1. The equation of a curve is

(i) Show that the whole of the curve lies above the x-axis. (ii)Find the set f values of x for which is a decreasing function.

The equation of a line is , where k is a constant.

(iii) In the case where k = 1, find the coordinates of the points of intersection of the line and the curve. (iv) Find the value of k for which the line is a tangent to the curve.

2. The function f is defined as for

(i) Express f(x) in the form of where a, b and c are constants. (ii)State the range of f.(iii)Explain why f does not have an inverse.

The function g is defined as for where A is a constant.

(iv) State the largest value of A for which g has an inverse. (v) When A has this value, obtain an expression in terms of x for and state the range of

3. Let for

(i) Express f in the form of stating the numerical values of a, b and c. (ii)State the range of f.

4. The function f is defined as for the domain

(i)

(ii)

(iii)

Find the range of each domain above and state which domain gives the function f an inverse.

P1.C1_C2.E1.Quadratics and Functions

1

y = x2 − 2x+ 8.

x2 − 2x+ 8

y − 6x = k

x ∈ R.

a(x+ b)2 + c,

2x2 − 8x+ 11 x ≤ A,

g−1(x) g−1(x).

f : x �→ 2x2 − 12x+ 3 x ∈ R.

a(x+ b)2 + c,

f(x) = x2 − 6x+ 3

4 ≤ x < 6

x ≥ 3

1 ≤ x ≤ 4

2x2 − 8x+ 11

1. Functions f and g are defined by ! ! for ! where k is a constant.

! ! ! ! ! ! for

(i) Find the values of k for which the equation fg(x) = x has two equal roots. (ii)Determine the roots of the equation fg(x) = x for the values of k found in part (i).

2. Functions and g are defined by

(i) Find the value of x for which fg(x) = 7.

(ii)Express each of and in terms of x.

(iii) Show that the equation ! ! ! has no real roots.

(iv) Sketch, on a single diagram, the graphs of y = f(x) and y = ! , making clear the relationship between these two graphs.

3. The function !! ! where a is a constant, is defined for all real x.

(i) In the case where a = 3, solve the equation ff(x) = 21.

The function ! ! ! is defined for all real x.

(ii) Find the value of a for which the equation f(x) = g(x) has exactly one real solution.

The function ! ! ! is defined for the domain

(iii) Express !! in the form ! ! , where p and q are constants. (iv) Find an expression for and state the domain of

4. The function f is defined by

! ! ! ! ! ! ! for

(i) Sketch, in a single diagram, the graphs of y = f(x) and y = !! making clear the relationship between the two graphs.

The function g is defined by

! ! ! ! ! ! ! for

(ii) Express gf(x) in terms of x, and hence show that the maximum value of gf(x) is 9.

The function h is defined by

(iii) Express in the form where a and b are positive constants. (iv) Express in terms of x.

P1.C1_C2.E2. Quadratics and Functions

2

f : x �→ 4x− 2k x ∈ R,

g : x �→ 9

2− x x ∈ R, x �= 2.

f : x �→ 2x− 5, x ∈ R.g : x �→ 4

2− x, x ∈ R, x �= 2.

f−1(x) g−1(x)

f−1(x) = g−1(x)

f−1(x)

f : x �→ 3x− 2a,

g : x �→ 6− 5x− x2

h : x �→ 6− 5x− x2 x ≥ 3.

6− 5x− x2 −(x+ p)2 + qh−1(x) h−1.

f : x �→ 3x− 2 x ∈ R.

f−1(x),

g : x �→ 6x− x2 x ∈ R.

h : x �→ 6x− x2, x ≥ 3.

6x− x2 a− (x− b)2,h−1(x)

1.

The diagram shows a rectangle ABCD. The point A is (0,-2) and C is (12,14). The diagonal BD is parallel to the x-axis.

(i) Explain why the y-coordinate of D is 6.

The x-coordinate of D is h.

(ii) Express the gradient of AD and CD in terms of h. (iii) Calculate the x-coordinates of D and B. (iv) Calculate the area of the rectangle ABCD.

2.

