Higher Maths

Calculator Practice

Practice Paper A

1. K is the point (3, 2, 3) , L(5, 0,7) and M(7, 3, 1).

(a) Write down the components of KL and KM.

(b) Calculate the size of angle LKM.

2. (a) (i) Show that ( 2)x is a factor of 3 2( ) 2 3 3 2.f x x x x

(ii) Hence factorise ( )f x fully.

(b) Solve 32( 1) 3 ( 1)x x x .

3. (a) Find the equation of the tangent

to the parabola with equation

26y x x

at the point (1, 6).

(b) Show that this line is also a tangent to the circle with equation

2 2 4 14 51 0x y x y

4. In the right-angled triangle shown in

Diagram 1, 1

2tan .a

(a) Find the exact values of

(i) cos ;a

(ii) cos2 .a 3

In the right-angled triangle shown in

Diagram 2, 1

3tan .b

(b) Find the exact value of sin(2 ).a b

y

x 0 1

sin .y p q rx

sin .y p q rx

5. Solve 9 9

1log ( 2) log ( 5), 5.

2x x x

6. The diagram below shows part of the graph of sin .y p q rx

(a) Write down the values of p, q and r.

The graph of sin .y p q rx has a minimum turning point at A and a maximum turning point at

B.

(b) Calculate the shaded area in the diagram above.

y

y

x

x

0

0

3

3

4

2

A

B

7. Cobalt-60 is used in food irradiation and decays to Nickel-60, which is a stable substance.

Cobalt-60 decays according to the law 0

,kt

tm m e where

0m is the initial mass

of Cobalt-60 present and t

m is the mass remaining after t years.

The time taken for half the mass of Cobalt-60 to decay to Nickel-60 is 5 years.

(a) Find the value of k, giving your answer correct to 3 significant figures.

(b) In a sample of Cobalt-60 what percentage has decayed to Nickel-60 after

2 years?

8. A rectangular park measures 200 metres by 300 metres.

A path connecting the two entrances, at opposite corners of the park, is to be laid through the park as

shown.

The cost per metre of laying the path through the park is twice the cost, per metre, of laying the path

along the perimeter.

(a) Show that the total cost of laying this path can be modelled by

2( ) 2 200 300C x x x

(b) Find the value of x which would minimise the cost of laying the path.

Park

x metres

y metres

200 metres

300 metres

Entrance

Entrance

Practice Paper B

1. (a) A sequence is defined by the recurrence relation 1 0

0 4 6, 0.n n

u u u

Determine the values of 1 2 3, and .u u u

(b) Why does this sequence have a limit as ?n

(c) A second sequence, generated by 1

4n n

v pv , has the same limit as the sequence in (a).

Find the value of p.

2. A function f is defined on the set of real numbers by 3 2( ) 4 7 10.f x x x x

(a) Show that ( 1)x is a factor of ( )f x , and hence factorise ( )f x fully.

The graph shown has equation of the form 3 24 7 10.y x x x

(b) Calculate the shaded area labelled S.

(c) Find the total shaded area.

3. D has coordinates (7, 2,1) and F is ( 1, 2, 5).

(a) Find the coordinates of E which divides DF in the ratio 1: 3.

G has coordinates (6, 2, 5).

(b) Show that EG is perpendicular to DF.

3 24 7 10y x x x

y

x 0

4 S

T

4. P, Q and R have coordinates ( 4, 6), (8, 10) and (2, 28) respectively.

(a) Show that PQ is perpendicular to QR.

(b) Hence find the equation of the circle which passes through P, Q and R.

5. Two functions f and g are defined on the set of real numbers by

2( ) 2 and ( ) 2f x x k g x x k , where 0k

(a) Find (i) ( ( ));f g x

(ii) ( ( )).g f x

(b) Find the value of k for which ( ( )) ( ( ))f g x g f x has equal roots.

6. A closed wooden box, in the shape of

a cuboid, is constructed from a sheet of

wood of area 2600 cm .

The base of the box measures 2x cm by x cm.

The height of the box is h cm.

(a) Assuming the thickness of the sides of the box are negligible, show that

the volume (in cubic centimetres) of the box is given by

34

3( ) 200V x x x

(b) (i) Calculate the value of x for which this volume is a maximum.

(ii) Find the maximum volume of the box.

