PAPER 1 REVISION QUESTIONS (NO CALCULATOR ALLOWED)

Although these questions are sorted by topic, there are occasional questions involving

more than one topic. For example, there might be some differentiation as part of a

question involving polynomials.

STRAIGHT LINE

1) Triangle PQR has vertices P(-3, 5) , Q(7, 3)

and R(-1, -5) as shown in the diagram below.

a) Find the equation of the median RM. (3)

b) Find the equation of the altitude AP. (3)

c) Find the coordinates of the point of

intersection of RM and AP. (2)

2) A(-2, 4) , B(10, 4) and

C(4, 8) are the vertices

of triangle ABC as shown

in the diagram.

a) Write down the equation of the altitude from C. (1)

b) Find the equation of the perpendicular bisector of BC. (4)

c) Find the point of intersection of the lines found in (a) and (b). (2)

3) A line joins the points P(-4, 3) and Q(2, -7).

Find the equation of the perpendicular bisector of PQ.

(4)

4) Part of the line, L1 , with equation 4 3y x ,

is shown in the diagram. The line L2 , is parallel

to L1 and passes through the point (0, -1).

Point A lies on the x-axis.

a) Establish the equation of the line L2 and

write down the coordinates of the point A.

b) Given that the line AB is perpendicular to

both lines, find algebraically the coordinates

of point B. (5)

c) Hence calculate the exact shortest distance

between the lines L1 and L2 . (2)

5) The diagram shows triangle OAB with M being the midpoint of AB.

The coordinates of A and B are (-2, 6) and (20, 0) respectively.

a) Work out the coordinates of M. (1)

b) Hence find the equation of the median OM. (2)

c) A line through B, perpendicular to OM meets OM at C.

i) Find the equation of the line BC and hence find the coordinates of C. (4)

ii) What can you say about triangles OAM and BMC ? Explain your answer. (2)

6) Find the equation of the line passing through the point A(-2, 3) which is parallel to

the line with equation 2 4 5x y . (3)

7) Part of the line with equation 3 9x y is

shown in the diagram. B lies on this line and

has coordinates (3, k).

a) Find the value of k. (1)

b) Given that the line AB is perpendicular

to the line 3 9x y , find the equation

of AB. (3)

c) Hence write down the coordinates of A. (1)

d) Calculate the area of the shaded triangle. (4)

8) ABCD is a parallelogram. A, B and C have coordinates (2, 3) , (4, 7) and (8, 11).

Find the equation of DC. (3)

9) Triangle ABC has vertices A(-1, 12) , B(-2, -5) and C(7, -2).

a) Find the equation of the median BD. (3)

b) Find the equation of the altitude AE. (3)

c) Find the coordinates of the point of intersection of BD and AE. (3)

10) Find the equation of the line passing through the point P(5, -2) which is perpendicular

to the line with equation 3 2 1 0x y . (3)

11) Find the equation of the line ST,

where T is the point (-2, 0) and angle STO is 60°.

(3)

12) PQRS is a rhombus. Vertices P, Q and S have coordinates (-5, -4) , (-2, 3) and

(2, -1) respectively.

Work out the coordinates of the fourth vertex R, and hence, or otherwise

find the equation of the diagonal PR. (4)

13) Given that the points (3, -2) , (4, 5) and (-1, a) are collinear, find the value of a. (3)

14) Find the equation of the line which passes through the point P(3, -5) and is parallel

to the line passing through the points (-1, 4) and (7, -2) (4)

15) The perpendicular bisector of the line joining the points A(1, -2) and B(3, -4) passes

through the point (5, k). Find the value of k. (4)

16) Triangle PQR has vertices P(-6, -3) , Q(2, -7) and R(5, 9).

a) Find the equation of the median RS. (3)

b) Show that this median is perpendicular to side PQ. (2)

c) What type of triangle is PQR ? (1)

VECTORS

1) A is the point with coordinates (1, 1, 2), B(3, 0, 3) and C( 2, 3, 4).

a) Express and AB AC in component form. (2)

b) Find the size of angle BAC. (3)

2) In the triangle opposite, 2 unitsa b .

Find .( )a a b c .

(6)

3) The point Q divides the line joining P(-1, -1, 0) to R(5, 2, -3) in the ratio 2 : 1.

Find the coordinates of Q. (3)

4) P is the point (4,1, 2), Q is (5, 2, 0) and R is (7, 4, 4).

a) Show that P, Q and R are collinear. (3)

b) Find the ratio in which Q divides PR. (1)

5) In the triangle opposite :

2 and 2 3.p q r

Find .( )p p q r

(5)

6) Vectors u and v are defined by 3 2u i j and 2 3 4 .v i j k

Determine whether or not u and v are perpendicular to each other. (2)

7) Relative to a suitable coordinate system, A and B are the points (-2, 1, -1) and

(1, 3, 2) respectively.

A, B and C are collinear points and

C is positioned such that 2BC AB .

Find the coordinates of C. (4)

8) VABCD is a pyramid with rectangular base ABCD.

The vectors , and AB AD AV

are given by :

8 2 2AB i j k

2 10 2AD i j k

7 7AV i j k

Express CV in component form.

(3)

9) Vectors a and c are represented by two sides of

an equilateral triangle with sides of length 3 units,

as shown in the diagram.

Vector b is 2 units long and is perpendicular to both a and c.

