mathematics - larberthigh.com24427... · ab sinc. 1. solve the following inequality 3x − 5≤ 5x...

Please note … the format of this practice examination is different from the current format. The paper timings are different and calculators can be used throughout.

Credit Mathematics - Practice Examination A

MATHEMATICS Standard Grade - Credit Level

Read Carefully 1. Answer as many questions as you can. 2. Full credit will be given only where the solution contains appropriate working. 3. You may use a calculator

Time allowed - 2 hours 15 minutes

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: a b c bc A or A b c abc

= + − =+ −2 2

2 2

22

cos cos2

Area of a triangle: Area = 1

2 ab Csin

1. Solve the following inequality 3 5 5 19x x− ≤ + 3 2. A bus has a single large windscreen wiper as shown below.

The radius OA = 95cm and radius OC = 20cm.

If the wiper rotates through an angle of 11 about O , calculate the area 5oof window cleaned during this sweep.

4 3. The radius of the earth is metres. 4 8 106⋅ ×

]

Calculate the surface area of the earth, assuming that the earth is a sphere. Give your answer in scientific notation and in units of square kilometres . 3 [ Surface area of a sphere r= 4 2π 4. A rectangular sheet of paper measures 30cm by 20cm. A cut is made at an angle of xo , as shown, and the triangular piece is removed. The area of the triangular piece is found to be exactly 20% of the area of the whole sheet. Find the size of angle xo giving your answer correct to the nearest degree. 5

KU RA

x o

115o

20cm

95cm

C

A

. O B

30cm

20cm

5. A function f x( ) is given by f x x( ) = +2 3 (a) Find the value of i) f (5) ii) f ( )10 . 2 (b) Show that for any value of t , f t f t( ) ( )2 2 3= − 3 6. (a) Solve the quadratic equation 3 4 5 02x x+ − = , giving your answer correct to one decimal place. 5 (b) i) Factorise 2 7 42x x− − . 1

ii) Hence, simplify 2 74 2

2x x 4x− −+

. 2

7. A boy kicks a stone off a cliff. The path of the stone as it falls is part of a parabola whose equation is . y x= −98 2 2

(a) Given that the boy is standing at the maximum turning point of the parabola, find the height of the cliff. 2 (b) The stone lands at the point A . By establishing the coordinates of A , or otherwise, calculate how far out from the bottom of the cliff the stone lands. 3

KU RA

y (feet)

y x= −98 2 2

o x (feet) A

8. The time, T minutes , to whitewash a square wall varies directly as the square of the side of the wall, W metres , and inversely as the breadth, b centimetres , of the whitewash brush. (a) The painter takes 40 minutes to whitewash a certain wall. How long would it have taken him if his brush had been twice as broad as the one he actually used ? 2 (b) The next wall to be painted has a side which is 50% broader than the first wall. Using his original (smaller) brush, how long should it take him to whitewash this second wall ? 3 9.

(a) Find x , the arc length of the quarter - circle . Give your answer in centimetres , correct to one decimal place. 2 (b) The arc length , x centimetres , becomes the circumference of the base of the cone . Find r , the radius of this base circle in centimetres . 3 (c) Calculate the volume of this cone in cubic centimetres . 5 [ ] 10. The magnetic force, F, of a particle in a magnetic field is given by the formula

F mvr

=2

.

Make v the subject of the formula. 3

KU RA

A toy manufacturer makes a range of small castles for children . The basic turret is conical in shape. It is made by folding a quarter - circle of flexible plastic sheeting into a cone , as shown in the diagram below .

x cm

10 cm 10 cm

of a cone r h= 13

2πVolume

sh

r

11. A ship sails from port P and sails due east to an oil rig Q , 75km away. After delivering its cargo, it sails on a bearing of 120 for 60km until it o

reaches its destination, port R . (a) Write down the size of angle PQR . 1 (b) Calculate the distance between the ports P and R . Do not use a scale drawing 5 12. The sequence known as triangular numbers begins as follows : 1 , 3 , 6 , 10 , ...... . They can be represented in diagram form as shown below T1 T2 T3 T4 (a) How many dots will there be in T6 ? 1 (b) The formula for the number of dots , D , in the nth triangular number is given by the formula D n= +1

22 1

2 n . 78 dots are required for a certain triangular number Tn . Calculate n . 3

0

13. Solve algebraically the equation , for 07 5cosxo + = 360≤ <x . 3

. .. .

..

. .. .

..

. ..

.

..

.

75km

R

60km

P

N

Q

KU RA

.

14. A 3 - digit number , pq r , has the value 100p + 10q + r . For example ...... 563 = (100 × 5) + (10 × 6) + 3 .

(a) Write down the value of r q p . 1 (b) Consider the following subtractions which involve a number and its 'reverse' . 321 − 123 = 198 432 − 234 = 198 543 − 345 = 198

Show that, if you start with any three digit number in the form pq r , where q r= + 1 and p r= + 2 , then when you subtract its 'reverse' number from it, the answer is always 198 . 4 15. A small ' sampler ' tin of Ludux paint is similar in shape to the larger 2 5 litre tin. ⋅

Ludux Gloss

Paint

Ludux Gloss

Paint2.5 litres

c

If the diameters of the sampler and the larger tin are 4cm and 16cm respectively, how many millilitres will the sampler tin contain ? 4 16. All straight lines are in the form y mx= + , where m and c are numbers. Consider the diagram below (a) Given that the coordinates of A and B are (-5,-35) and (2,7) respectively, form a system of equations and then solve it to find the values of m and c . 5 (b) Hence, or otherwise, write down the coordinates of the point P . 1

.x

A(-5,-35)

B(2,7)

y

P.o

.

16cm4cm

KU RA

17. The graph of has a root between 3 and 4 . y x x x= − + +3 24 1

Use iteration to find this root correct to one decimal place. Show clearly all your working. 3 18. The point A ( 3 , 1 ) lies on a circle centre the origin and radius r as shown in the diagram below. (a) 4 r = 2

KU RA

3

x.4

..2

. o

y y x x x= − + +

3 24 1

(b)

Consider the diagram below for any point A ( x , y ) in the first quadrant. Show that and hence that the area of triangle OAB is 1 square unit.

Show that, if ( x , y ) is any point on the circumference of the circle, centre O and radius r , then the area (A) of the triangle OAB is given by the formula A y x= +1

22 2y .

r

y

o

.

B.

A (x , y )

x

y

3

r

.

B.

A ( , 1 )

x o


Credit Mathematics - Practice Examination B




FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =


= + − =+ −2 2

2 2

22

cos cos2


2 ab Csin

1. Solve the following equation : 2 18 7 3x x− = − 3 2. A garden is in the shape of a sector of a circle. If the angle at the centre of the garden is 13 and 5o the radius is 8 m , find the area of the garden to the nearest square metre. 3 3. A two-tone earing, in black and white, is in the shape of a right-angled triangle. Its dimensions are shown in the diagram opposite. If the black area is exactly 1

4 of the full triangular area, find the length w . 4 4. A hospital patient is given 200 mg of a blood-thinning drug at 1.00 pm. 12% of the amount of the drug at the beginning of any hour is lost to his blood stream by the end of that hour (due to natural body processes). For the drug to be successful, there must be at least 125 mg of it in his blood stream. Between which two hours in the day will the amount of the drug drop below 125mg ? SHOW YOUR WORKING CLEARLY. 4 5. A window cleaner is washing the windows in a block of flats. (a) If his 4 metre long ladder is placed 1.2 metres from the base of the wall , how far up the building will it reach ? 2 (b) The cleaner wishes the ladder to rest on a window sill which is 6 metres above the ground. 4 m He can extend his ladder by 2.1 metres. If he places the ladder at the same point on the ground as before, with the 2.1 metre extension in place, will it reach the window sill ? JUSTIFY YOUR ANSWER. 3

18 mm

24 mm w

RA KU

135o

8 m

1 2⋅ m

6. In the diagram above, which is not drawn to scale, the line DF bisects the angle GFE and the line GE bisects the angle DEF.

(a) In the triangle DEF , show that 2a + b = 117. 2 (b) Use triangle GFE to form another equation connecting a and b and hence, or otherwise, find the values of a and b . 5 7. Below is part of the graph of . f x x x( ) = − −3 2 It has a root between x = 1 and x = 2 . Use iteration to find this root, correct to 1 decimal place. SHOW ALL YOUR WORKING CLEARLY. 3 8. (a) Factorise . 2 2 9 42p p− +

(b) Express 31

2x x−

− as a single fraction in its simplest form. 3

(c) The planet Pluto is about from the sun. 5 91 109⋅ × km Light travels from the sun at a speed of . 3 105× km per ondsec How many hours does it take for the sun's light to reach Pluto ? 3

9. Show that , if x y z= and y zx

= 2 , then xy

=1

2 . 4

o-1

63o

72o

aobo

boao

D

E F

RA KU

x

f x( )

21

x x= − −3 2

G

f(x)

10. The ordinary match box and the giant “picnic” size match box, shown above, are mathematically similar. If the volume of the smaller box is calculate the volume of the larger box. 4 15 2 3⋅ cm 11. While rummaging in his attic a man finds a triangular sheet of stiff cardboard. The sheet has dimensions as shown in diagram 1. He decides to make a kite for his son out of the cardboard by making a cut at right angles somewhere along the line AC as shown in diagram 2. (a) Write down the length of AX. 1 (b) By considering two similar triangles, or otherwise, show that the exact length of the cut XY is 45 cm . 5 12. Find the equation of the straight line shown opposite in terms of T and x . 4

KU RA

13 5⋅ cm5 4⋅ cm

BRIGHTO MATCHES BRIGHTO

MATCHES

A

B C

90 cm

120 cm

A

B

X

diagram 1

diagram 2

CY

1

1

2

2

3

3

4

4 0

T

x

13. The outline of a buoy consists of an equilateral triangle of side 120 cm with a semi-circular base as shown opposite. AM is an axis of symmetry. (a) State the radius of the base and show that the vertical height, h , is equal to 164 cm when rounded to the nearest whole number. 4 1 3 h s 14. A skip is a metalic container, used for the collection and eventual disposal of rubbish from building sites, factories, etc. The cross-section of one such skip is shown below. The cross-section is symmetrical. (a) Find the area of this cross-section. 4

1

(b) If the skip is 2 metres in depth, calculate the volume of the skip, giving your 15⋅ answer correct to the nearest whole number of cubic metres. 2 15. From a window at M in a building , a man spots a wallet lying at W in the carpark. The same wallet is spotted by his colleague from the window at N which is 6 metres below window M . If the angles between the building and their lines of sight are 3 and respectively, find the distance o 49o

from the wallet to the base of the wall. 5

RA KU

A

h

B C X

XM in terms of s .

(b) Taking s as the length of AC, write down expressions for the lengths of XC and

(c) Hence or otherwise show that . = ⋅1 37

M

1 36⋅ m

2 00⋅ m

2 90⋅ m

W

M

N

O

6 m 31o

49o

16.

Because of a hill, there is no direct road from a motorist’s village, V , to his local main town , T . He must travel 700 metres to a roundabout at R and then make a 120 turn for a o

further 1,050 metres to get to T. It is proposed that a tunnel is driven through the hill, in order to create a direct route from V to T . How long will this direct route be ? 4 17. Solve the equation . 3 2 3 0 0 360+ = ≤ <cos ,x xo where 18. The sum of consecutive squares of ODD numbers can be found using the following number pattern : SQUARES PATTERN SUM (a) Write down a similar expression for the sum of odd squares up to 13. 2 (b) By examining the connection between the first 2 numbers of the numerator (top line) of the fractions under PATTERN , or otherwise, write down an expression for S , the sum of the first n odd squares, in terms of n . 3

RA KU R

120o700 m 1050 m

T V

1 1 1 33

1

1 3 2 3 53

10

1 3 5 3 5 73

35

1 3 5 7 4 7 93

84

2

2 2

2 2 2

2 2 2 2

=× ×

=

+ =× ×

=

+ + =× ×

=

+ + + =× ×

=

[ END OF QUESTION PAPER ]

Credit Mathematics - Practice Examination C Please note … the format of this practice examination is

different from the current format. The paper timings are different and calculators can be used throughout.




FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =


= + − =+ −2 2

2 2

22

cos cos2


2 ab Csin

1. Solve the following inequality 5(2t – 1) ≥ 4t – 23. 4 2. By the end of each week, a garden pond has lost 4% of the volume it had at the

beginning of that week. If its volume at the beginning of week 1 was 26,000 litres, and it continues to lose 4% of its previous volume per week, how many litres will it have by the end of the fourth week ? (Answer to the nearest 100 litres). 3

3. A man is building a kite for his son. Its sides are 40 cm

and 85 cm long and the angle between these two sides is 105°, as shown. He finds a rod one metre long and intends to use it for the long diagonal of the kite.

Will this rod be long enough ?

5

4. On his birthday, Albert decides to buy sweets for himself and his 6 friends at the

school tuckshop. When he buys 5 Venus bars and 2 Tropics, he receives 77 pence change from £3.

The next day, his friend Fred, decides to return Albert’s generosity. When Fred buys 4 Venus bars and 3 Tropics, he receives 81 pence change from £3.

Find the cost of each sweet. 6

5. The empty water jug shown across is being filled

with water at a constant rate. Which of the four graphs below best show how the water level, h, is changing with time, t ? Explain your answer fully.

A B C D 4

KU RA

40cm

85cm

105o

h

t o

h

t t

h

to o o

h

6. Solve the equation , giving your answers to 1 decimal place. 5 0722 2 =−− xx 7. Given that the earth is a sphere of circumference 40,000 kms and that the speed of

light is 3 x 105 km per second, calculate how many times light can travel round the earth in one hour. Give your answer in scientific notation. 4 8. The time, T seconds, taken for a piece of luggage to slide

down the luggage chute of an aeroplane varies directly as the length, l metres, of the chute and inversely as the square root of h, the height in metres of one end of the chute above the other.

If it takes 14 seconds for a suitcase to travel down a 20 metre chute with one end metres above 256 ⋅the other : (a) Find an equation connecting T , h and l.

(b) How long will it take for a suitcase to slide down a chute 15 metres long if h 252 ⋅= metres ?