2x cm

x cm

h cm

7. Whilst carrying out an experiment a scientist gathered some data.

The table shows the data collected by the scientist.

x 4 3 4 7 5 2 6 1

y 0 027 0 018 0 011 0 004

The variables x and y, in the table, are connected by a relationship of the form .bxy ae

Find the values of a and b.

8. Solve 2 3cos 4sin 0x x for 0 2x

Practice Paper C

1. (a) Given that ( 1)x is a factor of 3 22 3 6,x x kx find the value of k.

(b) Hence, or otherwise, solve 3 22 3 6 0.x x kx

2. OABC,DEFG is a rectangular prism as show.

OA is 8 units long, OC is 5 units and OD is 7 units.

(a) Write down the coordinates of B and G.

(b) Calculate the size of angle BEG.

3. A circle, centre C, has equation 2 2 4 2 20 0.x y x y

(a) Find the centre C and radius of this circle.

(b) (i) Show that the point P (5, 3) lies on the circumference of the circle.

(ii) Find the equation of radius CP.

(c) Find the equation of the chord which passes through (7,1) and is perpendicular to radius CP.

4. Solve 3cos2 11cos 6 for 0 2x x x

5. (a) Diagram 1 shows part of the graph

with equation 3 25 2 8.y x x x

Calculate the shaded area.

y

z

x A

B C

D E

F G

0 8

5

7

Diagram 1

(b) Given that

3 2

1

( 5 2 8) 12 13

p

x x x dx

find the total shaded area in diagram 2.

6. Find the smallest integer value of c for which

2( ) ( 2)( 2 )g x x x x c

has only one real root.

7. (a) Write 2sin 5 cosx x in the form sin( ),k x a where 2

0 and 0 .k a

(b) Sketch the graph of 4sin 2 5 cos for 0 2y x x x

8. For a particular radioactive isotope, the mass of the

original isotope remaining, m grams, after time t years

is given by 0 18

0

tm m e where 0

m is the original mass

of the isotope.

(a) If the original mass is 1000g, find the mass of the original isotope remaining after 20 years.

The half-life of the isotope is the time taken for half the original mass to decay.

(b) Find the half life of this isotope.

9. Find

4

6

sin 4

sin 2

xdx

x

.

Diagram 2

Practice Paper D

1. A sequence is defined by recurrence relation 1 0

6, 0.n n

u ku u

(a) Given that 2

8,u find the value of k.

(b) (i) Why does this sequence tend to a limit as n ?

(ii) Find the value of this limit.

2. 3 2( ) 2 4.f x x px qx

Given that ( 2)x is a factor of ( ),f x and the remainder when ( )f x is divided

by( 1)x is 9, find the values of p and q.

3. Security guards are watching a parked car,

via two CCTV cameras, in a supermarket

car park.

With reference to a suitable set of axes, the car is at C (5, 3, 2) and the cameras are at positions

A ( 1, 6, 4) and B(7, 9, 5) as shown.

Calculate the size of angle ACB.

4. Part of the graphs of 23y x x and 25 2y x are shown opposite.

The curves intersect at the points S and T.

(a) Find the coordinates of S and T.

(b) Find the shaded area enclosed between the two curves.

S

0

T

y

x

5. A circle with centre 1

C has equation 2 2 2 6 15 0.x y x y

(a) Write down the coordinates

of the centre and calculate the

length of the radius of this circle.

A second circle with centre 2

C

has a diameter twice that of the

circle with centre 1

C .

1

C lies on the circumference of this

second circle.

The line joining 1

C and 2

C is

parallel to the x-axis.

(b) Find the equation of the circle with centre 2

C .

6. A manufacturer of executive desks estimates that the weekly

cost, in £, of making x desks is given by 3 2( ) 6 560 800C x x x x .

Each executive desk sells for £2000.

(a) Show that the weekly profit made from making x desks is given by

3 2( ) 6 1440 800P x x x x

(b) (i) How many desks would the manufacturer have to make each week

in order to maximise his profit?

(ii) What would his annual profit be?

1C

1C

2C

y

y

x

x

0

0

7. The number of bacteria, b, in a culture after t hours is given by

0

ktb b e where 0

b is the original number of bacteria present.

(a) The number of bacteria in a culture increases from 800 to 2400 in 2 hours.