Evaluate the scalar product .( )a a b c . (4)

10) A and B are the points (-2, -1, 4) and (3, 4, -1) respectively.

Find the coordinates of the point C, given that 3

2

AC

CB . (5)

11) The picture below shows a small section of a larger circuit board.

Relative to rectangular axes the points P, Q and R have as their coordinates

(-8, 3, 1) , (-2, -6, 4) and (2, -12, 6) respectively.

Prove that the points P, Q and R are collinear, and find the ratio PQ : QR. (4)

12) Points E, F and G have coordinates (-1, 2, 1) , (1, 3, 0) and (-2, -2, 2) respectively.

a) Given that 3EF GH , find the coordinates of the point H. (3)

b) Hence find EH (2)

13) Three vertices of the quadrilateral PQRS are P(7, -1, 5) , Q(5, -7, 2) and R(-1, -4, 0).

a) Given that QP RS , find the coordinates of S. (2)

b) Hence show that SQ is perpendicular to PR. (3)

14) The parallelogram OABC, where O is the origin, has two more of its vertices at

A(3, -2, 1) and C(5, 6, -3) as shown in the diagram.

a) Find AC in component form. (1)

b) D is a point such that

2

6

3

BD

. Show that A, C and D are collinear. (4)

15)

a) Given that Q divides PR in the ratio 1 : 2 find the coordinates of Q. (3)

b) Hence prove that angle SQR is a right angle (4)

DIFFERENTIATION

1) Find the stationary points on the curve 3 29 24 2y x x x and determine their

nature. (7)

2) Find the values of x for which the function 2 3( ) 5 24 3f x x x x is decreasing. (5)

3) A function f is defined by 3 2( ) 2 3 ,f x x x where x is a real number.

a) Find the coordinates of the points where the curve with equation ( )y f x

crosses the x and y-axes. (3)

b) Find the stationary points on the curve ( )y f x and determine their nature. (6)

c) i) Sketch the curve ( ).y f x

ii) Hence solve 3 22 3 .x x (3)

4) A function f is defined by 3 2( ) 2 4 1f x x x x , where 0 3x .

Find the maximum and minimum values of f . (5)

5) The diagram shows part of the

quartic with equation ( ).y g x

There are stationary points at

2, 0 andx x x a .

On separate diagrams sketch the graph of :

a) ( ).y g x (3)

b) ( 3).y g x (2)

6) Find the equation of the tangent to the

curve with equation 4

yx

at the point

where 2.x

(5)

7) A function, f, is defined on a suitable domain as 21f x x x

x .

a) Differentiate f x , with respect to x, expressing your answer with positive indices (4)

b) Hence find x, when 5f x . (3)

8) A curve has the equation 2 4x x

yx

, where and 0x R x .

Find the gradient of the tangent to this curve at the point where 4x . (6)

9) The diagram shows part of the

diagram of ( ).y f x

The function has stationary points

at P(0, 8) Q(5, 0) and R(10, -8)

as shown.

Sketch a possible graph for ( )y f x ,

where ( )f x is the derivative of ( )f x .

(4)

10) A curve has the equation 3 21 3 4 1y p x px x , where p is a positive integer.

a) Find dy

dx. (2)

b) Hence find the value of p given that this curve has only one stationary point. (5)

11) a) A function f has as its derivative 3 2 4f x x ax ax .

Find a, if the function has a stationary point at 4x . (4)

b) Hence find the rate of change of this function at 2x and comment on

your result. (2)

12) Find f x when 2 2x x

f xx

, expressing your answer with positive indices,

and hence calculate the value of the gradient of the tangent to the curve

y f x at 1

.4

x (6)

13) A ball is thrown vertically upwards.

After t seconds its height is h metres, where 21.2 19.6 4.9 .h t t

a) Find the speed of the ball after 1 second. (3)

b) For how many seconds is the ball travelling upwards ? (2)

14) Given that 9 1

,where ,1 4 4

f x xx

find the value of 1f (4)

15) The graph of a function f intersects the x-axis at (-a, 0) and (e, 0) as shown.

There is a point of inflexion at (0, b) and a maximum turning point at (c, d).

Sketch the graph of the derived function f x .

(3)

16) A function f is defined on the set of real numbers by 3 3 2f x x x .

a) Find the coordinates of the stationary points on the curve y f x

and determine their nature. (6)

b) i) Show that 1x is a factor of 3 3 2x x . (3)

ii) Hence or otherwise factorise 3 3 2x x fully. (2)

c) State the coordinates of the points where the curve with equation y f x

meet both the axes and hence sketch the curve. (4)

17) Find algebraically the values of x for which the function 3 22 3 36f x x x x

is increasing. (4)

18) Given that 3 ,find 4 .f x x f (3)

19) The diagram shows the graph of the function 1

, 11

f x xx

.

Prove that the function f is decreasing for all values of x except 1x . (4)

20) Find 3

2

6, when

x xf x f x

x

(4)

21) Find the derivative of 31

1 2 sin 26

x x , with respect to x . (4)

22) Differentiate 2 1x

x

, with respect to x. Express your answer with positive indices. (4)

23) Given that 4

21, 1

2g x x

, find the value of 2g (4)

24) The graph of y f x is shown

opposite.

Sketch the graph of y f x (3)

25) Differentiate 14 1

2x

x with respect to x. (5)

26) The diagram opposite shows the graph

of y g x .