9. The outline of a children’s play area, which is not drawn to s

It consists of a square of side 7 metres and an obtuse angled Angle BDC = 110°.

Given that the area of the triangle is equal to the area of the 10. (a) Factorise 12112 2 +− aa

(b) Solve 4

25

2 xx −=

− , for x .

(c) Change the subject of the formula trP += 22π

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7m

7m 110o

E

D

B

A

h

cale, is shown abtriangle.

square, find the le

to r .

C

l

3

2

ove.

ngth of DC. 4

2

4

3

KU RA 18cm

11. A rectangular frame, 24 cm x 18 cm , contains a rectangular photograph. The frame gives the photograph a border of 3 cm all around, as shown opposite.

(a) The length – to – breadth ratio of the frame is 18

24 .

By considering the length – to – breadth ratio of the

photograph, show that the frame is not similar to the photograph. 2

(b) Show that, if there is a border of width w centimetres around the photograph, then the photograph and the frame can never be similar. 4

18cm

w

w

3

3

24cm

24cm

12.

cm52 ⋅

TbyCea

13. A (a)

(b) 14. T G

fi

14cm ⇒

he diagram above shows the net of a solid along with t this net.

alculate the volume of the solid, given that its net is foch 14 cm x cm , and two equilateral triangles. 52 ⋅

function is defined by 43)( += xxf where

Find the values of (i) (ii) . )1(−f )15(f

If , find the value of t . 16)( =tf

he diagram opposite shows part of the graph of

. 363 +−= xxy

iven that this curve has a root between 2 and 3 , nd this root correct to 1 decimal place.

he 3-dimensional solid formed

rmed by three equal rectangles, 5

34−≥x .

2

3

y

x o
1 2 3
3

KU RA

1242 ++−= xxh

o

h15. When a test rocket is fired, its trajectory follows the shape of a par

When this flight path is transferred to a coordinate diagram, as shopath is described by the formula

1242 ++−= xxh

(a) By establishing the coordinates of A, state how far the rocket horizontally from its starting point. (On the x – axis, 1 unit

(b) Find the maximum height of the rocket. (On the h – axis, 1

16. The sum of a series of numbers of the form 1 32 ..... +++++ xxx

found by using the formula

1232 .....1 −− ++++++ nn xxxxx11

−−

=x

xn

Example Find the sum of 1 5432 22222 +++++ by using

Solution 1 5432 22222 +++++1

16412126 −

=−−

=

(a) Use the method above to find the sum of 1 233 +++

(b) The sum 131333.....333 1232

−−

=++++++ −−p

nn1

Write down p in terms of n .

(c) Hence, by considering your answers to both (a) and

2656333.....333 121098 −

=+++++ −−n

nn

x A.

abola. wn above, then its

has travelled = 500 metres). 3

unit = 100 metres). 3

can be 12 −− + nn xx

the above formula.

63=

. 2 73 3.....3 ++

.

1

(b) above, show that

1 . 3

KU RA17. P

T

RQ O

10cm S

10cm In the diagram above, two tangents are drawn from the point P to the semi-circle

centre O. The semi-circle has a diameter of 10 cm. Angle PQR = 90° and PQ = 10 cm.

(a) Calculate the size of angle POQ , giving your answer correct to one decimal place. 3

(b) Hence, or otherwise, find the size of angle PRQ. 2

(c) Show that, if angle POQ = y and angle PRQ = x , then 902 −= yx . 3 18. Diagram 1 shows the outline of an ordinary cassette.

It consists of two identical "ring" shapes, each formed by two concentric circles with the outer circle having a radius of 12 mm as shown. The cassette tape is attached to the left hand ring and has a width of 8 mm.

8mm

.

diagram 1 diagram 2 5mm

(a) Show that the area of tape in

(b) After the tape has been play tape (on the left hand ring) h Find, in terms of π , the area

(c) When the tape on the left ha of 45 minutes. Calculate the number of min

(Assume that the time played

[ END O

12mm

diagram 1 is 256π square millimetres. 3

ing for some time, the 8 mm thick band of as been reduced to 5 mm. (see diagram 2)

of tape now attached to the left hand ring. 2

nd ring is 8 mm broad, it has a playing time

utes it takes for the tape to reduce its width to 5 mm. 3 is proportional to the area of tape used)

F QUESTION PAPER ]


Credit Mathematics - Practice Examination D




Pegasys 2005

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin

Pegasys 2005

KU RA1. Solve algebraically the following equation )12(4115 +=− xx 4 2. A function is defined as . 12)( 2 −= xxf (a) Find i) f (2) ii) f (√2 ) 2

(b) If f (t) = 17 , find the value(s) of t . 3

6cm

6cm

3. Circular tops for cans are often "stamped out" from a square of aluminium. The circular top for the can across is "stamped out" of a square of side 6 cm.

(a) Calculate the area of aluminium discarded for each top (i.e. find the shaded area). 3

(b) Express this discarded area as a percentage of the area of the original square. 2

)

(c) Show that, in general, for a circle stamped out of a square of side 2r , the shaded area, A, is given by = rA 4(2 π− (d) Hence, show that the percentage , P, of the origina

will be discarded is given by 4(25 −=P

4. The value of an antique grandfather clock is expected to inc

the value it had at the beginning of the previous year. If it was valued at £560 at the beginning of 1999, what willthe end of the year 2001, to the nearest £ ?

Pegasys 2005

2r

2r

3

)

l area of the square which

π 3

rease each year by 4% of

its expected value be at 3

KU RA5. A microwave oven has a 52 cm door which is designed in such a way that the door

will swing, at most, 48cm from the oven, as shown in the diagram below. (viewed from above)

52cm

48cm

(a) Show that the maximum angle the door can swing through is 67°

(given to the nearest whole number of degrees) 3 (b) What area will be swept out when the door swings through this angle ? 3 6. Solve algebraically the equation , giving your answers correct 0423 2 =−+ pp

to 1 decimal place. 4 B 7.

90m

T

60°

50m

A C

A farmer wishes to test the effect of a new fertilizer on his crops. To compare yields, he splits a triangular piece of land into two equal areas by constructing a fence

along BT. Each of the two smaller triangles is now "half" the area of the original triangle.

Given that AB = 50 metres, AC = 90 metres and that the angle BAC = 60°, calculate the length of AT. 4

Pegasys 2005

KU RA 8. An accurate formula for changing degrees Centigrade (C) to degrees Fahrenheit (F) is given as 3281 +⋅= CF

(a) A "rough" method of changing into o is to "double C and then add 30" . Co F Write down this "rough" method as a formula. 1

(b) There is only one temperature, in , where the "rough" formula gives exactly Co the same answer as the accurate formula.

Find this temperature algebraically. 3 9.

cm02 ⋅cm52 ⋅

(a) The volume of the larger bottle,

Calculate the volume of the smalle Give your answer correct to the (b) The manufacturers wish to make a If it is to be similar to the other tw nearest millimetre. 10. Establish the equation of the line opposite

in terms of d and t. 11. Solve the following equation for x , whe 6 tan4+ x

Pegasys 2005

“ Les Egouts de Paris ” is a perfume which is sold in two sizes, as shown opposite. The bottles are mathematically similar in shape.

which has a base diameter of cm, is 100ml. 52 ⋅r bottle which has a base diameter of cm. 02 ⋅nearest millilitre (ml). 3

20ml sampler bottle. o bottles, find its base diameter to the

4

9

4 (20,28)

42

t

d

o

re 0 ≤ x ≤ 360o.

3 =o

KU RA 12. A cargo ship sets out from its home port P with a load of machinery, bound for

port H , as shown in the diagram.

N

78o

N

T •

P

H 90km

Because of bad weather, the ship sails due north from P until it reaches point T , a distance of 90km, and then it travels on a bearing of 078° t

destination port H.

If the bearing of P from H is 222°, calculate the distance from P your answer correct to the nearest kilometre. 13. Western Gas charge their customers for the amount of gas that they use

of a year as follows :

(i) 1.6p per unit for the first 600 units use (ii) 1.1p per unit for any further units used;

(iii) a service charge of £8.50 is then addedmany units are used.

(a) John uses 2800 units of gas in one quarter. Calculate his bill fo (b) Find a formula for C , the cost in pounds, of using u units of g

Write the formula in its simplest form. 14. (a) Factorise 1092 2 +− xx

(b) Hence simplify 4

1092

2

−+−

xxx2

Pegasys 2005

o its

to H , giving

4

every quarter

d; , no matter how

r this usage. 3

as, where u > 600. 3

1

2

KU RA

15. The value of π21 , where ......14153 ⋅=π , can be shown to have an approximate value

which is given by evaluating a series of fractions, as shown in the examples below :

i) π21 ....4221]4[ 5

434

32

12 ⋅=×××= ii) π2

1 ....7061]5[ 56

54

34

32

12 ⋅=××××=

iii) π2

1 ....4621]6[ 76

56

54

34

32

12 ⋅=×××××=

(a) Write down and evaluate the series of fractions for π2

1 ]8[ . 2 (b) Let n be any even number. Write down the last two fractions of the series π2

1 [ , in terms of n. 2 ]n

(c) This time let n be any odd number. Again, write down the last two fractions of the series π2

1 ][n , in terms of n. 2 16.

h

5cm

A lady's hair colourant comes in acardboard. Its net is shown above.

The package has a square base of

sloping edge of length 6 cm . 5⋅ Find the volume of this pyramid , [ Volume of a pyramid is given and h is the vertical height ]

Pegasys 2005

6.5cm

package which is a square-based pyramid made of

side 5cm and each of the four isosceles triangles has a

to the nearest cubic centimetre.

as hA31V = , where A is the base area

5

KU RA

17.

1 LAYER 2 LAYERS 3 LAYERS Clare is using headless matchsticks to make simple triangular patterns, as shown above. (a) How many matches will she need to construct a similar pattern which

has 4 LAYERS ? 1

(b) The number of matches, M , required to complete a pattern which has L layers is given by the equation

M bLLa += 2

Find algebraically the values of a and b . 5

[ END OF QUESTION PAPER ]

Pegasys 2005

Credit Mathematics - Practice Examination E

Please note … the format of this practice examination is the same as the current format. The paper timings are the same, as are the marks allocated. Calculators may only be used in Paper 2.


Time a Read Carefully

1. Answer as many questions as 2. Full credit will be given only wh3. You may not use a calculator

Pegasys 2005

Paper I

llowed - 55 minutes

you can. ere the solution contains appropriate working.

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin

Standard Deviation: 1

)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

KU RE

1. Evaluate 2 7524538 ÷⋅−⋅

2. (a) Evaluate ab where and 3−=a 9=b 2

(b) Evaluate where kjj 42 + 5−=j and 2−=k 2

3. Evaluate

−

21

32

7of6 2

4. (a) 2 If )2(of value thefind,25)( 2 −−= hxxxh

(b) Factorise fully 15 1 yy 32 −

(c) Hence, or otherwise, express 125

3152

2

−−

yyy in its simplest form. 2

5. (a) Calculate the value of y° in the isosceles triangle opposite. 2

112°

y°

(b) Find the value of y° in terms

of a° in the diagram opposite. 3

a°

y°

Pegasys 2005

KU RE

6. (a) Remove the brackets and simplify 2 )23)(5( vtvt +− (b) Solve algebraically the equation 3 04133 2 =+− xx

(c) Solve algebraically the equation

( ) 12

33

=−

−mm 3

7. The diagram shows a dog's kennel. The dimensions of the kennel, in feet, are shown. If its two ends are congruent trapeziums and all its other faces are rectangles, find the volume of this kennel. 4 8.

2ft 5ft

4ft 3.5ft

A

O

C

The diagram above shows a circle, centre O, and E is the

Find the size of angle OEA.

Pegasys 2005

E

24°B

quadrilateral ABCO. BA and BC are tangents to the point where OB meets the circle.

3

KU RE

9. The diagram across shows the cross-section of the petrol tank of a Zephyr car. The tank is being filled at a constant rate. Which of the following 4 graphs, A to D, best describes how the depth of the fuel is increasing as the tank is filling up with petrol? Give reasons for your answer. 4

Depth

Time

Depth Depth Depth

Time Time Time O O O O

C D B A

FUEL

10. N = 1 N = 2 N = 3 Fred is making square patterns from headless matches.

N stands for the number of matches along the side of each square pattern.