Find the value of k correct to 3 significant figures.

(b) How many bacteria, to the nearest hundred, are present after a further

4 hours?

8. (a) Express 2cos 5sinx x in the form cos( ) ,k x a where 0k and 0 90.a

(b) (i) Hence write 2cos2 5sin2x x in the form cos(2 ) ,R x b where 0R

and 0 90.b

(ii) Solve 2cos2 5sin2 5x x in the interval 0 360.x

Practice Paper E

1. (a) A line, 1

l , passes through the

points A( 3, 0) and B (5, 4).

The line makes an angle of a

with the positive direction on

the x-axis.

Find the value of a.

(b) A second line, 2

l , with equation

4 3 12x y , crosses the line in (a).

The angle between the two lines

is b , as shown.

Find the value of b.

2. The rectangular based pyramid D,OABC has vertices A(6, 0, 0) , B(6, 8, 0) and D(3, 4, 7) .

(a) (i) Write down the coordinates of C.

(ii) Express AC and AD in component form.

(b) Calculate the size of angle CAD.

y

x 0 A

B

y

x 0 A

B

a b

1l

2l

3. (a) (i) Show that ( 2)x is a factor of 3 26 3 10.x x x

(ii) Hence factorise 3 26 3 10x x x fully.

The line with equation 3 10y x intersects the curve with equation 2 36y x x

at the points A, B and C.

(b) Find the x-coordinates of the points A and C. 3

The area between the curve and the line from A to C is shaded in the diagram

below.

(c) Calculate the total shaded area shown in the diagram. 7

y

x 0

A

B

C

A

B

C

y

x 0

4. Solve 2cos2 sin 1 0 for 0 2x x x

5. A new ’24 hour anti-biotic’ is being tested on a patient in hospital.

It is know, that over a 24 hour period, the amount of anti-biotic remaining in the bloodstream is reduced

by 80%.

On the first day of the trial, an initial 250 mg dose is given to a patient at 7 a.m.

(a) After 24 hours and just prior to the second dose being given, how much

anti-biotic remains in the patient’s bloodstream?

The patient is then given a further 250 mg dose at 7 a.m. and at this time each subsequent morning

thereafter.

(b) A recurrence relation of the form 1n n

u au b can be used to model this course of treatment.

Write down the values of a and b.

It is also known that more than 350 mg of the drug in the bloodstream results in unpleasant side

effects.

(c) Is it safe to administer this anti-biotic over an extended period of time?

6. The diagram shows part of the graph of 3cos(2 ) 1.y x

Find the equation of the tangent at the point T, where 4

.x

7. Solve 3

2log ( 2) log (2 3) 2, .

x xx x x

8. A circle has the following properties:

The x-axis and the line 20y are tangents to the circle.

The circle passes through the points (0,2) and (0,18).

The centre lies in the first quadrant.

Find the equation of this circle.

Practice Paper F

1. (a) Find the equation of the tangent to the curve 3 10y x x at the point where 1.x

(b) Show that this line is also a tangent to the circle with equation

2 2 6 8 5 0x y x y

and state the coordinates of the point of contact.

2. The diagram opposite shows a rhombus ABCD.

AC and BD are diagonals of the

rhombus.

Diagonal AC has equation 2 3 0.x y

D is point with coordinates ( 4, 0) .

E is the point of intersection of the diagonals.

(a) Find the equation of diagonal BD.

(b) Hence find the coordinates of E.

3. Given that 2 3 p i j k and 3 , q i j k find the angle between p and q.

4. If 7

cos2 ,15

x find the exact value of :

(a) 2cos .x

(b) 2tan .x

5. A sequence is defined by the recurrence relation 1 0

, 0.n n

u au b u

(a) If 1

25,u write down the value of b.

(b) Given that 3

31,u find the possible values of a.

This sequence tends to a limit as .n

(c) Find the limit of the sequence.

y

x 0

A

B

C

D

E

6. (a) Express 4cos 7sinx x in the form sin( ) ,k x a where 0k and 0 360.a

(b) Hence solve 4cos 7sin 3 for 0 360.x x x

7. A new fish farm is being established consisting of a number of rectangular enclosures.

Each enclosure is made up from eight identical rectangular cages.

Each cage measures x metres by y metres.

The total length of edging around the top of all of the caging is 480 metres.