The function has stationary points at

(0, 2) and (5, -2).

Sketch the graph of y g x

(3)

27) Part of the graph of the curve 213 9

4y x x

is shown opposite.

The tangent to the curve at the point

where 4x has been drawn.

Calculate the size of the angle θ, the

angle between the tangent and the

horizontal. (4)

28) Find the equation of the tangent to the curve 2 3 5y x x at the point where

the curve crosses the y-axis. (5)

POLYNOMIALS AND QUADRATICS

1) a) Show that 1x is a factor of 3 2( ) 2 3 5 6f x x x x . (3)

b) Hence, factorise ( )f x fully. (2)

2) a) Write 2 6 13x x in the form 2

x a b . (2)

b) i) Sketch the graph of 2 6 13y x x .

ii) State the range of values of y. (4)

c) Write down the maximum value of 2

1

6 13x x (1)

3) For what value of p, where p > 0, does the equation 2 2 211 12 0p x px p

have equal roots. (6)

4) What can you say about p, if the equation 9

1x

p px has no real roots ? (6)

5) An equation is given as 3 2 0x ax x b where a and b are constants.

a) It is known that 1x and 2x are two roots of this equation.

Use the above roots to find the values of a and b. (4)

b) Hence find the third root of this equation. (3)

6) Find the value of k such that the equation 2 6 0, 0kx kx k has equal roots. (4)

7) a) Express 27 2x x in the form 2

a x b . (2)

b) State the minimum value of 27 2x x and justify your answer. (2)

8) Show that the line with the equation 2 1y x does not intersect with the

parabola with equation 2 3 4y x x . (5)

9) Express 22 4 3x x in the form 2

.a x b c (3)

10) a) i) Show that 1x is a root of 3 28 11 20 0x x x (1)

ii) Hence factorise 3 28 11 20x x x fully. (3)

b) Solve 2

2 2log 3 log 5 4 3x x x (5)

11) Given that 1x is a factor of 3 2 5 6x kx x , find the value of k and

hence factorise the expression. (4)

12) The famous Gateway Arch (Figure 1) in the United States is parabolic in shape.

Figure 2 shows a rough sketch of the arch, relative to a set of axes.

Work out the equation of the parabola in terms of h and x. (3)

13) Given that 1x and 2x are two roots of the equation 3 2 2 0x ax x b ,

find the values of a and b and hence find the third root of the equation. (5)

14) What can you say about if the equation 4

x bx

has real roots ? (5)

15) Find the value of c, if 1x is a factor of the expression

3 21 2 1cx c x c x c (4)

16) Given that 2x is a factor of 3 2 1 4x x k x , find the value of k and hence

fully factorise the expression when k takes this value. (4)

17) The functions 21and 2 4

11

2

f x g x x

x

are defined on suitable domains.

a) Given that h x f g x , show that h x can be written as :

1

1 1h x

x x

(2)

b) State a suitable domain for .h x (1)

c) Show that there are two values of x for which the functions f and h have

the same image but that they are both irrational. (4)

1Figure

2Figure

LOGARITHMS AND EXPONENTIALS

1) The graph illustrates the law .by ax

The straight line joins the points

(0, 4) and (1, 0).

Find the values of a and b.

(4)

2) The diagram below shows part of the curve with equation log ( ).b

y x a

The curve passes through the points P(-4, 0) and Q(4, 2).

Find the values of a and b. (4)

3) The diagram shows part of the graph of log .b

y x a

Determine the values of a and b.

(3)

4) Evaluate 5 5 5log 2 log 50 log 4 (3)

5) a) Given that 2log log 2 2x xy y find a relationship connecting x and y. (4)

b) Hence find y when 1

, 0.4

x y y (2)

6) Solve the equation 1

log 3 2 12

xx where 1x . (4)

7) Given that 2 22log log ( 6) 5x x find the value of x where x > 0. (5)

8) a) Given that 23log log 2x xy y , find a relationship connecting x and y. (4)

b) Hence find the two values of y when 2x y (3)

9) Given that 3 3log 1 2log 2 2x , find the value of x. (3)

10) Find x, given that 4log 6 2log 4 1x x

(3)

11) Solve for x, 3 3log 6 log 2 2x x (3)

12) Evaluate 4 4 4

1log 10 3log 2 log 25

2 (2)

FUNCTIONS

1) a) Functions f and g are defined on suitable domains by :

22 5 and 1f x x g x x

Find ( ( ))f g x . (2)

b) Sketch the curve with equation ( ( ))y f g x (3)

2) Two functions, defined on suitable domains are :

2 1 and 1f x x x g x x

a) Show that the composite function h x f g x , can be written in the form

3 2h x ax bx cx , where, a, b and c are constants, and state the values

of a, b and c. (4)

b) Hence solve the equation 6h x , for x, showing clearly that there is only

one solution. (4)

3) The diagram shows part of the graph of y f x .

Sketch the graph of 3y f x .

Mark clearly the new positions of the

coordinates marked on the diagram.

(3)

4) Two functions, defined on suitable domains, are given as :

2 3 and 2 1g x x x g x x

Show that the composite function g h x can be written in the form

a ax b x b , where a and b are constants, and state the values of a and b. (4)

5) A quadratic function, defined on a suitable domain, is

given as 312 3f x x x .

