights. [2]

(b) Find the value of T. [3]

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of

traffic lights. The motorcycle moves from rest with constant acceleration, a m s–2, and

passes the car at the point A which is 990 m from the first set of traffic lights. When the

motorcycle passes the car, the car is moving with speed 22 m s–1.

(c) Find the time it takes for the motorcycle to move from the first set of traffic lights to

the point A. [4]

(d) Find the value of a. [2]




WORKSHEET 2 – MECHANICS MIXED QUESTIONS

1)

A box of mass 5 kg lies on a rough plane inclined at 30° to the horizontal. The box is held

in equilibrium by a horizontal force of magnitude 20 N, as shown in Figure. The force acts

in a vertical plane containing a line of greatest slope of the inclined plane. The box is in

equilibrium and on the point of moving down the plane. The box is modelled as a particle.

Find

(a) the magnitude of the normal reaction of the plane on the box, [4]

(b) the coefficient of friction between the box and the plane. [5]

2. A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed

20 m s–1 and is at the point A. The car maintains the speed of 20 m s–1 for 25 s. The car

then moves with constant deceleration 0.4 m s–2, reducing its speed from 20 m s–1 to

8 m s–1. The car then moves with constant speed 8 m s–1 for 60 s. The car then moves

with constant acceleration until it is moving with speed 20 m s–1 at the point B.

(a) Sketch a speed-time graph to represent the motion of the car from A to B. [3]

(b) Find the time for which the car is decelerating. [2]

Given that the distance from A to B is 1960 m,

(c) find the time taken for the car to move from A to B. [8]

3. A particle P is projected vertically upwards from a point A with speed u m s–1. The point

A is 17.5 m above horizontal ground. The particle P moves freely under gravity until it

reaches the ground with speed 28 m s–1.

(a) Show that u = 21 At time t seconds after projection, P is 19 m above A. [3]

(b) Find the possible values of t. [5]

The ground is soft and, after P reaches the ground, P sinks vertically downwards into the

ground before coming to rest. The mass of P is 4 kg and the ground is assumed to exert

a constant resistive force of magnitude 5000 N on P.

(c) Find the vertical distance that P sinks into the ground before coming to rest. [4]


4. A particle P of mass 2 kg is attached to one end of a light string, the other end of which is

attached to a fixed point O. The particle is held in equilibrium, with OP at 30º to the

downward vertical, by a force of magnitude F newtons. The force acts in the same vertical

plane as the string and acts at an angle of 30º to the horizontal, as shown in Figure 3

Find

(i) the value of F,

(ii) (ii) the tension in the string. [8]

5. A lifeboat slides down a straight ramp inclined at an angle of 15º to the horizontal. The

lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from

rest at the top of the ramp and is moving with a speed of 12.6 m s–1 when it reaches the

end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined

plane, find the coefficient of friction between the lifeboat and the ramp. [9]

6.

The velocity-time graph in Figure 4 represents the journey of a train P travelling along a

straight horizontal track between two stations which are 1.5 km apart. The train P leaves

the first station, accelerating uniformly from rest for 300 m until it reaches a speed of


30 m s–1. The train then maintains this speed for T seconds before decelerating uniformly

at 1.25 m s–2, coming to rest at the next station.

(a) Find the acceleration of P during the first 300 m of its journey. [2]

(b) Find the value of T. [5]

A second train Q completes the same journey in the same total time. The train leaves the

first station, accelerating uniformly from rest until it reaches a speed of V m s–1 and then

immediately decelerates uniformly until it comes to rest at the next station.

(c) Sketch on the diagram above, a velocity-time graph which represents the journey of

train Q. [2]

(d) Find the value of V. [6]

7. Four coplanar horizontal forces of magnitude 60 N, 85 N, 75 N and 40 N act on a particle

P, of mass 5 kg, in the directions shown in the diagram, where tan α = ¾.