(a) How many matches would be required to make the pattern where N = 4 ? 2

(b) The formula for M, the number of matches needed to make a square of side N matches, is given by M = aN2 + bN Find the values of a and b. 4

[END OF QUESTION PAPER]

Pegasys 2005

Credit Mathematics Practice Exam E Marking Scheme - Paper 1 1. ans : 35 2 KU 1 know order of calculations • 2 carry out calculations • 2. (a) ans: -3 2 KU 1 substitute into expression • 2 evaluate • (b) ans: 65 2 KU substitute into expression 1• 2 multiply out •

3. ans: 71 2 KU

1 add fractions • multiply fractions 2• 4. (a) ans -18 2KU 1 interpret function notation • 2 evaluate function • (b) ans 3y(5y - 1) 1KU

(c) ans 15

3+yy 2KU

factorise denominator 1• 2 simplify • 5. (a) ans: 44° 2KU 1 finding the 2 equal triangle angles • 2 for calculating the 3• rd angle

(b) ans: 2a - 180° 3RE

1 for supplementary angle • 2 for y = 180° - 2 x (supp. angle) • for answer 3•

Illustration(s) for awarding each mark Give 1 mark for each

Pegasys 2005

12

1• )3(9 −÷2 3−

12

12

1•2

1

1•2

1•2

123

• 24.5 /7 =… • 35 (a) • (b) • (-5)2 + [ 4 x (-5) x (-2) ] • 65 • 1/6 • 1/7 or equivalent

(a) 5 x (-2) – [ 2 x (-2) x (-2) ]

• -18

(b) • 3y (5y - 1) (c) (5y - 1)(5y + 1) • 3y/(5y + 1) (a) 180° - 112° = 68° • 180° - 2 x 68° = 44° (b) • for (180 - a)° • for y = 180° - 2(180 - a)° • for 2a - 180

6. (a) ans: 3 2KU 22 1013 vvtt −−

1 for finding 3 • 22 10vandt − 2 for finding − • vt13

(b) ans: 314 == xandx 3KU

for finding 1 factor 1• • 2 for finding the 2nd factor • 3 for solving the equation (c) ans: m = 3 3KU • 1 add the fractions • 2 multiply expressions • 3 solve equation 7. ans: 37.5 ft3 4KU 1 for calculating rectangle area • 2 for calculating triangle area • for calculating prism area 3• • 4 for volume 8. ans: 57° 3KU 1 for finding angle at centre • 2 for finding other angle at centre as equal • to • , using tangent kite (s/i). 1

1• 22 10vandt −2 vt13

1

23

1234

123

Give 1 mark for each Illustration(s) for awarding each mark

for final answer. 3• 9. ans: Graph C 4RE 1 for non- constant initial rise on graph • 2 for the nature of this rise (i.e. a curve like • 'D' as opposed to 'A') for constantly rising finish to the fill-up. 3• • 4 for final conclusion. 10. (a) ans 40 matches 2RE

•1 for knowing how to continue the pattern (stated/implied/drawn) •2 for correct conclusion (b) ans a =2, b = 2. 4RE

•1 for using a diagram to set up 1 equation . •2 for using a 2nd diagram to set up another

(a) 3 • −

(b) • 3x - 1 • 2 x - 4 • 3 x = 4 and x = 1/3 (c) • 1 multiply by 6 or take common denominator

• -m + 9 • m = 3 • 2 x 3.5 =7 • 0.5 x 2 x 0.5 =0.5 • 7.5 • 5 x 7.5 = 37.5 • 66° • 66° • 57° For •1 to •3, see candidates responses •4 Graph C (a) •1 e.g. a sketch •2 40 matches (b) •1 e.g. a + b = 4 •2 e.g. 4a + 2b = 12 •3 e.g. b = 2 •4 e.g. a = 2 KU - 30 RE - 13 Total 43 marks

equation •3 for finding 1 of the variables •4 for finding the other variable

Pegasys 2005

Credit Mathematics - Practice Examination E

Please note … the format of this practice examination is the same as the current format. The paper timings are the same, as are the marks allocated. Calculators may be used in this paper.


Time allowed - 80 minutes


Paper II

Pegasys 2005

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin


)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

KU RE 1. Wallace the Bruce, the famous Scottish hero, went into

battle with a pyke which was 52 ⋅ metres in length.

Its steel blade was in the shape of a sector of a circle whose edges met at an angle of 135°, as shown in the diagram below If the edges of the blade were 38cm in length, what was the area of 1 side of the steel blade? (Answer to the nearest cm2 )

38cm

135°

38cm

3 2. At any given temperature, the time,T minutes, taken to cook a steak varies directly

with the weight, W kg, of the steak and also directly with the square of the distance, D cm, between the grill and the steak.

(a) Write down a formula which connects T with W and D. 1

(b) When the distance D is cm, it takes 7 minutes to cook a steak. 52 ⋅

How long would it have taken to cook the same steak, at the same temperature, if the distance had been 5 cm ? 2

3. John conducted a survey of his second year pupils to find out the most popular flavour of potato crisps. Here are his results:

Cheese and Onion 11

Smokey Bacon 8

Prawn Cocktail 6

Others 5

Draw an appropriate statistical diagram to illustrate the information that John received. 4 4. Solve algebraically the following inequality 5 ( )8548 −−≥− tt 3

Pegasys 2005

KU RE 5. (i) Take any 3 consecutive numbers - e.g. 3, 4, 5 . (ii) Square the middle one - e.g. 42 = 16 (iii) Multiply the 2 remaining "outside" numbers e.g. 3 x 5 = 15

Note that the difference between (ii) and (iii), (i.e. 16 - 15), is 1.

(a) Use the same routine as above with 7, 8, 9 and show clearly again that the difference between (ii) and (iii) is 1. 1 (b) Let the first (i.e. smallest ) number be n.Write down the next 2 numbers in terms of n. 2 (c) Hence, or otherwise, show that the square of the middle number of 3 consecutive numbers will always be 1 more than the product of the other 2 numbers. 3 6. A toll barrier has been placed across a river to control the flow of shipping.

31m

36°

F

DEAD TREE

28m

Unfortunately, a dead tree has drifted downstream and lodged itself on the river-bed at a position 28 metres directly up-river from the fulcrum, F, of the barrier and at an angle of 36° to the river-bank, as shown in the diagram above. The diagram is not to scale. The lock-keeper in charge reckons that, if the barrier were to turn clockwise from F, then it would not strike the dead tree ( so the tree presents no immediate danger to the barrier).

Is he correct? (Show calculations to justify your answer.) 4 7. Solve the following trigonometric equation: 3 3600,12sin ≤≤=+° xx 3

Pegasys 2005

KU RE 8. A charter aeroplane, when full, can carry 96 passengers. Some of these passengers will be travelling 1st class while others will be travelling 2nd class. Let F be the number of 1st class passengers and S be the number of 2nd class passengers. (a) Given that the plane is full, use the information above to write down a simple equation involving F and S. 1 Each 1st class passenger is allowed to have 65kg of luggage but a 2nd class passenger is allowed only 35kg. The total weight of luggage allowed on board is 4140kg (b) Assuming that each passenger has taken their maximum amount of luggage, write down another equation involving F and S. 2 (c) Find the number of 1st and the number of 2nd class passengers. 3 9. In the diagram AB is parallel to DC.

If AT=120cm, DT=143cm and CT=156cm, calculate the length of DB. 3

D

156cm 143cm

120cm

T

B A

C 10. A coal mine has 2 main entrances to W, its underground workings. The entrance at O is used for the extracted ore and as the miners entrance. The other at S is used for supplies and machinery. The distance, OS, between the two entrances is 62m.

The entrances O and S are at thesame horizontal level.

V 62m

57 °

W

28° O S

The angle of depression of W from O is 28° and the angle of depression of W from S is 57°. The mine owner wants to build a vertical shaft down to the workings from a point V, which is somewhere along the line OS. Calculate the length of this vertical shaft, giving your answer to the nearest whole number of metres. 5

Pegasys 2005

KU RE

11. The national soft drink of Spain is called " Elaborado del Hierro" and it is sold in two main bottle sizes.


1350ml

400 ml

5cm 7.5cm The smaller bottle has a base diameter of 5cm and holds 400ml.

The larger bottle has a base diameter of 7.5cm and it holds 1350ml.

The bottles look alike but could they actually be mathematically similar? ( Show calculations to justify your answer.) 4 12. The blood pressure of 9 young women was measured in mm. The data gave the following summary totals: and ∑ = 1156x 1489802 =∑ x (a) Calculate the sample mean and the standard deviation, giving your answer to 1 decimal place. 3

(b) A group of older women had a mean blood pressure of 158 0⋅ mm and a standard deviation of mm. How does the blood pressure of these older women compare 68 ⋅ with that of the younger women? 2 13. Solve the equation .072 2 =−− xx Give your answers correct to 1 decimal place. 4 14. A searchlight is shining its beam at a building.

12m20°

0.5m

At present, the angle of elevation of the beam is 20° and the beam originates from a point 0 ⋅ metres above the ground. 5

A security guard thinks that doubling the angle of elevation of the beam would exactly double the height that the beam presently reaches up the building. Show that the guard is in fact wrong. (No marks for a scale diagram ) 4

Pegasys 2005


Credit Mathematics - Practice Examination F

Paper I



Read Carefully 1. Answer as many questions as you can. 2. Full credit will be given only where the solution contains appropriate working. 3. You may not use a calculator

Pegasys 2005

KU RE RE 1. Evaluate 2 1. Evaluate 2 ( 221016 ÷−( 221016 ÷− )) 2. Evaluate 2. Evaluate 4

332 24 − 2

3. Light travels at 3 x 105 kilometres per second. How many kilometres will it travel in 1 hour ? Give your answer in scientific notation. 3 4. If (f , find the value of 243) xxx −= ).3(−f 2 5. A number pattern, involving the difference between a given number and its cube, is shown below:

243331322202111

3

3

3

××=−

××=−

××=−

(a) Write down a similar expression for 6 63 − . 1

(b) Hence, write down an expression for nn3 − . 2 (c) By examining your answer to part (b), show that 10 103 − can also be expressed as 10 103 − ( ) 910102 ×+= . 2 6. (a) Factorise completely 3 1 .122 −x

(b) Hence, simplify the fraction 6

1232

2

−+−xx

x . 2

7. A function (f is defined as )x ( ) ( )( )523 +−= xxxf . (a) Remove the brackets and express ( )xf in standard form i.e. 2 .2 cbxax ++ (b) Hence, solve the quadratic equation ( )xf = 20. 4 Pegasys 2005

KU RE 8.

-320 o

100

200

d

The straight line in the diagram above passes through 200 and 100 on the t and d axes respectively.

t

(a) Find the equation of the line in terms of t and d . 4 (b) If the line were continued, would it pass through the point P(-320, 250)? Give reasons for your answer. 2 9. A survey was carried out amongst 200 4th year pupils in order to find out which flavour of crisps they preferred. The results of the survey are shown in the table below :

BOYGIR

(a) (b) (c) 10. Market they usu Cereal Numbe consum Choose

Pegasys 20

Salt & Vinegar

Prawn Cocktail

Cheese & Onion

Smokey Bacon Others TOTAL

S 25 22 30 18 15 110 LS 22 28 20 12 8 90

What is the probability that any pupil chosen at random prefers Cheese & Onion? 1

A boy is chosen at random.What is the probability that he prefers Prawn Cocktail? 2

What is the probability that any pupil chosen at random does not prefer Salt & Vinegar? 2

research was carried out amongst 60 consumers to find out what type of cereal ally ate for breakfast. The outcome of the research is shown below.

Type Flakes Puffs Muesli Porridge Others/none

r of 21 15 4 11 9 ers

an appropriate statistical diagram and display these results. 4

End of question paper

05

Credit Mathematics Practice Exam F Marking Scheme - Paper 1 1. ans : 121 2 KU know order of calculations 1• carry out calculations 2•

2. ans: 1223 2 KU

1 changing to improper fractions and • choosing suitable denominator 2 evaluate • 3. ans: 1.08 x 109 3 KU 1 change units • multiply 2• leave in scientific notation 3• 4. ans -45 2KU 1 interpret function notation • 2 evaluate function • 5. (a) ans: 6x7x5 1RE 1 interpret •

(b) ans: n (n+1) (n-1) 2RE 1 for interpreting with n •

2• for n+1 and n-1

(c) ans: proof 2 RE •1 for interpreting n = 10

•2 for opening the brackets, etc 6. See next page.

1233

1256

411

314

−=−

212111

1223 or=

123

12

1•

12

1•2

1• (16 -5)2 =… 2• 121

1• =

•

• 1 hour = 3600secs • 3 x 105 x 3.6 x 103 • 1.08 x 109 • 3 x (-3) - 4 x (-3)2 • -45 (a) 6x7x5

(b) • n • n+1 and n-1 (c) 103 -10 = 10 x (10 + 1)x (10 - 1) • = (102 + 10 ) x 9


Pegasys 2005

Pegasys 2005

6. (a) ans: 3(x - 2)(x + 2) 1 KU • = 3(x - 2)(x + 2)

• (x+3)(x -2) • answer. • 3x2 … -10 or…+ 13x • 3x2 + 13x - 10 • 3x2 + 13x - 30 = 0 • (3x-5)(x+6) •3 x = -6 •4 x = 5/3 •1 m = (100-0) / (0-200) •2 …. = - ½ or equiv. •3 c = 100 •4 d = -1/2t + 100

•1 ? 250 = - 1/2x (-320) + 100 •2 250≠ 160 + 100, so point not on line. (a) •1 50/200 (b) • 110 as denominator • 22 as numerator (c) 153 as denominator • 200 as numerator Take account of e.g. proper scales, correct heightof bars, proper spacing etc

• factorise fully 1

3x2)3(x

++

1••

••

1•

•2•

•2•

1

12

12

12

12

1•2


(b) ans; 2KU

factorise denominator 2 simplify and stop. 7. (a) ans: f(x) =3x2 + 13x - 10 2 KU 1 for partial answer 2 for fully correct answer (b) ans: x = 5/3 or -6 4 KU •1 for standard form •2 for factorising •3 for one correct answer •4 for 2nd correct answer 8. (a) ans: d = -1/2t + 100 4 KU •1 for starting to find m •2 for calculating m •3 for finding c •4 for equation with d and t (b) ans: No - point does not satisfy 2 RE

equation •1 for substituting co-ordinates •2 for consistent conclusion 9. (a) ans: 50/200 ( = 1/4) 1 KU for choosing the 2 correct numbers (b) ans: 22/110 ( = 1/5) 2 KU 1 for knowing to select from boys for completing answer (c) ans: 153/200 or equiv. 2 KU 1 for correct number not prefering S/V for completing answer 10. Barchart, Histogram or Pie-chart 4 RE

Total : KU 27 RE 11

Credit Mathematics - Practice Examination F



Paper II


Read Carefully

1. Answer as many questions as you can. 2. Full credit will be given only where the solution contains appropriate working. 3. You may use a calculator

Pegasys 2005

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin


)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

KU RE 1. Solve the equation ( ) 43315 −=− xx . 3 2. The logo of Cheeses-R-Us, a cheese superstore, is shown across. The logo is a sector of a circle of radius of 86 cm. The larger angle at the centre is 250°. The manager decides to improve the sign by applying gold tape around the full

86cm

3. During the Fre

cut off the heawith dimension

To ensure a qthat the obtu

Does Pierre's Your answer

Pegasys 2005

250°
perimeter of the logo.
What length of tape will be required? 4

nch Revolution , Jacques LeMon and Co. made guillotines with which to ds of the aristocracy. His apprentice, Pierre Le Punk, brought Jacques a guillotine blade s as shown in the figure below.

48cm

20cm

33cm

uick and relatively clean decapitation, Jacques always made sure se angle of the blade was more than 110o

.

guillotine fulfil this condition? 5 must include appropriate working.