The diagram shows part of the graph of this quadratic

function .y f x

The graph passes through the points P(2, 12) and Q(4, 0)

as shown in the diagram.

a) Sketch the graph of 6y f x , marking clearly

the image points of P and Q and stating their coordinates. (3)

b) Given that 6g x f x , write down a formula for .g x (2)

6) Two functions are defined on suitable domains as 21 and 6 13.f x x g x x x

Given that the function, h is such that h x g f x , express h in the form

2

h x x a b , where a and b are integers, and hence write down the

minimum value of h and the corresponding value of x. (6)

7) Functions 1

and 2 34

f x g x xx

are defined on suitable domains.

a) Find an expression for 1g x. (2)

b) i) Find an expression in its simplest form for h x , where h x f g x . (2)

ii) Write down any restrictions on the domain of h. (1)

8) Two functions are defined on suitable domains as 2

2 1 and 2 .f x x g x x

Given that the function, h is such that h x g f x , express h in the form

2

h x a x b c , where a, b and c are integers, and hence write down the

minimum value of h and the corresponding value of x. (6)

9) The diagram shows a sketch

of part of the graph of a

trigonometric function whose

equation is of the form

sin .y a bx c

Determine the values of a, b and c. (3)

10) The diagram shows a sketch of the function .y f x

a) Copy the diagram and on it sketch

the graph of 2 .y f x (2)

b) Copy the diagram again, and sketch

the graph of 1 2 .y f x (3)

11) Functions are defined on suitable domains as 2 1and 1 2 .f x x g x x

Find in its simplest form :

a) g f x (2)

b) g g x (2)

12) A function is given as 2

.g xx

a) State a suitable domain for this function on the set of real numbers (1)

b) Evaluate 1

.2

g g

(2)

c) Find a formula for g g x in its simplest form. (3)

13) The diagram shows the graph of

a function f .

f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2).

a) Sketch the graph of .y f x (2)

b) On the same diagram, sketch the graph of 2 .y f x (2)

14) The diagram shows the graph of .y f x

Make a sketch of the graph of 6.y f x (4)

15) Two functions f and g are defined on the set of real numbers as follows :

9

2 3 ,4

xf x x g x

a) Evaluate 3f g . (1)

b) Find an expression for g f x , in its simplest form. (2)

16) Two functions are defined on suitable domains and are given as :

23 , 1f x x g x x .

a) Find an expression for the function h, when h x g f x (2)

b) Find the value(s) of a, given that 1h a f a . (3)

INTEGRATION

1) a) Given that 2( ) 3 2 10f x x x and ( 2)x is a factor of ( ),f x find a formula for ( ).f x (4)

b) Hence, factorise ( )f x fully (1)

c) Solve ( ) 0f x (1)

2) a) Write 2cos x in terms of cos 2x. (1)

b) Find 24 cos x dx (3)

3) A curve has as its derivative 23 4dy

x xdx

.

Given that the point (3, -7) lies on this curve, express y in terms of x. (4)

4) Find 3

2

4 1, 0.

xdx x

x

(4)

5) The graph of y f x passes through the point ,19

.

Given that sin3f x x , express y in terms of x. (4)

6) The gradient of the tangent at any point (x, y) on a curve is given by :

2

82

dy

dx x

Given that the point (1, 4) lies on this curve, express y in terms of x. (4)

7) Work out

2

6

3 2dx

x (2)

8) Work out cos 3 2x dx (2)

9) Work out 2sin2 3x x dx (3)

10) Work out sin 2 1x dx (2)

11) Work out 2 2 1x x

dxx

(4)

12) Work out

21

z dzz

(4)

RECURRENCE RELATIONS

1) Two sequences are generated by the recurrence relations

1

1

0 4 8 4

2

n n

n n

U U

V kV

The two sequences approach the same limit as .n

a) Evaluate this limit. (2)

b) Hence determine the value of k. (2)

2) A sequence is defined by the recurrence relation 120

n nU aU

where

a is a constant.

a) Given that 010u and 1

26u find the value of a. (2)

b) Find the value of 2S , if 2 1 2

S U U . (2)

3) Two sequences are defined by the following recurrence relations :

1 10.6 20 and 0.9

n n n nU U U U b

, where b is a constant.

a) Explain why both sequences have a limit as n. (1)

b) Find the value of b, if both sequences have the same limit. (4)

4) A recurrence relation is given as 10 8 60

n nU U

.

a) Given that 1220U , find the initial value, 0

U , of this sequence. (2)

b) State why this sequence has a limit and hence find the difference between the

initial value and the limit of the sequence. (4)

5) a) A sequence is defined by 1

1

2n n

U U with 0

16U .

Write down the values of 1 2 and U U . (1)

b) A second sequence is given by 4, 5 7, 11, …

It is generated by the recurrence relation 1n nV pV q

with 1

4V .

Find the values of p and q. (3)

c) Either the sequence in a) or the sequence in b) has a limit.

i) Calculate this limit. (2)

ii) Why does the other sequence not have a limit ? (1)

PAPER 2 REVISION QUESTIONS (CALCULATOR ALLOWED)

As with the paper 1, there are questions which involve more than one topic.