Calculate the magnitude of the resultant force and determine the angle it makes with the

85 N force. [5]

8. A person of mass 64 kg is standing in a lift which is of mass M kg. When the lift is

accelerating downwards at a constant rate of 0.425 ms–2, the tension in the lift cable is

7500 N.

(a) Calculate the value of M. [3]

(b) Find the reaction between the person and the floor of the lift. [3]

9. A skydiver drops from rest from a hot air balloon and falls vertically under gravity for 5 s

before his parachute opens. After the parachute has opened, his speed of descent

reduces with uniform retardation for a further 10 s until his speed is 4 ms–1. He then

continues to travel at a constant speed of 4 ms–1 until he reaches the ground 2 minutes

after he left the hot air balloon.


(a) Calculate the speed of the skydiver just before his parachute opens. [3]

(b) Draw a sketch of the velocity-time graph for the skydiver’s descent. [3]

(c) Determine the height of the skydiver above the ground when he drops from the hot

air balloon. [3]

10.

Three forces of magnitude 8 N, 12 N and 9 N act at a point in the directions as shown in

the diagram.

Find the magnitude of the resultant of these three forces and the angle between the

resultant and 9 N force. [5]

11) A ball is hit vertically up into the air from a point A, which is 1·75 m above the ground.

The ball hits the ground for the first time after 2·5 s. Ignoring air resistance,

(a) find the initial speed of the ball. [3]

(b) Find the greatest height above the ground reached by the ball, [1]

(c) Calculate the speed of the ball as it hits the ground, [3]

12) Two cars, P and Q, are being crashed. At start P travels towards Q with the speed of

8 m/s. Q is initially at rest and moves towards P with an acceleration of 4 m/s2. The cars

collide T seconds after the start.

i) Find expression in terms of T for how far each of the cars has travelled since the

start and at the time they collide. [3]

At the time of start P is 90 m from Q.

ii) Show that T2 + 4T – 45 = 0 and find the value of T. [4]

13) a) A particle travelling in a straight line at 15 m/s is brought to rest with a constant

deceleration in a distance of 22.5 m. Show that this takes 3 seconds. [1]

b) A car and a bus are travelling along a straight road towards traffic lights. The traffic

lights change at time t = 0 seconds. At this instant the car has a speed of 15 m/s. The

car then decelerates uniformly to rest in 22.5 m. It had to wait at the traffic lights for 7


seconds after which it accelerates uniformly up to 15m/s in 5 seconds and then travels

at 15 m/s down the road.

i) Sketch a v-t diagram for the motion of the car in the interval 0 ≤ t ≤ 20. [4]

ii) Calculate the distance travelled by the car in the interval 0 ≤ t ≤ 15. [3]

When t = 0 the car is level with the bus which is travelling at a constant speed of 20 m/s

along a bus lane. The bus is not required to stop at the traffic lights and continues at this

speed down the road.

iii) Show that the bus has travelled 240 m further than the car at the time that the car

again reaches 15 m/s. [4]

At the instant that the car reaches its constant speed of 15 m/s, the bus begins to

decelerate uniformly at 0.2 m/s2.

It takes T seconds for the car to catch up with the bus after begins to decelerate. Show

that the car must travel (240 + 20 T – 0.1T2) m in this time. [3]

14)

The diagram shows four horizontal forces acting at a point P.

Given that the forces are in equilibrium, calculate the value of T and the size of the angle

. [6]

15)


Velocity time graph of a particle is shown above.

i) At what times the acceleration of the particle is 0. [2]

ii) Describe the motion of the particle. [3]

iii) Find the distance travelled by the particle in the motion. [2]

iv) How much is the change in position of the particle from start until 7 sec. [2]

v) Explain the difference in the answers of part iv) band v). [1]

“You could give me 43 years to do homework and I still wouldn't do it until the

night before”

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A1 MATHEMATICS HOLIDAY HOME WORK

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