4. Carbon dioxide is the gas put into "fizzy" drinks to give the drink its"fizz". The makers of Steel-Bru, a soft drink, estimate that, by the end of a week, a bottle of their drink will have lost 2% of whatever volume of carbon dioxide that the bottle had contained at the beginning of that week. If the bottle loses more than 15% of its carbon dioxide, the contents have to be "re-carbonated" i.e. more carbon dioxide must be put into the drink. If 250 cm3 of carbon dioxide is put into a bottle at the beginning of a week, how many weeks can the bottle lie unused without needing to be re-carbonated? 4 (show your working) 5.

A woman is having a shower but the sprinkler is a bit too close to her head. To give herself more headroom, she would like to increase the vertical height of the sprinkler by 9cm. This can be achieved by rotating the shower attachment AB by x° in an anti-clockwise direction so that B is now at position C.

If the shower attachment is 22cm long and is, at present, at an angle of 30° to the horizontal, as shown in Figure 2 above, find the angle x° required to produce this 9cm increase in height. 5

6. (a) Express 1

33+

−xx

as a single fraction in its simplest form ( )1,0 −≠≠ xx . 3

(b) Change the subject of the formula to in vc

uvb −= 2

A

22cm

22cm

30°

x°

B

C

9cm

RE KU

A 30°

22cm B

Figure 2

Pegasys 2005

KU RE RE 7. A girl is using headless matches to construct hexagonal patterns, as shown in the 7. A girl is using headless matches to construct hexagonal patterns, as shown in the three diagrams below. three diagrams below. Pattern 1 (P = 1) P =2 P =3 Pattern 1 (P = 1) P =2 P =3 Matches = 6 (m = 6) m=16 m =30 Matches = 6 (m = 6) m=16 m =30 (a) How many matches will be needed for P = 4, i.e. Pattern 4? 2 (a) How many matches will be needed for P = 4, i.e. Pattern 4? 2 (b) The number of matches, m , required to construct Pattern P is given by (b) The number of matches, m , required to construct Pattern P is given by the formula the formula m = aP2 + bP m = aP2 + bP

Find algebraically the values of a and b. 4 Find algebraically the values of a and b. 4

8. After finishing a glass of lemonade, the drinker discovers that the circular area 8. After finishing a glass of lemonade, the drinker discovers that the circular area of the base of his glass is half of the area of the circular beer-mat on which of the base of his glass is half of the area of the circular beer-mat on which the glass is resting. the glass is resting.

6cm

d

If the diameter of the base of the glass is 6cm, what is d, the diameter of the beer-mat? If the diameter of the base of the glass is 6cm, what is d, the diameter of the beer-mat? ( Give answer in centimetres, to 1 decimal place.) 4 ( Give answer in centimetres, to 1 decimal place.) 4 9. Solve the equation . 9. Solve the equation . 0925 2 =−− xx 0925 2 =−− xx Give your answers correct to 2 significant figures. 4 Give your answers correct to 2 significant figures. 4

Pegasys 2005

KU RE RE 10. "HAPPY-COLA" have decided to issue a “limited edition” cone-shaped can to 10. "HAPPY-COLA" have decided to issue a “limited edition” cone-shaped can to celebrate their 50th anniversary. celebrate their 50th anniversary. Their normal can is a cylinder whose height is 11 Their normal can is a cylinder whose height is 11 55⋅ cm and whose diameter is 6 cm. 5⋅

511 ⋅ cm511 ⋅

cm 56 ⋅

If the height of the cone is to be the same as the height of the cylinder, i.e.11 cm, 5⋅ and the volume of the cone is to be the same as the volume of the cylinder, calculate the diameter of the cone? [ Volume of a cone : hr 2

31 π=V ]

(Answer in centimetres giving your answer correct to 1 decimal place) 4 11. The time, T minutes, taken to mow a square lawn varies directly as the square of its length s metres and inversely as the breadth b cm of the blade in the lawnmower. A lawnmower whose blade is 30 cm in breadth takes 18 minutes to mow a square of length 6 metres. (a) Find a formula connecting T , s and b. 3 (b) The gardener has just mowed a square lawn in 20 minutes. Another lawn has a side which is twice as long and the gardener wants to mow this lawn in the same time. By what number would the blade's length in the lawnmower have to be multiplied in order to achieve this? 2

12. The heights, in centimetres, of 30 pupils at the end of their 3rd year at secondary school were measured and the following data was obtained: and ∑ = 4545x ∑ = 6984502x (a) Calculate the mean and standard deviation. 3 (b) The previous year, a similar survey had been done. The mean had been 152 cm and the standard deviation had been 13.5cm. How do these figures compare with the present figures? 2

Pegasys 2005

KU RE RE 13. In a bearing diagram, the bearings of a ship from 2 different ports are often given. To 13. In a bearing diagram, the bearings of a ship from 2 different ports are often given. To make other useful calculations, it is often necessary to find the angle between the lines make other useful calculations, it is often necessary to find the angle between the lines

of these 2 bearings. In the diagram below, a ship A lies on a bearing of 102° from a of these 2 bearings. In the diagram below, a ship A lies on a bearing of 102° from a port P and on a bearing of 235° from port Q. port P and on a bearing of 235° from port Q.

N

102°

A

P N

N

Q 235°

S R

(a) In the diagram above, show that angle PAQ = 133° 2 (a) In the diagram above, show that angle PAQ = 133° 2

A

a° P N

R

N N Q

b° S

(b) Show that , when A has a bearing of a° from P and b° from Q, as in the (b) Show that , when A has a bearing of a° from P and b° from Q, as in the diagram above, then the angle between the bearings, angle PAQ, is always diagram above, then the angle between the bearings, angle PAQ, is always

equal to b°− a°. 3 equal to b°− a°. 3

End of question paper End of question paper

Pegasys 2005


Credit Mathematics - Practice Examination G

Paper I



Read Carefully 1. Answer as many questions as you can. 2. Full credit will be given only where the solution contains appropriate working. 3. You may not use a calculator

Pegasys 2005

KU RE RE 1. Evaluate 1. Evaluate 505424 ⋅÷⋅ 505424 ⋅÷⋅+⋅ . 2 2. Evaluate 4

332 12 ÷ . 2

3. Solve the inequality 4 2( )3 4 1 3− − ≥ −x x , where x is a whole number. 4

4. Evaluate a b

c

2 2− c when 2,4,6 −==−= cba . 2

5. (a) Factorise 2 5 122x x+ − . 2

(b) Hence, simplify the fraction x

x x

2

2

162 5 1

−+ − 2

. 2

12cm

20cm

18cm

6.

-0⋅5

0⋅5 The diagram shows the graph of , y a bx= cos 0 0 360≤ ≤x . Find the values of a and b. 2 P Q 7. Triangles PQR and RST, with some of their measurements, are shown in the diagram opposite. R PQ is parallel to TS. 15cm Calculate the length of TQ. 4 T S

Pegasys 2005

KU RE 8. (a) Find the equation of the line in terms of p and q . 4

-40 o

18

24

q The straight line in the diagram shown passes through 24 and 18 on the p and q axes respectively.

p

(b) Does the point A(4,12) lie above or below the line? Give a reason for your answer. 3 9. Two adults and three children pay £17.40 for admission to their local school concert. One adult and two children pay £10.20 for admission to the same concert. How much would 3 adults and 1 child have to pay to be admitted to the concert ? 5 10. Two functions are defined as follows :

f x x x( ) = + −2 2 6 g x x( ) = +7 8

Find the value(s) of x for which ( ) 0)()(3 =+ xgxf . 4 11. Megan conducted a survey. She asked her school friends how they travelled to their Summer holiday. Here are their answers: Coach 12 Car 8 Aeroplane 7 Boat 3 Draw an appropriate statistical diagram to illustrate this information 4

End of Question Paper

Pegasys 2005

Credit Mathematics Practice Exam G Marking Scheme - Paper 1

Give 1 mark for each • 1. ans : 13.2 2 KU know order of calculations 1• carry out calculations 2•

2. ans: 3221

11121

or 2 KU

1 changing to improper fractions and • changing to multiplication 2 evaluate • 3. ans: 0, 1, 2, 3 4 KU 1 removing brackets • • collecting like terms 2

solving inequation 3• stating solution • 4

4. ans -26 2KU 1 correctly substituting • 2 evaluate expression • 5. (a) ans: ( )(2 3 4x x− + 2KU ) 1&2 factorising correctly •

(b) ans: 32

4−

−x

x 2KU 1 factorising numerator •

2• simplifying fraction

6. ans: a = 0.5, b = 2 2KU

•1 recognizing max/min • 2 recognizing period

Illustration(s) for awarding each mark

• 1 4 5 0 5 9⋅ ÷ ⋅ = • 2 answer

• 1 83

47

×

• 2 answer

• 1 4 6 8 1 3− + ≥ −x x • 2 − ≥ −3 11x

• 3 x ≤113

• 4 answer

• 1 36 16

2+

−

• 2 answer

• 1 ( )2 3x − • 2 ( )x + 4

• 1 ( )( )x x+ −4 4 • 2 answer

• 1 a = 0.5 • 2 b = 2

Pegasys 2005

Give 1 mark for each • Illustration(s) for awarding each mark 7. ans: 24cm 4KU

• 1 recognising similar triangles • 2 calculating scale factor • 3 calculating RQ • 4 calculating TQ

8. (a) ans: q p= − +34

18 4 KU

• 1 knowing how to calculate gradient • 2 correctly calculating gradient • 3 finding y intercept • 4 stating equation in terms of p and q

(b) ans: below the line. 3 RE •1 substituting p = 4 into line equation •2 comparing y coordinates •3 conclusion

9. ans: £15.60 5 RE

• 1 creating two equations • 2 knowing to solve system of equations • 3 evaluating one variable • 4 evaluating second variable • 5 calculating cost

10. ans: x = − 4 RE 523

,

•1 substituting correctly • 2 creating standard quadratic equation • 3 factorising • 4 solving equation

11. Barchart, Histogram or Pie-chart 4 RE

• 1 PQTS

PRRS

QRRT

= =

• 2 S.F. = 35

• 3 RQ = 35

15 9=×

• 4 TQ = 9 + 15 = answer

• 1 mverthoriz

= or equivalent

• 2 m = −34

• 3 c = 18 • 4 answer

• 1 ( )q = − + =34

4 18 15

• 2 12 < 15 • 3 conclusion

• 1 2A + 3C = 17.40 A + 2C = 10.20 • 2 solving simultaneously • 3 A = 4.20 • 4 C = 3.00 • 5 3(£4.20) + £3.00 = answer • 1 ( )3 2 6 7 82x x x+ − + + =

2

0 • 2 3 13 10 0x x+ − = • 3 ( )( )3 2 5x x 0− + = • 4 answer

Take account of e.g. proper scales, correct height of bars, proper spacing etc

Pegasys 2005

Total : KU 24 RE 16

Credit Mathematics - Practice Examination G


Paper II




Pegasys 2005

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin


)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

KU RE RE 1. The circumference of the earth is approximately 41. The circumference of the earth is approximately 4 01 10401 104⋅ × km. Calculate, correct to three significant figures, the radius of the earth, expressing your answer in standard form. 3 2. House prices are predicted to rise approximately 2 5%⋅ per year, for the next few years. A cottage bought in January 2002 cost £87 000. How much, to the nearest £, would the same cottage be worth in January 2005 ? 3 3. Solve the equation 2 4 32 0x x+ − = .

Give your answers correct to 1 decimal place. 4

4. Primary 7 are making medieval hats as part

of their History project. A few of the girls decide to make a hat like

the one shown , which consists of a cone shaped body made of card with a ribbon attatched to the top.

40 cm

250o

The sector of card used to make the cone is cut from a square piece of card of side 40 cm as shown.

If the angle at the centre of the sector is 250o, calculate the percentage of card wasted, to the nearest percent. 5

Pegasys 2005

KU RE 5. Solve, algebraically, the equation 3

5 3 0 0 3600cos , .x + = ≤ < for x

6. In the triangular shaped swimming pool shown below a swimmer dives in at A and swims directly to the opposite side BC. Angle ABC = 37o and angle BCA = 66o. The length of BC is 34⋅1 metres.

Calculate, correct to three significant figures, the shortest possible distance the swimmer has to cover. 5

66o

37o

34⋅1 m

C

A

B

7. (a) Express yyxy

x 6)2(

6−

− as a single fraction in its simplest form . 3

(b) Change the subject of the formula to H in V d 2 H= 2

8. The volume of a square based pyramid, of base side e and height h, as shown, is given by the formula he2

31=V .

The base length is doubled and the height is halved. What happens to the volume of the cone ? 3

e

h

Pegasys 2005

KU RE 9. A health survey is carried out between a group of 200 males and 200 females. The number of smokers of different ages is recorded. The tables below show the results.

Females Males

Age No. of smokers Age No. of smokers

14 6 14 5 15 10 15 8 16 19 16 11 17 17 17 20 18 14 18 26 19 4 19 11

(a) Construct separate cumulative frequency tables for both females and males. 2

(b) On the same set of axes draw a cumulative frequency diagram for both groups. 3 (c) Use the cumulative frequency diagram to compare smoking between the groups of males and females. 3 10. A toy for toddlers is designed in such a way that it never falls over. The base is a hemisphere and the top is a cone, with some added decorative enhancements as shown in the diagram. To prevent the toy from falling it must have the base completely filled and 24% of the upper body filled with sand. Calculate the amount of sand needed, if the radius of the hemisphere is 8 cm and the height of the cone is 12 cm. Give your answer correct to 3 significant figures. Volume of a cone = hr 2

31π

Volume of a sphere = 334 rπ 4

Pegasys 2005

KU RE 11. KENNOLAUT, the dog food specialists, have recommended that dogs should drink at least three times the volume of food they consume. To promote this the company have designed a new feeding dish with two sections, as shown in the diagrams below. side and above view top view The larger section is for water and the smaller section is for food.

The dish is cylindrical in shape with a radius of 15 cm. The dividing strip is 26 cm long.

If filled to capacity will the dish satisfy the company’s recommendation? 6 12. A company sells boxed chocolates in two different sizes. The boxes are mathematically similar truncated cones, as shown in the diagram below.

14 cm 10 cm

The cost of the chocolates should be in direct proportion to their weight. The chocolates in the larger box have been weighed and are priced at £5 . ⋅45 The company is considering pricing the smaller box at £2 ⋅25 . Is this a fair price ? Your answer must be accompanied with appropriate working. 3

End of Question Paper

Pegasys 2005

Credit Mathematics - Practice Examination H Please note … the format of this practice examination is the

same as the current format. The paper timings are the same, as are the marks allocated. Calculators may only be used in Paper 2.