(a) Show that the total surface area, in square metres, of the top of the eight cages is given by

220

3( ) 320A x x x .

(b) (i) Find the value of x which maximises this surface area.

(ii) Hence find the dimensions of each enclosure.

8. The diagram shows the graphs of 2cos(2 ) 3y x and 7 5cos( )y x for 0 360.x

The two graphs intersect at T, which has coordinates ( , ).p q

(a) Find the exact value of cos .p

(b) Determine the value of p.

ANSWERS

PAPER A

1. (a)

2 4

2 and 1

4 4

KL KM

(b) 110 8 or 1 934 radians

2. (a) (i) Show that (2) 0f (ii) ( ) ( 1)( 2)(2 1)f x x x x

(b) 1

21, , 2

3. (a) 7 0x y

(b) Proof

4. (a) (i) 2

cos5

a (ii) 3

cos 25

a

(b) 3

sin(2 )10

a b

5. 17

2x

6. (a) 3, 1 and 2p q r

(b) 3

4 7122

7. (a) 0 139k

(b) 24 3%

8. (a) Proof

(b) 20x

PAPER B

1. (a) 1 2 3

6, 8 4 and 9 36u u u

(b) 1 0 4 1

(c) 3

5 (or 0 6)p

2. (a) Show that (1) 0;f ( ) ( 1)( 5)( 2)f x x x x

(b) 65

(or 5 416)12

(c) 289

(or 48 16)6

3. (a) E(5, 1, 2)

(b) Show that EG . DF 0

4. (a) Show that PQ QR

1m m

(b) 2 2( 1) ( 17) 130x y

5. (a) (i) 2( ( )) 2 3f g x x k (ii) 2 2( ( )) 4 4 2g f x x kx k k

(b) 1k

6. (a) Proof

(b) (i) 5 2 7 071x cm (ii) Maximum volume : 32000 2942 8

3cm

7. 0 7 2 0, 1a e b

8. 2 086, 6 051

PAPER C

1. (a) 5k

(b) 3

22, 1,

2. (a) B(8, 5, 0) and G(0, 5, 7)

(b) 72 1 (or 1 258 rads)

3. (a) Centre : (2, 1) Radius : 5 units

(b) (i) Proof

(ii) 4 3 11 0x y

(c) 3 4 17 0x y

4. 1 231, 5 052

5. (a) 63

4

square units (or 15 75)

(b) 19 37

6. 2c

7. (a) 3sin( 0 841)x (b)

8. (a) 27 3g

(b) 3 85 years

9. 3

21 0 134

y

x 0

6

PAPER D

1. (a) 1

3k

(b) 1

31 1

(c) 9

2. 1, 8p q

3. 75 8 or 1 323 radians

4. (a) S( 1, 3) and T(2, 3)

(b) 9

2 square units

5. (a) 1

C (1, 3) and radius 5 units

(b) 2 2( 11) ( 3) 100x y

6. (a) Proof

(b) (i) 24 (ii) £1216 384

7. (a) 0 549

(b) 21600

8. (a) 29 cos( 68 2)x

(b) (i) 29 cos(2 68 2)x (ii) 135, 156 8, 315, 336 8

PAPER E

1. (a) 26 6a

(b) 100 3b

2. (a) (i) C(0, 8, 0) (ii)

6 3

AC 8 and AD 4

0 7

(b) 54 5 or 0 951 radians

3. (a) (i) Proof (ii) ( 2)( 5)( 1)x x x

(b) A : 1x C : 5x

(c) 81

2 square units

4. 3

20 848, 2 294, 4 712

5. (a) 50 mg

(b) 1

0 2 250n n

u u

(c) 312 5 350 safe to administer long term

6. 3

26 1 0x y

7. 2x

8. 2 2( 6) ( 10) 100x y

PAPER F

1. (a) 2 12 0x y

(b) (7, 2)

2. (a) 2 4 0x y

(b) E( 2, 1)

3. 108 8 or 1 899 radians

4. (a) 11

15

(b) 4

11

5. (a) 25b

(b) 1 6

5 5 or a

(c) 125

4

6. (a) 65 sin( 209 7)x

(b) 7 9, 231 5

7. (a) Proof

(b) (i) 24 (ii) 24 20m m

8. (a) 3

4cos p

(b) 221 4