In the diagram, AOB is a sector of a circle with centre O and radius 12cm. The point A lies on the side

CD of the rectangle OCDB. Angle AOB = radians.

Express the area of the shaded region in the form stating the values of the integers a and b

3.

In the diagram, the circle has centre O and radius 5cm. The points P and Q lie on the circle and the arc length PQ is 9cm. The tangents to the circle at P and Q meet at the point T. Calculate

(i) angle POQ in radians,(ii) the length of PT,(iii) the area of the shaded region

P1.C3_C4.E1.Coordinate Geometry and Circular Measure

π

3

a(√3)− bπ,

x

y

A(0,−2)

B

C

D

1

3π

O

A

B

C D

12cm

12cm

O

5cm

9cm

P Q

T

1.

In the diagram, AB is an arc of a circle, centre O and radius r cm, and angle AOB = radians. The pointX lies on OB and AX is perpendicular to OB.

(i) Show that , of the shaded region AXB is given by

(ii) In the case where r = 12 and , find the perimeter of the shaded region AXB, leaving your answer in terms of and

2.

The diagram shows a rhombus ABCD. Points P and Q lie on the diagonal AC such that BPD is an arcof a circle with centre C and BQD is an arc of a circle with centre A. Each side of the rhombus has length 5cm and angle BAD = 1.2 radians.

(i) Find the area of the shaded region BPDQ(ii) Find the length of PQ.

3. Three points have coordinates A(2,6), B(8,10) and C(6,0). The perpendicular bisector of AB meets the line BC at D. Find

(i) the equation of the perpendicular bisector of AB in the form ax + by = c ,(ii) the coordinates of D.

P1.C3_C4.E2.Coordinate Geometry and Circular Measure

√3 π.

θ =1

6π

A cm2

A =1

2r2(θ − sin θ cos θ)

θ

A

B

C

D

P

Q

A

BO X

rcm

θ rad

1. P is the point (3,2) on the curve ! ! Find the coordinates of the point of intersection ofof the tangent to the curve at P with the line ! ! ! ! ! ! ! ! ! ! ! ! [3]

2.

The diagram shows a circle of radius 10cm with the arc length 13cm. Calculate

(i) the angle AOB in radians [2](ii)the area of the shaded region [3]

3. ABC is a sector of a circle, centre O, and radians. Given that AB is perpendicular toBD and AD = 10cm, calculate,

(i) the perimeter of the shaded region(ii)the area of the shaded region.

4. The line intersects the curve at the point A and B. Find

(i) the coordinates of A and B,(ii)the distance AB.

5. The figure shows a circle, centre O, radius 10cm and a chord AB such that the angle radianCalculate

(i) the length of the major arc, ACB,(ii)the perimeter of the triangle AOB,(iii)the area of the shaded region.

P1.C3_C4.E3. Coordinate Geometry and Circular Measure

y = x2 − 5x+ 8.x+ y + 3 = 0.

∠BAC =π

6

2y + x = 5 y + xy = 6

AOB =2π

5

[3][3]

[3][2]

[2][3][3]

O

A

B

θ 13 cm

10cm

π

6A

B

CD

O

A

B

2π

6

10cm

1. In the circle with centre O and radius 5cm, AB is a chord of length 8cm,

(i) find the arc length of the sector AOB.(ii)area of the minor segment AOB.

2. Write down the coordinates of the midpoint M of the line joining A(0,1) and B(6,5). Show that the line passing through M and is perpendicular to AB.

3. The diagram shows a sector AOB of a circle with centre O, radius OA = 5cm and angle AOB = 0.7 radiansCalculate

(i) the length of the arc AB,! ! ! [1](ii)the length of the chord AB, giving your answer correct to 2 decimal places [2]

P1.C3_C4.E4. Coordinate Geometry and Circular Measure

3x+ 2y − 15 = 0

[3][3]

[3]

O

A

B

0.7 rad

5cm

1. (a) Find the solution of the equation where ! ! ! ! [4](b) Solve the equation ! ! ! for ! ! ! ! [4]

2. Find all the values of x for which! ! that satisfy the equation

! ! ! ! ! ! ! ! ! ! ! ! [3]

3. Prove the following:

(a) ! ! ! ! ! ! ! ! ! ! [3]

(b) ! ! ! ! ! ! ! ! ! ! [3]

4. Sketch the graph of where ! ! ! ! ! ! [4]

5. Solve ! ! ! for ! ! ! ! ! ! ! [4]

P1.C5.E1.Trigonometry

5 cos 2θ = 3, 0 ≤ θ ≤ 2π5 cos θ + 2 sin2 θ = 4, 0◦ < θ < 180◦.