STRAIGHT LINE

1) Triangle PQR has as its vertices P(-7, -2), Q(3, 8) and R(9, -10) as shown.

a) Find the equation of side PR. (2)

b) Find the equation of the altitude QS. (3)

c) Hence find the coordinate of S, the

point where the altitude QS meets

side PR. (4)

2) The diagram shows a line joining the points

A(-3,-1) and D(6,5). B has coordinate (9,-1)

and C is a point on AD.

a) Find the equation of the line AD. (2)

b) Hence find the coordinates of C

given that triangle ABC is isosceles. (3)

c) Calculate the size of angle BCD, to

the nearest degree. (3)

3) Triangle ABC has vertices A(-5,5) , B(11,-3)

and C(7,9). Q(4, 8) lies on AC and AM is a

median of the triangle.

a) Given that A, P and B are collinear,

find the value of k. (4)

b) Hence find the equation of PQ. (2)

c) Find the coordinates of R, the point

of intersection between the line PQ

and the median AM. (5)

4) In the diagram, triangle ABC has vertices A(8,4) B(20, p) and C(2, 17) as shown.

AD is perpendicular to BC.

a) Given that the gradient of BC

is 1

2 ,find the value of p. (3)

b) Find the equation of the

altitude AD in the form

0Ax By C . (3)

c) By considering the gradients

of side AB and the altitude AD,

calculate the size of the

shaded angle DAB. (3)

5) Triangle ABC has vertices A(-18, 6) , B(2, 4) and C(10, -8).

L1 is the median from A to BC.

L2 is the perpendicular bisector of

side AC.

a) Find the equation of L1 (3)

b) Find the equation of L2 (4)

c) Hence find the coordinates of T. (3)

6) Triangle PQR, shown below has vertices P(-4, -3) , Q(4, 6) and R(11, 3).

The altitude has been drawn from Q to PR.

a) Side PR has as its equation 5 2 7y x . Find the equation of the altitude QT. (4)

b) Hence find the coordinates of T. (2)

c) Calculate the area of the triangle in square units. (4)

7) In the diagram below, triangle PQR has vertices as shown.

a) Find the equation of the median from P to QR. (3)

b) Find the equation of the altitude from Q to PR. (4)

c) Hence find the coordinates of the point T, where the two lines cross. (3)

VECTORS

1) Three military aircraft are on a joint training mission. Their positions relative to

each other, within a three dimensional framework, are shown in the diagram below :

a) Show that the three aircraft are collinear. (3)

b) Given that the actual distance between Z and Y is 42 km, how far away from

Z is X ? (1)

c) Following further instructions, aircraft Y moves to a new position (50, 15, -8).

The other two aircraft remain where they are.

For this new situation, calculate the size of angle XYZ. (5)

2) Two forces are represented by the vectors 12 2F i j k and 2

3 .F i k

Calculate the angle between these two forces. (5)

3) A cuboid is placed relative

to a set of coordinate axes

as shown in the diagram.

The cuboid has dimensions

8 cm by 4 cm by 6 cm.

The origin of the axes is at the intersection point of the diagonals ED and HA and

1 unit represents 1 cm.

a) Find the coordinates of B, D and G (3)

b) Hence calculate the size of angle BDG. (6)

4) A metal casting is in the

shape of two cuboids.

The casting is positioned

relative to a set of rectangular

axes as shown opposite.

Both cuboids have been

centred along the x-axis.

a) Given that corners P and Q have coordinates (-8, -4, 6) and (6, -10, -15)

respectively, write down the coordinates of corners R and S. (2)

b) Hence show that the corners P, R and S are collinear. (2)

c) Find and RP RQ (1)

d) Calculate the size of angle PRQ. (5)

5) A cuboid with dimensions

12 cm by 4 cm by 4 cm is

placed relative to a set of

coordinate axes as shown

in the diagram.

F has coordinates (12, 4, 4) .

M is the midpoint of OA and

N is the midpoint of AB.

a) Write down the coordinates of M and N (1)

b) Calculate the size of angle MFN. (6)

6) Triangle ABC has vertices

A(4, -1, 2) , B(13, 2, -10)

and C(15, 3, -5) as shown.

Point D lies on side AB.

a) Given that D divides the line AB in the ratio 2 : 1 , find the coordinates of D (3)

b) Hence calculate the size of angle CDA. (5)

7) In the diagram, P, Q and R

have coordinates (3, 4, -1),

(0, 6, -6) and (k, 8, -10)

respectively.

a) Given that the angle PQR is a right angle, find the value of k. (4)

b) Calculate the size of angle RPS where S is the midpoint of QR. (6)

DIFFERENTIATION

1) The diagram shows part of the graph of the curve

with equation 2 8

, 0.2

xy x

x

a) Find the coordinates of the stationary point A. (5)

b) Also shown is the line with equation 2 7 8y x

which is a tangent to the curve at B.

Find the coordinates of B. (6)

2) A function is defined on a suitable domain as 21.f x x x

x

a) Find the derivative of the function f x , expressing your answer with

positive indices. (4)

b) Find the equation of the tangent to the curve 21y x x

x at the point

where 1x . (3)

3) The diagram below shows part of the curve with equation 3 22 5 4 3y x x x .

a) Find the coordinates of the stationary point marked A. (4)

b) Find the coordinates of B, one of the points where the curve crosses the x-axis. (2)

4) A group of soldiers decide that the best way to scale a cliff is to fire a metal hook

with a rope attached, over the top of the cliff, and hope it catches on to something

solid.