MATHEMATICS

Standard Grade - Credit Level Time a Read Carefully

1. Answer as many questions as 2. Full credit will be given only wh3. You may not use a calculator

Pegasys 2005

Paper I

llowed - 55 minutes

you can. ere the solution contains appropriate working.

FORMULAE LIST

The roots of ( )

aacbb

xcbxax2

4are0

22 −±−

==++

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin


)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

1. Evaluate 40428532 ÷⋅−⋅ . 2. Evaluate: 5

3 of ( )75

311 − .

3. The function is given by the formula , where x is a real number. )(xf 52)( 2 −= xxf (a) Find the value of ).3(−f (b) Find the values of a for which 45)( =af .

4. Solve the equation 53

42

13=

+−

+ xx , where x is a real number.

5. The graph below shows the relationship between the number of hours (h) a swimmer trains per week and the number of Championship medals (m) they have won. A best fitting straight line AB has been drawn. Swimmer A does not train but has won 3 medals this year. Swimmer B who trains for 14 hours per week has won 31 medals this year. (a) Find the equation of the straight line AB in terms of m and h.

KU RE

2

1

4

3

3

2

2

m

h Number of hours training per week

(h)

•

•

A

B•

•••

•

•••

• •

•

••

Number of

medals (m)

(b) How many medals would you expect a swimmer who trains 10 hours per week to have won ?

Pegasys 2005

6. Uranium is a radioactive isotope which has a half-life of years. This means 91054 ×⋅ that only half of the original mass will be radioactive after years. 91054 ×⋅ How long will it take for the radioactivity of a piece of Uranium to reduce to one eighth of its original level? Give your answer in scientific notation.

7. The Scottish Tourist Group carried out a survey amongst 500 adults from Great Britain to find out what would influence them most when choosing a holiday. The results of the survey are shown in the table below.

Age Cost Weather Amenities Scenery 30 and under 180 75 28 5

Over 30 90 35 12 75

(a) What is the probability that any adult chosen at random would have scenery as their main priority when choosing a holiday ?

(b) A 40 year old adult is chosen at random. What is the probability that the weather is his/her main concern when choosing a holiday ? (c) What is the probability that any adult chosen at random will not have cost as their main concern when choosing a holiday ? 8. The area of the triangle shown is 30 cm2. Show that 6

5=Bsin .

KU RE

3

1

2

4

2

A

C

8cm

B 9cm

Pegasys 2005

9. The ground floor vestibule area in a large office block is to be tiled with a mixture of two types of ceramic tile. The contractors left two samples, with their cost per square metre, as shown in the diagrams below. (a) Using Diagram 1 write down an equation in g and w, where g is the cost of a grey tile and w is the cost of a white tile. (b) Using Diagram 2 write down a second equation in g and w. Unfortunately the manager did not like any of the samples left and decided to use one of his own. His choice is shown in the diagram below. (c) How much per square metre would this design cost ?

KU RE

1

1

4

Diagram 1 Diagram 2

Cost: £22⋅70 Cost: £23⋅90

Pegasys 2005

10. Sandy found a small photo-frame and decided to put one of her favourite photographs in it. The diagram below shows the dimensions of the frame. Unfortunately the glass in the centre of the frame was cracked and had to be replaced. (a) Show that the area of glass needed for the centre of the frame can be given by the formula ( ) 22 cm108424 +−= xxA (b) If the area of glass needed was 54cm2, find a possible value for x.

KU RE

3

4

x cm

x cm

12cm

9cm

The width of the wooden surround is x cm.


Pegasys 2005

Credit Mathematics Practice Exam H Marking Scheme - Paper 1 1. ans : 25⋅49 2 KU

1 know order of calculations • 2 carry out calculations •

2. ans: 3513 2 KU

1 subtract fractions •

• 2 multiply fractions

3. (a) ans: 13 2 KU

1 interpret function notation •• 2 evaluate function

(b) ans: -5, 5 3 KU

• 1 substitute correctly • 2 attempts to solve equation • 3 correctly solves equation

4. ans: 3 KU 5=x 1 subtract fractions • 2 multiply expressions • 3 solve linear equation • 5. (a) ans: 4 RE 32 += hm

• 1 interpreting information • 2 calculating gradient • 3 identifying y – intercept • 4 correctly stating equation (b) ans: 23 medals 1 KU

• 1 sustituting into equation of line


Pegasys 2005

••

•2113

75

311 =−

•3513

2113

53

=×

• ( ) 532 2 −−•

• 4552 2 =−a• 25=a• 5±=a

• ( ) ( ) 53

422

133=

+−

+ xx

• 308239 =−−+ xx• 5=x

•

• 2014331grad =

−−

=

••

•

1 28⋅04/4=……. 2 25⋅49

1

2

1 2 13

1 2 3

1

2 3

1 Points (0, 3) and (14, 31)

2

3 c = 3 4 answer 1 m = 2(10) + 3 = 23

6. ans: 3 KU 1010351 ×⋅ 1 knowing to multiply by 3 •

• 2 correctly multiplying • 3 leaving answer in scientific notation

7. (a) ans:

254

50080 1 KU

(b) ans: 21235 2 KU

1 knowing to select from ‘Over 30’ • 2 completing answer •

(c) ans:

=

5023

500230 2 KU

1 calculating number not concerned • 2 completing answer • 8. ans: Proof 4 RE

• 1 knowing to use correct formula • 2 substituting correctly • 3 knowing to makesin the subject B• 4 completing the proof

9. (a) ans: 9 702216 ⋅=+ wg 1 KU 1 stating equation • (b) ans: 90231213 ⋅=+ wg 1 KU 1 stating equation • (c) ans: £23⋅30 4 RE 1 knowing to solve equations • simultaneously 2 evaluating one variable • 3 evaluating the second variable • 4 calculating cost •

•

•

•

••

••

BacsinArea 21=

• Bsin9830 21 ×××=

•65sin =B

•

•

••••

1 3 x 4⋅5 x 109 • 2 13⋅5 x 109 3 answer

1 answer

1 212 as denominator 2 35 as numerator

1 500 as denominator 2 230 as numerator

• 1 2

3 & 4

1 answer 1 answer 1 solving simultaneously 2 g = 1⋅10 3 w = 0⋅80 4 11(1⋅10) + 14(0⋅180)=23⋅30


Pegasys 2005

• x212 −• x29 −• ( )( )xxA 29212 −−=•

• 054424 2 =+− xx• ( )( ) 03292 =−− xx• cm51⋅=x

1 2 3 4 answer 1 2 3

10. (a) ans: Proof 4 RE

• 1 finding an expression for length • 2 finding an expression for breadth • 3 calculating area • 4 completing the proof

(b) ans: 1⋅5 cm 3 KU 1 equating expression to 54 • 2 attempting to solve the quadratic equation • 3 correctly solving equation •


KU - 26 RE - 16 Total 42 marks

Pegasys 2005

Credit Mathematics - Practice Examination H Please note … the format of this practice examination is the

same as the current format. The paper timings are the same, as are the marks allocated. Calculators may be used in this paper.

MATHEMATICS

Standard Grade - Credit Level Paper II

Time allowed - 80 minutes Read Carefully

1. Answer as many questions as you can. 2. Full credit will be given only where the solution contains appropriate working. 3. You may use a calculator

Pegasys 2005

FORMULAE LIST

The roots of ( )

ax bx c xb b a

a2

2

04

2+ + = =

− ± −are

c

Sine rule: aA

bB

cCsin sin sin

= =

Cosine rule: bc

acbAorAbccba2

coscos2222

222 −+=−+=


2 ab Csin


)(

1)(

222

−

∑−=

−

−= ∑∑

nnxx

nxx

s

Pegasys 2005

1. The speed of light is approximately 8 times faster than the speed of sound in air. 510× If the speed of sound in air is 372 metres per second, calculate the speed of light. Give your answer in scientific notation correct to 3 significant figures. 2. A farm was put on the market in January 2002. The land is extremely fertile and prime for farming so its value has appreciated since then by 4⋅2% per year. Unfortunately the farmhouse and outbuildings were in a state of disrepair and have depreciated by 3⋅5% per year. The value of the land was £360 000 and the value of the farmhouse along with the outbuildings was £135 000 in January 2002. What would be the expected value of the complete farm in January 2004 ? 3. A cat is trapped in a tree and a ladder is placed against the tree in an attempt to rescue it. The ladder rests against the tree making an angle of 60o with the horizontal and reaching 13 metres up the tree, allowing the rescuer to reach the cat. Unfortunately just a it jumps to a branch place. Calculate the size o with the horizontal 4. Solve the equation

Give your answer c

KU RE

5

2

60o

Pegasys 2005

13m

s the cat is about to be rescued 1 metre above its original resting

f the angle, to the nearest degree, that the ladder now has to make to allow the rescuer to reach the cat.

. 0742 =−+ xxorrect to 2 significant figures. 4

5

1m

5. A group of fifth year students from Scotia High School were asked how many hours studying they did in the week prior to their exams. The results are shown below. 13 8 10 11 18 9 15 (a) Use an appropriate formula to calculate the mean and standard deviation of these times. (b) A similar group of students from Scotia Academy were asked the same question

The mean number of hours studied was 14 and the standard deviation was 2⋅8. How did the number of hours studied by students from Scotia High School compare with the number of hours studied by students from Scotia Academy ?

6. Rainwater is collected in a rectangular based tank on top of a flat roof and is drained periodically to a cylindrical tank on the ground where it is used for watering plants in dry weather. The tank on the roof measures 4 metres by 8 metres and has a depth of 0⋅2 metres. The tank on the ground is 1⋅75 metres high and has base radius of 0⋅45 metres. Both tanks were empty, but after a heavy shower all the rainwater from the roof tank was drained to the ground tank and completely filled it. Calculate the depth of rainwater, to the nearest millimetre, in the roof tank immediately before it was drained to the ground tank. 7. The median of five consecutive even integers is 22 +p . (a) Write down, in terms of p , expressions for the five integers. (b) Show that the mean can be expressed as ( )12 +p .

KU RE

2

2

5

2

3

Pegasys 2005

8. A large boardroom table is in the shape of a rectangle with a circle segment at both ends, as shown in the diagram below. The rectangle at the centre measures 6 metres by 3⋅5 metres. AC and BC are radii of the circle and angle ACB is 120o.

(a) Show that AC, the radius of the segment, is 2⋅0 (b) To sit comfortably at this table it is estimated th person requires 80 cm of table edge. How many people can sit comfortably at the ta 9. The two boxes below are mathematically similar and decorative paper. If it requires 2⋅08 m2 of paper to cover the large box, c needed to cover the smaller box.

KU RE

C

A

120o

B 6m

40cm

Pegasys 2005

3⋅5m

2 m correct to 3 significant figures.

at an average

ble described above?

both have to be wrapped with

alculate the amount of paper 3

4

3

30cm


10. Two boats leave port together. Boat A sails on a course of 052o at 11 miles per hour. Boat B sails on a bearing of 108o at 14 miles per hour. After 45 minutes Boat A receives a distress call from Boat B requesting their help as soon as possible. How far, to the nearest mile, would Boat A have to travel to reach Boat B? 11. The graph shown has equation . 1032 −+= xxy (a) Find the coordinates of A, the point where the curve cuts the y – axis. (b) Find the coordinates of B and C, the points where the curve cuts the x – axis. (c) Find the coordinates of the minimum turning point.

REKU

North

Port

B

A

4

A

0B C x

y

1

3

2

.

12. Solve the equation 3600for,03tan5 ≤≤=+ xxo

4

Pegasys 2005

Credit Mathematics - Practice Exam A Marking Scheme 1. For − ≤24 2x .......... (2)

x ≥ − 12 (or equiv.) .......... (1) [ 3 marks KU ]

2. For using 115360

.......... (1)

For A sq. cm .......... (1) larger sector = 9052 6⋅⋅ For sq. cm .......... (1) Asmall sector = 401 2

For shaded area = 9052 6 401 2 8651 4⋅ − ⋅ = ⋅ sq. cm .......... (1) ( N.B. i) Ignore rounding errors ii) For consistent use of C r= 2π , instead of A r marks ) [ 4 marks KU ] = π 2 3

4,

3. For .......... (1) radius km= ⋅ ×4 8 103

.......... (1) S A. . = ⋅ ×289 4 106

.......... (1) [ 3 marks KU ] = ⋅ ×2 894 108 sq km.