0◦ < x < 360◦,

y = 2 cos 3θ, 0◦ ≤ x ≤ 360◦.

sinA

1 + cosA≡ 1− cosA

sinA

sinA

1 + cosA+

1 + cosA

sinA≡ 2

sinA

sin

�1

2x

�=

1

4

2 cos2 θ + sin2 θ = 2, 0◦ ≤ θ ≤ 360◦.

1. Show that the equation ! ! ! may be written as a quadratic equation in [2]

Hence solve this equation, for ! ! ! ! [5]

2. Show that

3. Find all the angles, between and which satisfy

(i)

(ii)

4. Find all the angles between and inclusive which satisfy the following equations.

(a)

(b)

5. Sketch the graph of for ! ! ! ! ! ! [3]

P1.C5.E2.Trigonometry

15 cos2 θ = 13 + sin θ sin θ.

0◦ ≤ x ≤ 360◦.

(tan θ + sin θ)(tan θ − sin θ) ≡ tan2 θ sin2 θ.

0◦ 360◦,

4 sin2 x = 6− 9 cosx

3 cos y +cos y

sin y= 0

0◦ 360◦,

cos(2x− 40◦) = sin 70◦

tanx+ 2 sinx = 0

y = 3 cos1

2x, 0 ≤ x ≤ 2π

[4]

[4]

[4]

[4]

[4]

1. The position vector of A, B and C relative to the origin are -i, -j, 2i + 3j and 11i - 6j respectively. Evaluate and hence find cos

2. Given that and , find

(i) the value of a.b(ii) the angle between a and b(iii)the value of p for which b and c are perpendicular

3. The position vectors of three points A, B and C with respect to a fixed origin O are 2i - 2j + k, 4i + 2j + k and i + j + 3k respectively. Find unit vectors in the direction of CA and CB.

Calculate angle ACB in degrees.

4. In the diagram OABCDEFG is a cuboid. The midpoints of AB and FG are M and N respectively.

Given that and

(i) Show that ! ! ! ! ! ! [3]

(ii) Show that ! ! ! ! ! ! [3]

(iii) By using scalar product, find the angle between ON and MG, giving your answer to the nearest degree.

P1.C6.E1.Vectors

−−→BA ·−→CA ∠BAC.

a =

�31

�, b =

�−52

�c =

�p

p+ 2

�

−→OA = 6i,

−−→OC = 2j

−−→OD = 2k.

−−→ON = 3i+ 2j+ 2k−−→MG = −6i+ j+ 2k

[4]

[2][3][3]

[3]

[4]

[4]

i

jk

A

BC

DE

FG N

M

1. The points A, B and C have position vectors

respectively, relative to a fixed origin O.

(i) Find the vectors (a - b) and (c - b). Hence evaluate (a - b).(c - b).! ! [3](ii)Calculate the size of the angle ABC, giving your answer to nearest degree [3]

2. In triangle OAB, O is the origin, ! ! ! and

(a) Find the unit vector in the direction of [2]

(b) Show that triangle OAB is isosceles. [2]

(c) Find angle AOB. [2]

(d) Hence or otherwise, find the area of triangle OAB. [2]

3.

The figure shows a cuboid in which OA = 1m, OC = 3m and OD = 2m taking O as origin and unit vectorsi, j, k in the directions of OA, OC, OD respectively.

Express in terms of i, j, k the vectors

(i) OF(ii)AG

By considering an appropriate a scalar product, find the acute angle between diagonals OF and AG.

4. In the diagram OABCDEFG is a cube in which the length of each edge is 2 units. Unit vectors i, j, k are parallel to OA, OC, OD respectively. The midpoints of AB and FG are M and N respectively.