It is known that the firing device will launch the rope in such a way that the height,

H feet, above the ground is given by the formula :

2

330

dH d d , where d feet is the horizontal distance travelled.

Given that the height of the cliff is 84 feet, will this device be able to throw the

rope high enough ? Justify your answer with the appropriate working. (6)

5) The curve shown below has as its equation 5 33 5y x x .

Find algebraically the coordinates of A and B. (7)

6) The diagram below shows the parabola

with equation 28 3y x x and the line

which is a tangent to the curve at the point

T(1, 5).

Find the size of the angle marked θ , to the nearest degree. (4)

7) The curve shown below has as its

equation 3 48 2y x x .

a) Find the coordinates of the points A and B. (7)

b) Find the equation of the tangent to the curve at the point where 1

2x (5)

8) The line which passes through the

origin with gradient -3 intersects the

curve with equation 3 24y x x at

two further points A and B as shown

in the diagram below.

a) Find the coordinates of A and B. (5)

b) Hence show that OA is half the length of AB. (3)

9) A curve has as its equation 3 214 15 .

3y x x x

Part of the graph of this curve is shown

in the diagram opposite.

The tangent at the point R on the curve

is also shown.

a) Find the coordinates of the stationary

points. (6)

b) Find the coordinates of point R, given

that PR is parallel to the x-axis and that

the x-coordinate of R is a whole number.

Hence find the equation of the tangent at R. (5)

c) This tangent meets the curve at a second point.

State the coordinates of this second point. (1)

10) The diagram shows a sketch of the graph of 3 21 268

3 3y x x x .

a) Find algebraically the coordinates of the stationary points A and B. (5)

b) Find the equation of the line AB and hence prove that this line passes

through one of the points where the curve crosses the x-axis. (4)

11) Find the coordinates of the point on the curve 3 2 4 2y x x x where the

gradient of the tangent is 1 and x < 0 (4)

POLYNOMIALS AND QUADRATICS

1) A curve has as its equation 2

6 8y x .

Given that the line with equation 2 5y x is a tangent to this curve, establish

the coordinates of the point T, the point of contact between the curve and the line. (4)

2) Part of the graph of the curve with equation

3 3 2y x x is shown opposite.

The curve passes through the point (-1, 0).

Find algebraically, the coordinates of A and B. (7)

3) An equation is given as 1 1ax x c x , where 0, 0a c and a and c are constants.

a) Show clearly that this equation can be written in the form :

2 0ax a c x c (2)

b) What condition needs to be met for this quadratic equation to have equal roots ? (4)

4) The diagram below shows a rectangle and an isosceles triangle.

The letter p is a constant. All lengths are in centimetres.

a) Taking 1A as the area of the rectangle, and 2

A , as the area of the triangle, show

clearly that the difference between the two areas can be written in the form :

2

1 28 4 8A A x p x p (4)

b) Given that 2

1 21A A cm , find the value of p, where p > -1 , for this

equation to have only one solution for x. (6)

c) Hence find x when p takes this value. (2)

5) In the diagram below, the triangle and the rectangle have equal areas.

a) Show clearly that the following quadratic equation, in x, can be constructed from

the information given. (4)

2 22 2 0x p x p

b) For what value of p does the above quadratic equation have equal roots ? (4)

6) A function is defined as 2

3

18 87

pf x

x x

, for x R and p is a constant.

a) Express the function in the form

2

3pf x

x a b

and hence stat the maximum

value of f in terms of p. (4)

b) Given that 2

2 1p

show that the exact maximum value of f is 2 1 (2)

7) The two cuboids below have the same volume.

Cuboid 1 has dimensions 5 by x by 2x k and cuboid 2 is 4 by x k by x k .

All lengths are in centimetres.

a) By writing down expressions for the two volumes, show that the following

equation can be constructed :

2 2 25 4 0x k x k (4)

b) Given that 0k , find the value of k for which the equation 2 2 25 4 0x k x k

has equal roots. (4)

c) Hence solve the equation for x and calculate the volume of each cuboid when

k takes this value. (3)

8) The two diagrams shown are design logo backgrounds.

Design 1 is a square of side 2x .

Design 2 is a rectangle measuring 1x by c.

All lengths are in centimetres.

a) The area of Design 2 is 24 cm2 more than that of Design 1. By equating the areas

show that the following equation can be constructed :

2 4 28 0x c x c (3)

b) Hence find the value of c if the equation 2 4 28 0x c x c has equal roots. (5)

c) Using this value for c, solve the equation for x and hence calculate the area of

each design. (3)

9) Given that 1x and 3x are both factors of 3 22 5x x ax b , find a and b. (4)

LOGARITHMS AND EXPONENTIALS

1) A radioactive substance decays according to the formula 0.3

08 t

tM M , where

0M is

the initial mass of the substance and t

M is the mass remaining after t years.

Calculate, to the nearest day, how long a sample would take to half its original mass. (5)

2) Solve the equation 2 110 36x

3) Solve the equation 1 2 38 4x x

4) In a scientific experiment, corresponding readings of pressure (P) and temperature (t)

were taken and recorded.

The data was put into logarithmic form and the graph of 10log p against 10

log q .