4. For Rect. Area = 600 sq. cm and Tria. Area = 120 sq. cm .......... (1) For Are .......... (1) a h∆ = × × =1

2 20 120 For h = 12 .......... (1) For Tan x .......... (1) = 12

20

For answer (accept ) .......... (1) [ 5 marks RA ] x = 31o 30 9⋅ o

5. (a) i) f (5) = 13 ii) f (10) = 23 .......... (2) [ 2 marks KU ] (b) For f t t t( )2 2 2 3 4( ) 3= + = + .......... (1) For 2 3 2 2 3 3f t t( ) ( )− = + − .......... (1) .......... (1) = +4 3t ( Other proofs of course are acceptable, however, no marks for a series of examples i.e. f(1) and f(2) considered , then f(2) and f(4) etc.) [ 3 marks RA ] 6. (a) For a b and c= = = −3 4, 5 .......... (1)

For x = .......... (1) − ± − × × −4 4 4 3 5

6

2 ( )

For correct square root number i.e. 76 .......... (1) For calculation to x or= ⋅ − ⋅0 78 2 11..... ..... .......... (1) For rounding x or= ⋅ − ⋅0 8 2 1 .......... (1) (b) i) For ( )(2 1 4)x x+ − .......... (1)

ii) For ( ) .......... (1) (( )

2 1 42 2 1x x

x+ −

+)

For cancelling to x − 42

.......... (1) [ 8 marks KU ]

7. (a) Maximum occurs at x = 0 (stated or implied) .......... (1) For @ .......... (1) x y= =0 9then feet8 (b) At A , y = 0 ∴ = .......... (1) −0 98 2 2x For x .......... (1) 2 49= x = 7 .......... (1) (if a is calculated then given as (7,0), full marks) [ 5 marks RA ]

8. (a) Knowing what inverse means .......... (1) Applying and then answer 20 mins. .......... (1) (N.B. 0

2 for 80 minutes) (b) Evidence of .......... (1) ' '1 5⋅ Evidence of ' ' .......... (1) 1 52⋅ For answer 2 mins. .......... (1) [ 5 marks KU ] 25 40 90⋅ × =

9. (a) For using 90360 (or equiv.) .......... (1)

For answer x = cm .......... (1) [ 2 marks KU ] ⋅15 7 (b) For 2 15 7π r = ⋅ .......... (1)

r =⋅15 7

2π .......... (1)

r = ⋅2 5 cm .......... (1) (c) For s = cm ( stated or implied ) .......... (1) 10 For pyth. 10 .......... (1) 2 2= +r h2

For h = ⋅ ⋅9 7 9 68( )or etc. .......... (1) For V = × .......... (1) × ⋅ × ⋅1

322 5 9 7π

For answer V = ⋅ .......... (1) [ 8 marks RA ] 63 5 2cm

10. For F r .......... (1) mv= 2

F rm

v= 2 .......... (1)

F rm

v= .......... (1) [ 3 marks KU ]

11. (a) For ∠ ......... (1) =PQR 150o

( N.B. Pupils may still achieve full marks for part (b) even if ∠ PQR is wrongly stated )

)

⋅

(b) For use of the cosine rule .......... (1) For correct sub. ..... (1) PR2 2 275 60 2 75 60 150= + − × × ×( cos o

For bracket ( 7794 ) .......... (1) 2⋅ For PR .......... (1) 2 17 019 2= For PR .......... (1) km= ⋅130 5 ( N.B. i) Ignore rounding errors. ii) Use of pythagoras' 0

5 . iii) Use of sin 150, instead of cos150, correctly followed through 3

5 marks. [ 5 marks KU ]

12. (a) For 21 dots .......... (1) (b) For 78 .......... (1) 1

22 1

2= +n n For n n .......... (1) 2 156 0+ − = For n = 12 .......... (1) ( N.B. answer of n = 12 without working ... 1

3 marks ) [ 4 marks RA ]

13. For cosxo =−57

.......... (1)

.......... (1) x = ⋅135 6o

or .......... (1) [ 3 marks KU ] x = ⋅224 4o

14. (a) For r q p r q p= + +100 10 .......... (1) (b) For pqr rpq p q r r q p− = + + − + +100 10 100 10( ) .......... (1) = −99 99p r eventually( ) .......... (1) = + −99 2 99( )r r .......... (1) For answer .......... (1) [ 5 marks RA ] = 198

15. For Scale .......... (1) factor or used correctly= 14 4( )

For S .......... (1) F or used correctly. . ( )3 164 64=

For 1 litre = 1000 ml (stated or implied) .......... (1) For answer 39 ml .......... (1) ( N.B. i) 39.06 or other approx. .... full marks ii) S F with answer 156 ml ..... 3 marks ) [ 4 marks KU ] . . 2 ≈

16. (a) For strategy i.e. knowing to sub. into y mx c= + .......... (1) For − = − +35 5m c (or equiv.) .......... (1) 7 .......... (1) 2= +m c ( .or equiv ) Solving system ....then m = 6 .......... (1) and c = -5 .......... (1) [ 5 marks RA ] (b) For answer P (0 , -5) .......... (1) [ 1 mark KU ]

17. For correct strategy ( i.e. checking f f( ) , ( )3 1 3 2⋅ ⋅ , etc.) .......... (1) For deducing that root lies between 3 6 3 7⋅ ⋅and .......... (1) For establishing correct value of .......... (1) [ 3 marks KU ] 3 7⋅

18. (a) For r .......... (1) 2 23= +( ) 21 For r = 2 .......... (1) For r OB= (stated or implied) .......... (1) Then Area = × .......... (1) × =1

2 2 1 1 Answers/solutions involving trig. should be checked for exactness. Approx. solutions i.e. r = 1.966.... means 1 mark off. (b) For OB .......... (1) r x y= = +2 2

For using y as y-coordinate at A .......... (1) Then Area = 1

2 × ×OB AT

= 12

2 2y x y+ .......... (1) [ 7 marks RA ]

R A K U

42 47 Totals

Credit Mathematics - Practice Exam B Marking Scheme

1. For -5x = 15 .......... (2) x = -3 .......... (1) [ 3 marks KU ]

2. For using 135360

.......... (1)

For correct subst. .......... (1) For answer A = m75 36⋅ 2 .......... (1) [ 3 marks KU ] (Accept all correct roundings) 3. For A (large) = 216 .......... (1) ∆ mm2

For 1

4 A (large) = 54 ........... (1) ∆ mm2

For 1

2 18× ×w = 54 .......... (1)

For w = 6 .......... (1) [ 4 marks RA ] mm 4. For 12% of 200mg = 24mg (or 0 88⋅ as a multiplier) .......... (1) For Compound use of 12% .......... (1) For < 125 by the end of 4 hours (between 3 and 4) .......... (1) For answer between 4 pm and 5 pm .......... (1) [ 4 marks KU ] 5. (a) For 4 1 22 2= 2⋅ + d .......... (1) For d m= ⋅3 8 .......... (1) [ 2 marks KU ] 2(b) For 6 1 1 22 2⋅ = ⋅ + d .......... (1) For d = 5.98 m .......... (1) For consistent conclusion .......... (1) [ 3 marks RA ] 6. a) For 2a + 63 + b = 180 (or equiv) .......... (1) For 2a + b = 117 .......... (1) [ 2 marks RA ] b) For 2b + a + 72 = 180 ........... (1) For 2 b + a = 108 ........... (1) For knowing to use Sim Eqns ........... (1) For a = 42 ........... (1) For b = 33 ........... (1) [ 5 marks RA ] ________________________________________________________________________ 7. For methodical approach .......... (1) For f root f( ) (1 5 1 6)⋅ < < ⋅ .......... (1) For root to 1 d.p. .......... (1) [ 3 marks KU ] = ⋅1 5

8. (a) For each factor ... (1 mark) i.e. (2p - 1) (p - 4) .......... (2) [ 2 marks KU ]

(b) For correct denom. i.e. x(x - 1) .......... (1) For correct num. i.e. 3x - 2(x -1) .......... (1)

For answer x .......... (1) [ 3 marks KU ] x x

+−21( )

(Expansion of brackets is O.K. but 1 off for further cancelling) (c) For , stated or implied .......... (1) (5 ) ( )⋅ × ÷ ×91 10 3 109 5

For 1 .......... (1) 97 10 197004⋅ × or ondssec For answer = 5 .......... (1) [ 3 marks KU ] 5⋅ hours 9. For eliminating z e.g. .......... (1) yx z2 = For x y .......... (1) yx= ( )2

For 1 = y .......... (1) 2x

For 1 x= .......... (1) [ 4 marks RA ] 2y 10. For S.F. = .......... (1) 2 5⋅ For Vol SF = .......... (1) 2 5 15 6253⋅ = ⋅ For V = 15 625 15 2⋅ × ⋅ .......... (1) For V = .......... (1) [ 4 marks KU ] 237 5 3⋅ cm 11. (a) For AX = 90 cm .......... (1) [ 1 mark RA ] (b) There are a number of solutions. However, almost all will depend on finding AC = 150 For AC = 150 (by Pyth.) .......... (1) Possible Solution (Similar ∆ ' s) For use of similar triangles (stated/implied) .......... (1)

For XY90

60120

= .......... (2)

For ans. x = 45 cm .......... (1) [ 5 marks RA ] N.B. Last mark unavailable if previous error has occured. For solutions involving the use of trig functions which lead to an approximate, but inexact, value of XY = 45cm , give 3

5 for part (b) _______________________________________________________________________

12. For m =−−

4 00 3

(stated or implied) .......... (1)

For m = − 43 (accept -1.33 but not 1.3) .......... (1)

For C = 4 .......... (1) For T = − +4

3 4x (accept y = ... ) .......... (1) [ 4 marks KU ]

13. (a) For r = 60 cm .......... (1) For AX = 104 cm .......... (2) For h = 164 cm .......... (1) [ 4 marks RA ]

(b) For XC XM s= = 12 .......... (1) [ 1 mark RA ]

(c) For AX s s2 2 1

22= − ( ) .......... (1)

For AX = (or equivalent) .......... (1) 0 87⋅ s For h = .......... (1) [ 3 marks RA ] 1 37⋅ s 14. (a) For Area (side ) =∆ 1

221 36 0 45 0 306× ⋅ × ⋅ = ⋅ m .......... (1)

For area of triangle = .......... (1) 2 × 0 612 2⋅ m For area of rectangle = .......... (1) 2 72 2⋅ m For total area = (accept approx.) .......... (1) [ 4 marks KU ] 3 332 2⋅ m (b) For V = area x length (stated or implied) = 3 332 2 15⋅ × ⋅ .......... (1) = (or a rounded off figure) .......... (1) [ 2 marks KU ] 7 1638 3⋅ m 15. For ∠ = (stated or implied) .......... (1) ∠ =MNW and MWN131 18o o

For 6 .......... (1)

For 18 31 131sin sin sin

=

NW or MW

oNW r MW= ⋅ = ⋅10 0 14 7( ) .......... (1) For OW = × .......... (1) or ⋅ ×10 41 14 7 59cos ( cos )o o

For OW = ⋅ .......... (1) m7 5 N.B. Do not penalize approximations which, when carried through, may lead to - e.g. OW m= ⋅7 6 [ 5 marks RA ] 16. For VT .......... (1) 2 2 2700 1050 2 700 1050 120= + − × ×( co os ) For = ....................... - (-735000) .......... (1) For = 2327500 .......... (1) For VT = 1525 (or reasonable approximation) .......... (1) [ 4 marks KU ] 6⋅ m 17. For cosx = − 2

3 ........ (1) For ........ (1) x = ⋅131 8o

For ........ (1) [ 3 marks KU ] x = ⋅228 2o

18 (a) For 1 3 5 7 9 11 13 7 13 153

2 2 2 2 2 2 2+ + + + + + =× × .......... (2)

(b) S n n n=

× − × +( ) (2 1 2 13

) .......... (3) [ 5 marks RA ]

N.B. Do not penalise bad form - e.g. 2n - 1+ 2 for 2n + 1 , KU RA42 44

only 1 mark off for lack of brackets. Totals

Credit Mathematics - Practice Exam C Marking Scheme

1. For 10t - 5 ≥ 4t - 23 .......... (1) For 6t ≥ - 18 .......... (2) For t ≥ -3 .......... (1) [ KU 4 ] 2. For 0.96 .......... (1) For ( 0.96 ) ( or equiv.) .......... (1) For 22 100 litres (ignore roundings ) .......... (1) [ KU 3 ]

4

3. For knowing to use Cosine Rule .......... (1) For d 2 = 40 2 + 85 2 - 2x 40 x 85 x Cos 105 .......... (1) For = 1600 + 7255 - ( - 1760 ) .......... (1) For = 10585 .......... (1) For d = 102.9cm and " No " .......... (1) Accept any reasonable roundings. [ KU 5 ] 4. For ... 5v + 2t = 223 and 4v + 3t =219 (or equiv.) .......... (1) For 15v + 6t = 669 (or equiv.) .......... (1) For 8v + 6t = 438 .......... (1) For 7v = 231 .......... (1) For v = 33 pence .......... (1) For t = 29 pence .......... (1) [ RA 6 ] 5. For answer .... Graph C .......... (1) For explaining how each of the 3 parts of the jug are related to the 3 respective parts of the graph .......... (3) [ RA 4 ] 6. For knowing to use the quadratic formula .......... (1) For calculating discriminant 7.75 (or equiv.) .......... (1) For correct sub. to 4)75.72( ÷±=x (or equiv) .......... (1) For 2.44 and -1.44 .......... (1) For 2.4 and -1.4 .......... (1) [ KU 5 ] 7. Give any correct version ..... 4 marks. For example : For changing to km/h i.e. 10800 km/h .......... (1) 510× For dividing .......... (1) For 27 000 .......... (1) For times .......... (1) [ KU 4 ] 41072 ×⋅

8. (a) For T = k L / √h .......... (1) For 14 = k x 20/ √6.25 .......... (1) For k = 1.75 .......... (1) [ KU 3 ] (b) For T = 1.75 x 15 / √2.25 .......... (1) For T = 17.5 secs .......... (1) [ KU 2 ]

9. For area of square = 49 .......... (1) For area of triangle = 0.5 x 7 x DC x sin 110° .......... (1) For = 3.3 x DC .......... (1) For DC = 14.9 m .......... (1) [ RA 4 ]

10. (a) For ( 2a - 3) and ( a - 4) .......... (2) [ KU 2 ] (b) For 4( x - 2) = 5( 2 - x) .......... (1)

For 4x - 8 = 10 - 5x .......... (1) For 9x = 18 .......... (1) For x = 2 .......... (1) [ KU 4 ] (c) For P – t = 2 π r2 .......... (1) For ( P – t) / 2π = r2 .......... (1) For r = √ (P – t / 2π ) .......... (1) [ KU 3 ] 11. (a) For photo ratio of 18 / 12 or 1.5 .......... (1) For noting that 18 /12 ≠ 24 / 18 so, no similarity .......... (1) [ KU 2 ]

(b) For 24 - 2w and 18 - 2w .......... (1)

For 34

2182

=24

−−

ww (or equivalent equ. const.) .......... (1)

For cross mult. to solve (or equiv.) .......... (1) For w = 0 and conclusion .......... (1) [ RA 4 ] 12. For Area = Cba sin2

1 (or equiv.) .......... (1) For = 0.5 x 2.5 x 2.5 x Sin 60 .......... (1) For = 2.7 sq. cm .......... (1) For V = 2.7 x 14 .......... (1) For V = 37.8 cm3 .......... (1) [ RA 5 ]

13. (a) For f ( -1 ) = 1 and f (15 ) = 7 .......... (2) [ KU 2 ] (b) For 16 = √ (3t + 4 ) .......... (1)

For 256 = 3t + 4 .......... (1) For 84 = t .......... (1) [ KU 3 ]