The resulting graph, with the line of best fit added, can be seen below.

a) Explain why the original data takes the form nP kt , where k and n are constants. (4)

b) Given that the points A and B on the graph have coordinates (1.6, 11.98) and

(1.844, 13.45) respectively, work out the equation of the line and stat the values

of n and k. (4)

5) Water flows from a small reservoir according to the law kt

t oh h e ,

where oh is the original height of the water level and t

h the

height of the water after t days.

If the water level decreases by one fifth of its original height in 10 days,

calculate the number of days required to reduce the water level to

half its original height. (6)

6) The mass of radium-226 remaining after a decay period

of t years can be calculated

using the formula 0

kt

tM M e where

0M is the initial mass,

t

M is the mass remaining after t years and k is a constant.

a) Find the value of k, given that a sample of radium-226 takes 500 years to decay

to 80% of its initial mass.

Give your answer to 2 significant figures. (5)

b) Hence calculate the approximate percentage mass remaining, of a sample of

radium-226, after a period of 5000 years.

Give your answer to the nearest percent. (2)

7) The diagram shows part of a graph of 2log P against 2

log t .

The straight line has a gradient of 1

2and passes through the point (0, 2)

a) Find an equation connecting t and P. (3)

b) Hence show clearly that when 8 4P ,t takes the value 13 2 2

2 (3)

8) A Baryon particle decays according to the formula 0.0009

0

t

tM M e

where 0

M is the initial mass of the substance and t

M is the mass

remaining after t seconds.

Calculate, to the nearest ten seconds, how long a sample would take

to lose 30% of its original mass. (5)

9) The number of bacteria present in a beaker during an experiment can be measured

using the formula 1.2530 tN t e where t is the number of hours passed.

a) How many bacteria are in the beaker at the start of the experiment (2)

b) Calculate the number of bacteria present after 5 hours. (2)

c) How long will it take for the number of bacteria present to treble ? (3)

10) The mass, M grams of a radioactive isotope after a time of t years, is given by the

formula 0

ktM M e where 0

M is the initial mass of the isotope.

In 5 years, a mass of 10 grams of the isotope is reduced to 8 grams.

a) Calculate the value of k. (3)

b) Calculate the half-life of the substance. (The time taken for half of the

substance to decay). (3)

FUNCTIONS

1) The functions, f and g, defined on suitable domains, are given as :

2 3 5

and2 4 4

x axf x g x a , where a is a constant.

a) Given that 1f a g , find the value of a, where a < 0. (4)

b) With a taking this value, find the rate of change of g. (2)

2) Three functions are defined on suitable domains as :

2 31 , 3 3 and 6f x x g x x h x x x

a) Given that y g f x h x , find a formula for y in its simplest form. (4)

b) Hence find the coordinates of the maximum turning point on the graph of

y g f x h x , justifying your answer. (2)

3) A function is defined on a suitable domain as 2f x x a , where a is a constant.

a) Find a formula for h given that h x f f x . (2)

b) Given now that 2 8h , find a. (3)

4) Two functions are defined as 2 51 and h ,

2

x qf x px x

where p and q are

constants.

a) Given that 2 2 7f h , find the values of p and q. (3)

b) Show that 2110 1 .

2h f x x (2)

c) Find the value of the constant k, when 2 4h f x k f x (3)

RECURRENCE RELATIONS

1) A recurrence relation is defined as 10 75 12

n nU U

.

Given that 0

32U , find the difference between the limit of the sequence and

the third term 3

U . (5)

2) A sequence is defined by the recurrence relation 1n n

U aU b where a and b

are constants.

a) Given that 02 and 1U a b , show that 2

12 1U a a . (2)

b) Hence find an expression for 2U in terms of a. (2)

c) Given that 2

37U , find an expression for 2U in terms of a.

Explain why there is only one possible answer for a . (4)

3) A scientist studying a large colony of bats in a cave has noticed that the fall in the

population over a number of years has followed the recurrence relation

10.75 200

n nU U

, where n is the time in years and 200 is the average number of

bats born each year during a concentrated breeding week.

a) He estimates the colony size at present to be 2100 bats with the breeding week

just over.

Calculate the estimated bat population in 4 years time, immediately before that

years breeding week. (3)

b) The scientist knows that if in the long term, the colony drops, at any time

below 700 bats, it is in serious trouble and will probably be unable to sustain

itself. Is the colony in danger of extinction ?

Give an appropriate explanation for your answer. (4)

4) A sequence of numbers is defined by the recurrence relation 1n nU kU c

,

where k and c are constants.

a) Given that 0 1 210, 14, 17.2U U U , find algebraically, the values of k and c. (3)

b) Calculate the value of E, given that 3

24

E L where L is the limit of this

sequence. (3)

5) A sequence is defined by the recurrence relation 10.8 3

n nU U

.

a) Explain why this sequence has a limit as n. (1)

b) Find the limit of this sequence. (2)

c) Taking 010U and L as the limit of the sequence, find n such that 2.56

nL U (3)

6) Over a period of time the effectiveness of a standard spark plug slowly decreases.

It has been found that, in general, a spark plug will loose 8% of its

burn efficiency every two monts while in average use.

a) A new spark plug is allocated a Burn Efficiency Rating (BER)

of 120 units. What would the BER be for this plug after a

year of average use. Give your answer correct to 1 decimal place. (3)

b) After exhaustive research, a new fuel additive was developed.

This additive, when used at the end of every 4 month period, immediately

allows the BER to increase by 8 units.