14. For organised approach e.g. trying ... f (2 .0) , f (2.1) , etc. .......... (1) For 2.1 ≤ x ≤ 2.2 .......... (1) For x = 2.1 to one decimal place .......... (1) [ KU 3 ] 15. (a) For ( -x -2 ) ( x - 6 ) = 0 .......... (1) For x = - 2 or x = 6 .......... (1) For A is ( 6,0 ) and distance = 3000m .......... (1) [ KU 3 ] (b) For e.g. axis of symmetry is x = 2 .......... (1) For h = 16 when x = 2 .......... (1) For the maximum height is 1600m .......... (1) [ KU 3 ]

16. (a) For 1 + 3 + 3 2 + 3 3 + … = ( 3 8 - 1) /( 3 - 1) .......... (1)

For = 3280 .......... (1) [ RA 2 ] (b) For p = n .......... (1) [ RA 1 ] (c) For realising that the solution involves the subtraction of ( 3n - 1 ) /( 3 - 1 ) - 3280 [ from (a) and (b) ] .......... (1) For 38 + 39 + 10 + …. + 3 n – 2 + 3 n – 1 ...... = ( 3n - 1) / 2 - 3298 .......... (1) For reaching the final answer of ( 3n - 6561 ) / 2 .......... (1) [ RA 3 ] 17. (a) For OQ = 5 and using tangent .......... (1) For tan POQ = 10 / 5 ( or 2 ) .......... (1) For angle POQ = 63.4° .......... (1) [ RA 3 ] (b) For angle OPQ = angle OPR ( kite !) .......... (1) For angle PRQ = 36. 8° .......... (1) [ RA 2 ] (c) For angle QPO = 90° - y .......... (1) For e.g. 90° + 2 ( 90 - y )° + x° = 180 .......... (1) For x = 2y - 90 .......... (1) [ RA 3 ] 18. (a) For Area of circle (large ) = 400π .......... (1) For Area of circle (small) = 144π .......... (1) For Area of tape 256π .......... (1) [ RA 3 ] (b) For Area of circle (larger) = 289 π .......... (1) For new area of tape = 145π (i.e. 289π - 144π ) ..... (1) [ RA 2 ] (c) For tape used up during playing = 111π .......... (1) For e.g. correct ratio of tape used up i.e. 111π / 256 π ..... (1) For 111π / 256 π x 45 mins = 19.5 mins .......... (1) [ RA 3 ] (If 145π is used in ratio for ans. of 25.5 minutes , 2 out of 3 marks) * ( If π = 3.14 is used for multiplying / division in a correct manner … 1 mark off. )

R A K U

45 51 Totals

Credit Mathematics - Practice Exam D Marking Scheme 1. For …. = 8x + 4 …. (1) For -15 = 3x or 15 = -3x …. (2) For -5 = x …. (1) [KU 4] 2 a) (i) For f(2) = 7 …. (1) (ii) For f ( 2 ) = 3 …. (1) [KU 2]

b) For …. (1) 1712 2 =−t For …. (1) 92 =t For two answers 33 =−= tort …. (1) [KU 3] 3 a) For radius = 3 cm stated / implied …. (1) For area of circle = 28.24 …. (1) For area of square = 36 and shaded area = 7.74sq. cm. …. (1) [KU 3] b) For 7.74 /36 x 100 …. (1) For %ge = 21.5 …. (1) [KU 2]

c) For area of square = 4r2 …. (1) For area of circle = πr2 …. (1) For A = 4r2 - πr2 = r2( 4 - π ) …. (1) [RA 3] d) For P = r2( 4 - π )/ 4r2 x 100 …. (2) For canceling down to P = 25 (4 - π ) …. (1) [RA 3] 4. For 1.04 stated / implied …. (1) For 1.04 3 stated / implied …. (1) For £630 ( or the unrounded £629.92 ) …. (1) [KU 3] 5. a) For hypotenuse = 52cm, stated / implied …. (1) For sin x = 48/52 …. (1) For x = 67° …. (1) [KU 3]

b) For A = 67 / 360 ×…. …. (1) For 3.14 × 522 …. (1) For Area = 1581cm2 (accept correct, unrounded answers) …. (1) [KU 3] 6 For a = 3, b = 2, c = -4 , stated / implied …. (1) For correct substitution of above into quadratic formula (s/i) …. (1) For (-2±√52) / 6 …. (1) For x = 0.9 and -1.5 …. (1) [KU 4]

Pegasys 2005

7. For e.g. Area of ∆ ABC = ½ x 50 x 90 x sin60° = 1948.6m2 (or 1949) .... (1) For ½ of above area = 974.3 m2 (or 974.50 …. (1) For re- using area formula i.e. 974.3 = ½ x 50 x AT x sin60° …. (1) For AT = 45m …. (1) [RA 4] 8. a) For F = 2C + 30 or equivalent …. (1) [KU 1] b) For 2C + 30 = 1.8C + 32 …. (1) For 0.2C = 2 …. (1) For C = 10 …. (1) [RA 3] "C = 10" unsupported by equation work…0/3 ; C = 10" checked in/ into both equations..1/3 9. a) For scale factor = 2.0 / 2.5 = 0.8, stated or implied …. (1) For V = 0.83… …. (1) For V = 0.83 x 100 = 51ml ( Accept 51.2ml) …. (1) [KU 3] b) For e.g. 20 = ( S.F.) 3 x 100 …. (1) For 20 / 100 = ( b / 2.5 )3 …. (1) For 0.58 = b / 2.5 …. (1) For 1.45cm = b and b = 15mm …. (1) [RA 4] Accept legitimate rounding differences leading to 14mm. 10. For c = 42 …. (1) For m = (42 - 28) ÷(0 - 20), stated or implied …. (1) For m = - 0.7 …. (1) For d = - 0.7t + 42 …. (1) [KU 4] N.B. For y = - 0.7x + 42 ...... 0/1 11. For tanx° = 0.75 …. (1) For x° = 36.9° (or 37°) …. (1) For x° = 216.9° (or 217°) …. (1) [KU 3] 12. For deducing that angle THP = 36° …. (1) For knowing to use sine rule and attempting to substitute values …. (1)

For °

=° 36sin

90102sin

PH or equiv. …. (1)

For PH = 149.7km (or 150km) …. (1) [KU 4]

Pegasys 2005

13. a) For 600 x 1.6p = 960p (or £9.60 ) …. (1) For 2200 x 1.1p = 2420p (or £24.20) …. (1) For adding service charge of £8.50 to previous charges to get total bill of £42.30 …. (1) [RA 3] b) For appearance of both 9.60 and 8.50 (1st 600 units + s/c) …. (1) ( or 18.10 -£ sign may be included.) For " of (U - 600) x 0.011 (no £ sign required) …. (1) For "tidying up" to give C = 0.011U + 11.5 or equiv. …. (1) [RA 3] 14. a) For (2x - 5)(x - 2) …. (1)

b) For (x - 2 )(x + 2) …. (1)

For simplifying to get 252

+−

xx (Ignore further cancelling) …. (1) [KU 3]

15. a) For 98

78

76

56

54

34

32

12

××××××× =… …. (1)

For …= 1.48 (or 1.5) …. (1)

b) For 11 +

×− n

nn

n (1 mark for each fraction) …. (2)

c) For n

nn

n 11 +×

− (1 mark for each fraction) …. (2) [RA 6]

16. For Area of base = 25cm2 …. (1) For s2 = 6.52 - 2.52 where s = sloping height …. (1) For s = 6cm …. (1) For vertical height h = 5.5cm (or 5.45cm) …. (1) For Volume = 1/3 x 25 x 5.5 = 46cm3 …. (1) [RA 5] 17. a) For 30 matches …. (1) b) For attempting to use sim. equations …. (1)

For e.g. 3 = a x 12 + b x 1 i.e. 3 = a + b and then similarly, with e.g. the 2nd diagram 9 = 4a + 2b …. (2) For solving to find a = 1.5 , b = 1.5 ( 1 mark each) …. (2) [RA 6]

Totals KU RA 45 40

Pegasys 2005

Credit Mathematics Practice Exam E Marking Scheme - Paper 2 1. ans: 1700 sq cm. 3 KU 1 for ratio • 2 for formula • for answer 3• 2. (a) ans : T = kWD2 1 KU 1 for formula • (b) ans: 28 mins 2 RE 1 for idea that doubling D will square T • •2 for answer 3. ans: see candidates work 4 RE 1 proper labels • 2 - •4 properly completed diagram. • 4. ans : t ≥ 2 3 KU 1 for terms collected • 2 for numbers collected • for answer 3• 5. (a) ans: difference is 1 1 RE 1 for clear use of the algorithm • (b) ans: n + 1 and n + 2 2 RE 1 for 1• st term 2 for 2• nd term (c) ans: proof 3 RE •1 for applying 1st part of algorithm. •2 for applying 2nd part. •3 for final proof 6. ans: lock-keeper correct since 34.6 > 31 4 RE •1 for use of trigonometry •2 for use of trigonometry where x is the distance from dead tree to F. •3 for x = 34.6 or 35 •4 for correct conclusion.


Pegasys 2005

1

•3

1•

1

1••

1•

3•

1•

1••

1•

3•

• 135/360 2 3.14 x 382 … • 1700sq cm - do not penalise lack or errors in rounding (a) T = kWD2 or equiv. (b) • e.g. (5 / 2.5)2 •2 28 mins 2 - •4 e.g. correct rectangles on barchart • 10t or - 10t

2 20 or - 20 t ≥ 2

(a) 8x8=64, 9x7=63 and 64 - 63=1 (b) n + 1 2 n + 2 (or n + 1 + 1) (c) •1 (n +1)2 = n2 + 2n + 1 •2 n ( n + 2) = n2 + 2n •3 difference is 1 clearly shown. • cos 36°=… 2 …=28/x x = 34.6 or 35 •4 lock-keeper is correct (34.6 > 31, so barrier will miss the tree.)

7. ans: 199.5°, 340.5° 3 KU 1 for manipulation to sinx =… • 2 for calculation • for 23• nd answer 8. (a) ans: F+ S = 96 1 RE 1 for equation • (b) ans: 65F + 35S = 4140 2 RE 1 for part of equation • 2 for part of equation • (c) ans: F = 26, S = 70 3 RE for setting up the sim. equations 1• for calculating 1 variable 2• • 3 for calculating the other variable 9. ans: DB = 253cm 3 KU and •2 for scale factor or fractions •1

1••

3•

1•

1••

1•2

•

3•

••

3•


sinx = - 1/3

2 x = (- 19.5° =) 340.5° x = 199.5°

N.B. x = - 19.5° is not acceptable for •2

(a) F + S = 96

(b) e.g. 65F + 35S… 2 65F + 35 = 4140 (c) 65F + 35 = 4140, F + S = 96 • F = 26 • 3 S = 70 1- •2 S.F. = 120/156 or equiv. or BT/143 = 120/156 BT = 110cm and DB = 253cm. 1 e.g. OW / sin57° = 62 / sin95° 2 OW = sin57° x 62 / sin95° OW = 52.2 •4 e.g. sin 28° = VW / 52 •5 VW = 25m. N.B. Ignore premature rounding - this will usuallylead to a rounded answer of 24m. Also, do not penalise unrounded answers. Note: There are other ways to this solution, mark

at your own discretion.

for final answer. 3• 10. ans: VW = 25m 5 RE 1 for attempting to find OW or WS • using the sine rule 2 for OW = … • for finding OW. 3• •4 for using trig. to find VW. •5 for finding VW.

Pegasys 2005

11. ans : Yes, bottles could be similar 4 KU 1 calculating the linear scale factor • 2 knowing to cube the S.F. • • 3 for calculating the new volume • 4 for consistent conclusion 12. (a) ans: 128.4 and 7.9 3 KU 1 calculate mean to 1 d.p. • 2 substitute into formula for standard form • • 3 calculate the standard deviation. (b) ans: older women have higher blood pressure and s.d. is higher

2 RE compare blood pressure 1• 2 compare standard deviation. • 13. ans: - 2.6 or 2.1 4 KU 1 identify constants • substitute constants into quadratic 2• formula • 3 calculation of 1 value • 4 calculation of the other value 14. ans: proof (4.87 x 2 ≠ 10.57) 4 RE 1 use correct trig. function to calculate • height 2 calculation , adding the 0.5m • • 3 knowing to double the angle and to re-calculate the "new"height • 4 compare the 2 heights and clearly show that 1 is not exactly the double of the other.

12

1•

219

91156148980

2

−

−

1

2

1

24

72411 2 )(xx)()(x

−−−±−−=

61.x −=12.x =

12

• 7.5 / 5 = S.F. • 1.53 = 3.375 s/i •3 V = 400 x 3.375 =1350 •4 bottles could be similar since volumes are consistent with similar shape (a) 128.4

•

•3 7.9 (b) • e.g. these women have higher blood pressure • e.g. the standard deviation is "higher" for these older women( accept "about the same as ") • a=2, b=-1,c=-7

•

• 3 • 4 • tan20 = x / 12 • x1 = 4.37 + 0.5 = 4.87m • 3 x2 =10.07 + 0.5 = 10.57 • 4 4.87 x 2 ≠ 10.57


KU - 24 RE - 33 Total 57 marks

Pegasys 2005

Credit Mathematics Practice Exam F Marking Scheme - Paper 2 1. ans : x = 3 3 KU

•1 for opening brackets •2 for gathering terms •3 for solution 2. ans: 547.1cm 4 RE •1 for ratio •2 for C = π x 172 •3 for calculation •4 for adding 2x and completing calculation 3. ans: No, angle = 105.2° < 110° 5 RE •1 creating a R.A.T, sides = 13 and 48cm •2 for tanx° •3 for calculating x° •4 for adding to 90° •5 for clear conclusion 4. ans: 8 weeks 4 KU e.g. •1 for 2% lost = 0.98 •2 for 250 x (0.98)8, stated or implied. •3 for minimum volume •4 250 x (0.98)9 < 212.5 and conclusion 5. ans: 35.4° 5RE

•1 for knowing horizontal dist. from A to B •2 for calculation •3 for finding horizontal dist. from A to C •4 for calculation

•5 for subtraction to give answer

6. (a) ans: )1( +xx

3 3 KU

•1 for numerator •2 for denominator •3 for simplifying numerator

•1 5 - 15x •2 48 = 16x •3 3 = x •1 250/360 x… •2 172 x π •3 …= 375.1 •4 547.1cm •1 e.g. diagram •2 tan x° = 13/48 •3 x° = 15.1° •4 angle = 105.2° •5 not big enough •1 0.98 x … •2 212.69…cubic cm •3 212.5 cubic cm •4 208.4 < 212.5, so 8 •1 sin30° = d / 22 •2 d = 11cm •3 11 + 9 = 20cm, sin •4 full amgle = 65.4 •5 required angle = 35 •1 3( x+ 1) - 3x •2 x(x+1) •3 = 3


Pegasys 2005

N.B. Pupils may simply use successive calculations or may solve ......

85.0980 <⋅ n

weeks max.

(x+30)° = 20/22

.4°

6. (b) ans : v = u + bc 2 KU

•1 for removing fractions •2 for transferring u 7. (a) ans: 48 matches 2 RE •1 for correct answer, give 2/2 but… •2 for e.g.a sketch leading to wrong answer (b) ans: a = 2, b = 4. 4 RE •1 for knowing to use sim.eqns. •2 for forming 2 eqns. •3 for calculating 1 letter's value •4 for calculating the other value 8. ans: 8.5 cm 4 RE

•1 for area of base and area of beermat •2 for r2 •3 for r •4 for diameter and rounding Ignore premature rounding 9. ans: x = 1.6 or -1.2 4 KU

•1 for finding a,b and c for use in formula and for correct substitution •2 for square root calculation •3 for 2 answers unrounded •4 for correctly rounded answers

10. ans: Diameter 11.2 cm 4 RE ≈ •1 for volume of cylinder •2 for vol. of cyl. =vol. of cone (strategy) •3 for calculating r2 •4 for final answer


Pegasys 2005

1095442 −××−+

=x

•1 bc = v-u •2 bc+u=v •1 48 matches •2 diagram •1 see working •2 e.g. 6 = a + b, 16 = 4a +2b •3 a = 2 •4 b= 4 •1 glass area= 28.6,beermat area = 56.52 •2 r2 =56.52 / 3.14 =18 •3 r = 4.24 •4 D = 8.48 = 8.5cm

•1 a=5, b=-2,c=-9 and

•2 184 = 4 - 4 x 5x (-9) •3 x = 1.56.. or - 1.16… •4 x = 1.6 or -1.2 •1 volume of cyl. = 381.4 cm3 •2 381.4 = 1/3πr2h •3 r2 =31.7cm •4 D = 11.2cm NB Ignore rounding, information to 1 d.p. is only given as a guide to help the pupil through the calculation.

11. (a) ans: T = 15s2 / b 3 KU

•1 for interpreting variation statement •2 for substituting •3 for finding k, then constant of variation. No need for full equation (b) ans: Blade x 4 2 RE •1 for interpreting " ..as the square of.." •2 for answer 12. (a) ans: mean = 151.5 and 3 KU s.d. = 18.5 cm •1 for calculating mean •2 for correct substitution into standard form. •3 for answer (b) ans: same average (or mean) height but more spread than last year 2 RE •1 for comparing means •2 for interpreting s.d. as the idea of "spread" 13. (a) ans: 78° + 55° = 133° 2KU

•1 for 180° - " alternate angle " •2 for bearing - 180° then adding to •1 (b) ans; proof 3RE e.g. •1 for angle PAN •2 for angle NAQ •3 for adding the 2 parts above and clearly simplifying to get answer Accept -a° + b°

( )29

304545698450..2 ÷−

=ds

•1 T = ks2/b •2 18 = kx62/30 •3 k = 15 •1 s x 2⇒ b x 22 •2 b x 4 •1 4545 / 30 = 151.5cm

•2

•3 = 18.5 •1 mean or average the"same-ish" as at present.•2 last year's heights less spread out •1 180° - 102° = 78° •2 235° - 180° = 55°. Then 55° + 78° = 133° •1 PAN = 180°- a° •2 NAQ = b° - 180° •3 PAQ = PAN + NAQ = 180° - a° + b° - 180° = -a° + b° i.e b° - a° or PAN = 180 – a , NAQ = 180 – (360 – b), etc.


Total : KU 24 RE 35

For PI & PII Totals : KU 51 RE 46

Pegasys 2005

Credit Mathematics Practice Exam G Marking Scheme - Paper 2


• 1 ×

=4 01 10

12764 234.

.π

• 2 r = =12764 23

2638211

..

• 3 answer

• 1 1.025 × £87000 • 2 1025. × previous answer • 3 1025. × previous answer

• 1 a = 2, b = 4, c = -3

• 2 x =− ± − × × −4 4 4 2

4

2 3

• 3 x = 0.6 • 4 x = -2.6

• 1 250360

2536

=

• 2 A = ×π 202

• 3 Sect A= ×2536

=872.66

• 4 40 872 66 727 342 − =. .

• 5 727 341600

100%.

×

• 1 3

• 2 2nd 180 – ans, 3rd 180 + ans • 3 answer

d =

1. ans : 3 KU

6 38 103⋅ × km

cos x = −55. ans: 126.9o, 233.1o 3 KU

Give 1 mark for each •

• 1 using d =π

C

• 2 calculating radius • 3 answer in standard form

2. ans: £93689 3 KU

• 1 knowing 2.5% rise has M.F.= 1.025 • 2 knowing 3 years increase M.F. = 10253. • 3 calculating answer

3. ans: 0.6 and –2.6 4 KU

• 1 identifying a, b, c • 2 substituting correctly into formula • 3 calculating one value • 4 calculating second value

4. ans: 45% 5 RE

• 1 stating fraction • 2 calculating area of circle • 3 calculating area of sector • 4 calculating area of card not used • 5 calculating percentage

• 1 rearranging to find cos x = • 2 identifying quadrants • 3 calculations

Pegasys 2005

Give 1 mark for each • 6. ans: 19.2 m 5 RE

• 1 attempting to calculate side AC or AB • 2 calculating AC or AB using Sine Rule • 3 knowing shortest dist is at 90o to BC • 4 using SOH to calculate shortest dist • 5 calculating correctly

7. (a) ans: 12

2x y− 3 KU

• 1 numerator • 2 denominator • 3 simplifying

(b) ans: Hvd

=2

4 2 KU

• 1 removing root sign by squaring • 2 dividing

8. ans: Volume is doubled 3 RE

• 1 replacing e with 2e, and h with 12

h

• 2 simplifying expression • 3 conclusion

9. (a) ans: Females 6, 16, 35, 52, 66, 70 Males 5, 13, 24, 44, 70, 81

2 KU

• 1&2 knowing how to construct a cumulative frequency column


• 1 361

77 37 66.

sin sin sin= =

AC AB

• 2 AC = 21.1 m

• 3&4 sin.

66211

=dist

• 5 answer

• 1 ( )6 6 2 12x x y− − = y

• 2 ( )y x y− 2 • 3 answer

• 1 V d H2 4= • 2 answer

• 1 ( )V e= × ×

13

212

2 h

• 2 V e=23

2h

• 3 answer

• 1 Female column • 2 Male column

Pegasys 2005

×

Give 1 mark for each •

9. (b) ans: 3 KU

(c) ans: Statement 3 RE • 1 comparing totals • 2 considering differences at different ages • 3 recognising a critical age

10. ans: 1270 cm3 4 RE • 1 calculating volume of hemishere • 2 calculating volume of cone • 3 calculating 24% • 4 total volume to 3 sig figs

11. ans : Yes . Vol of water > 3 vol of food. 6 RE

• 1 splitting the top view into two sectors, the smaller containing an isosceles triangle.

• 2 calculating the angle at the apex of isosceles triangle

• 3 calculating the area of the minor sector • 4 calculating the area of the triangle • 5 calculating area of the food section • 6 comparing the areas of both sections

ans: No, as £2.25 > £1.99

3 RE • 1 finding scale factor for reduction • 2 calculating cost • 3 comparing cost with £2.25

Ages

Males

Females Number Of Smokers

12.

Pegasys 2005

For PI & PII


• 1 axes labelled correctly • 2 points plotted correctly • 3 lines drawn and identified

• 1 more males smoked than females• 2 up to age 17 there were more female smokers • 3 at age 18 and over there were more male smokers

• 1 V = × × × =12

43

8 1072 333π . cm3

• 2 V = × × × =13

8 12 804 252π . cm3

• 3 24% = 193.02 cm3 • 4 1265.35 = 1270 cm3

• 1&2 sin .x x= ⇒ =1215

60 07

angle of minor sector = 120.1o

• 3 min sect=1201

360152 23582

..× × =π

• 4 ∆ = × × × =1

215 15 120 1 97.33sin .

• 5 food section = 138.49 cm2 water section = 568.37 cm2

• 6 3 138 49 568 37× <. .

• 1 linear S.F. =75

1410

=

• 2 cost = 99.1£45.5£7

3

=

5×

• 3 answer

Total : KU 23 RE 29

Totals : KU 47 RE 45

Credit Mathematics Practice Exam H Marking Scheme - Paper 2 1. ans: 2 m/sec 2 KU 81098×⋅

1 multiplication • 2 answer in scientific notation •

2. ans : £ 516 590 5 KU 1 for 4.2% increase = 1.042 • 2 for , stated • 20421360000 ⋅× or implied 3 for 3.5% decrease = 0.965 • 4 for135000 , stated • 29650 ⋅× or implied 5 adding two sums together • 3. ans: 690 5 RE 1 using • o60sin 2 calculating length of ladder • 3 creating R.A.T. with sides 15 • and 14 4 using sin • ....ox 5 • ( )....sin 1−=x 4. ans : x = 1⋅3 or –5⋅3 4 KU 1 identifying a, b, c • 2 substituting correctly in formula • 3 calculating one value • 4 calculating second value • 5. (a) ans: 12=x , sd = 3⋅6 3 KU 1 calculating mean • 2 calculating • ∑∑ 2and xx 3 calculating standard deviation • (b) ans: Academy had higher mean no. of hours and their times were less spread out. 2 RE • 1 comparing means • 2 interpreting s.d. as the idea of spread


Pegasys 2005

• 810982372 ×⋅×•

• ......0421 ×⋅• 20421360000 ⋅×• .....9650 ×⋅• 29650135000 ⋅×•

•l

o 1360sin =

•

•1514sin =ox

•

•

• ( )2

71444 2 −××−±−=x

••

•7

159181110813 ++++++=x

• 10842 =∑ x

•6

1084.s.d 7

842−=

••

1 2 answer + rounding 1 2 =390875.04 3 4 =125715.38 5 390875.04 + 125715.38

1

2 l = 15m

3&4

5 answer

1 a = 1, b = 4, c = -7

2

3 x = 1⋅3 4 x = -5⋅3

1

2

3

1 mean is higher for Scotia Acad 2 results less spread out for Scotia Acad

(ignore rounding)

6 ans: 35 mm 5 RE 1 calculating volume of cylinder • 2 calculating volume of cuboid • in terms of d, the depth 3 knowing to equate the two • volumes 4 solving for d • 5 converting to millimetres • 7. (a) ans: 62p4,2p2,2p2p,2,2p +++− 2 RE 1 indicating a progression of • ± 2 2 stating sequence • (b) ans: proof 2 RE 1 attempting to calculate mean • 2 completing the proof • 8. (a) ans: proof 3 RE • 1 creating R.A.T. angle 600 and side 1⋅75 • 2 using sin .....60o

• 3 completing proof (b) ans: 25 people 4 RE • 1 knowing how to calculate length of arcs • 2 correctly calculating arcs • 3 calculating perimeter of table • 4 calculating no. of people 9. ans: 1⋅17m2 3 KU

1 calculating linear scale factor • 2 calculating area scale factor • 3 calculating area of paper •


Pegasys 2005

• 1131751450 2 ⋅=⋅×⋅×= πcylV• dd 3284 =××• 113132 ⋅=d••

• pp 2,22 −• 62,42 ++ pp

•5

624222222 +++++++− ppppp

• ( )125

1010+=

+ pp

•

•AB

o 75160sin ⋅=

•

• 044360120

⋅××π

••• 255825804620 =⋅=⋅÷⋅

•43

4030s.f.linear ==

•169

43.s.farea

2

=

=

• 171082169

⋅=⋅×

1 m3 2 Vcuboid = 3 4 d = 0⋅03478 m 5 d = 35 mm

1 2

1

2

1 triangle

2

3 AB = 2⋅02m

1 (or equivalent)

2 Arc AB = 4⋅229 m 3 Perimeter = 2(4⋅23) + 2(6) = 20⋅46m 4

1

2

3

10. ans: 9 miles 4 RE • 1 calculating distances using SDT • 2 correctly interpreting info. • 3 knowing to use Cosine Rule • 4 correctly using Cosine Rule. • 11. (a) ans: (0, -10) 1 RE 1 substitute x = 0 • (b) ans: B(-5, 0), C(2, 0) 3 RE 1 knowing to equate to 0 and solve • 2 solving correctly • 3 stating coordinates • (c) ans: (-1⋅5, -12⋅25) 2 RE • 1 finding the axis of symmetry • 2 substituting correctly 12. ans: 149o, 329o 4 KU 1 rearranging to find • ....tan =ox 2 identifying quadrants • 3&4 calculating angles •

o56cos5102582510258 22

×⋅×⋅×−

⋅+⋅

miles9481 =⋅

• ( ) 10100302 −=−+

• 01032 =−+ xx• ( )( ) 025 =−+ xx•

••

•53tan −=ox

•••

Illustration 1 Boat A = 8⋅25miles Boat B = 10⋅5 miles

• 2&3

• 4 1 1 2 3 answer 1 x = -1⋅5 2 y = -12⋅25

1

2 2nd and 4th 3 x = 149o

4 x = 329o

(ignore rounding)

(s) for awarding each mark Give 1 mark for each

KU - 21 RE - 33 Total marks 54 For PI & PII TOTALS KU – 47 RE – 49 Total marks 96

Pegasys 2005

mathematics - larberthigh.com24427... · ab sinc. 1. solve the following inequality 3x − 5≤ 5x...

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