A plug which falls below a BER of 93 units should be replaced.

What would be the maximum recommended lifespan for a plug, in months,

when using this additive ? (5)

c) The manufacturer is close to developing a new plug which contains a revolutionary

double core made from titanium.

Because of its “hot burn” qualities it can operate effectively down to a BER of

50 units.

Would this new plug ever need to be replaced if it is used in conjunction with

the additive ?

Your answer must be accompanied with the appropriate working. (3)

7) A sequence is defined by the recurrence relation 1n nU aU b

, where a and b

are constants.

a) Given that 04U and 8b , express 2

U in terms of a. (2)

b) Hence find the value of a when 288U and a > 0 (3)

c) Given that 3 1 2 3S U U U , calculate the value of 3

S . (2)

8) A sequence is defined by the recurrence relation 1n nU aU k

, where a

and k are constants.

a) Given that 06U , write down an expression for 1

U in terms of a and k. (1)

b) Hence show that 2U can be written as 2

26U a ka k (1)

c) Given that 20U , find k , where k > 0, if 26 0a ka k has equal roots. (3)

d) Find a when k takes this value. (2)

9) Industrial coolant is a water/oil based liquid used to cool down metal components

during their manufacture. It is continuously poured over the component and the

cutting tool.

The diagram below shows how the coolant is pumped from a main holding tank to

each machine in the factory and is then recycled back to the tank.

The coolant has one other important and expensive ingredient namely an

anti-bacterial agent. This agent, while active, hinders the growth of bacteria which

thrive in the heated coolant.

A company works the following system : At the beginning of each week (Monday

morning) the liquid is drained from the system and replaced by new coolant, 40 units

of the anti-bacterial agent is immediately added via the filter system.

It is known that the anti-bacterial agent decays at the rate of 13% per working day

and that this decayed amount is now said to be non-active.

a) How many units of the anti-bacterial agent are still active at the end of a normal

day (Friday evening) ?

Give your answer to the nearest hundredth of a unit. (3)

b) New government guidelines have just been issued which state that the minimum

amount of active anti-bacterial agent which should be present in industrial coolant

is 28 units.

The company decides to meet these new guidelines by topping up the agent at the

end of each working day.

Calculate the minimum number of whole units which should be added at the end of

each day in order to keep the active agent above the 28 units until the end of

the working week. (5)

c) The company had looked at the alternative of simply adding more that the 40 units

of the anti-bacterial agent at the beginning of the week.

Consider this option and comment. (3)

10) In a marine tank, the amount of salt in the water is crucial for the health of the fish.

Recommended limits give a salt solution of between 41 and 55 grams per gallon.

It is known that the strength of the salt solution decreased by 15% every day.

To combat this, salt is added at the end of each day, which effectively increases the

strength of the solution by 8 grams per gallon, thus creating a closed system.

To allow the plants to acclimatise, the initial strength in the tank has to be

45 grams per gallon.

a) For how many days should the system be run before the introduction of fish ? (3)

b) In the long term, will the strength of the solution remain within safe limits ? (3)

11) The diagram opposite shows a small bar magnet which is part of

an electrical control circuit.

When first placed in the circuit, the magnetic strength of the

magnet is rated at 100 mfu (magnetic flux units).

a) When the circuit is switched on, heat is produced.

This heat disturbs the dipoles within the domains of the magnet, producing

a demagnetization of the magnet. (a decrease in magnetic strength).

During any six hour period, when the circuit is running, the magnet is known to

lose 4% of the magnetic strength it had at the beginning of the period.

Calculate the magnetic strength of the magnet after the circuit has been running

continuously for 24 hours. (3)

b) At the end of each 24 hour period, the circuit (with the magnet in place) is

automatically passed through a very intense electric field. This allows the magnet

to regain some of its lost strength.

A single pass through the field and the magnet regains 12 mfu of strength.

The 24 hour cycle described above is now left to run uninterrupted for a number of

weeks.

Given that a magnet which falls below a strength of 77 mfu will not function properly

in the circuit, comment on whether the above conditions are satisfactory. (4)

c) Although an expensive option, the company decide to change their strategy.

They buy new equipment so that at the end of each 24 hour period, the circuit is

passed through an electric field may times stronger than the original.

The result for the magnet is a gradual increase in magnetic strength until “magnetic

saturation” is reached.

(It is know that the strength of a magnet cannot be increased beyond a certain limit)

Calculate the strength of this particular magnet when “magnetic saturation” occurs,

given that the new equipment is now adding 16 mfu at the end of each 24 hour period. (3)

12) In a commercial laundry a certain material is washed in a solution of water and

bleach. This washing solution is kept in a large vat and is continuously recycled

through the cleaning machines. This solution is renewed each day.

At the beginning of each day, fresh bleach and clean water are mixed in the vat.

The initial strength of the bleach is 20 units/gallon.

It has been found that in every hour of use, the bleach loses 12% of its strength.

a) Calculate the strength of the bleach after 4 hours of use.

Give your answer correct to 2 decimal places. (3)

b) It is known that if the strength of the bleach drops below 10 units/gallon, the

solution will no longer be an efficient cleaning agent.

The company decide to add a quantity of fresh bleach to the solution every 4

hours. This has the immediate effect of raising the overall strength of the

bleach by 6 units/gallon.

Comment on whether or not this policy is effective. (4)