(i) Express each of the vectors and in terms of i, j and k.

(ii) Show that the acute angle between the direction of and is , correct to the nearest 0.1 degrees.

P1.C6.E2.Vectors

a = 2i+ j− k b = 3i+ 4j− 2k c = 5i− j+ 2k

−→OA = 4i− 3j+ 4k

−−→OB = i+ 6j− 2k

−→OA.

−−→ON

−−→MG

−−→ON

−−→MG 63.6◦

[2]

[3]

[2]

[3]

O A

BCD E

FG

1 m

2 m3 m

1. (a) Find the first four terms in the expansion, in ascending powers of x, of Use this expansion to obtain an estimate of

(b) Find the constant term in the expansion of

2. (a) An arithmetic progression is such that the fifth term is three times the second term.

(i) Show that the sum of the first 8 terms is 4 times the sum of the first 4 terms. (ii) Given further that the sum of the fifth, sixth, seventh and eighth terms is 240, calculate the value of the first term.

(b) The first term of a geometric progression is a and the common ratio is r. Given that a + 96r = 0 and that the sum to infinity is 32, find the 8th term.

(c) The first and second terms of a geometric progression are 10 and 11 respectively. Find the least least number of terms such that their sum exceeds 8000.

P1.C7.E1.Series

�1− x

2

�7.

(0.95)7.�x3 − 1

2x2

�10

.

[4]

[3]

[3]

[3]

[3]

[3]

1. (a) Find the coefficient of x in the expansion

(b) Obtain the first 4 terms in the expansion of

2. Find the term independent of x in the expansion of

3. (a) The third term of a GP is and the sixth term is find the sum to infinity.

(b) If the sum of the first 9 terms of an A.P is 63 and the sum of the next four terms is -24, find the second term.

4. (a) Expand using binomial theorem or otherwise, and reduce the terms to their simplest form.

(b) Given that the first three terms in the expansion of are

Find the values of a and b.

P1.C7.E2.Series �x2 − 3

x

�5

.

(x− x2)7.

�1

x− 2x

�6

.

−1

3− 1

81,

�2− x

2

�5

�2− x

2

�5(1 + ax) 32− 16x+ bx2.

[3]

[3]

[3]

[3]

[3]

[3]

[3]

1. A curve is such that! ! ! Given that it passes through (1,3), find the equation.

2. Find the area bounded by the y-axis, the lines y = 1 and y = 2 and the curve

3. The diagram shows part of the curve The straight line cuts the line y = 1 at A(3,1) and the curve at B(2,4). The line y = 1 cuts the curve at C(1,1). Calculate

(i) the area of the shaded region,(ii)the volume generated when this region is rotated through about the x-axis.

4. Find the equation of the curve which passes through the point (3,6), and for which

P1.C8_C9.E1.Differentiation and Integration

dy

dx= 3x(x− 2).

x =8

y2.

y = x2. y + 3x = 10

360◦

dy

dx= x(x− 2).

[3]

[3]

[3]

[4]

[3]

y + 3x = 10

x

y

y = x2

y = 1C(1, 1)

B(2, 4)

A(3, 1)

1. The gradient at any point on a particular curve is given by the expression where x > 0.

Given that the curve passes through the point P(4,18), find the equation of the normal to the curve at P.! ! ! ! ! ! ! ! ! ! ! [3]

2. The diagram shows part of the curve . Find

(i) the area of the shaded region A,(ii)the volume obtained when the shaded region B is rotated through about the x-axis.

3. Find the point on the curve where the gradient of the tangent to the curve at that point is parallel to the x-axis.

4. The diagram shows a triangle ABC with angle ABC = and

(i) Show that the area of triangle [2]

(ii) By completing the square, find the greatest possible area of the triangle ABC as x varies, and the corresponding value of x. [4]

P1.C8_C9.E2.Differentiation and Integration

x2 +16

x2,

y =12

(x+ 3)2

360◦

y = (2 + x)(6− x),

30◦, AC = 4− x BC = 2 + x.

ABC =1

4(8 + 2x− x2).

[3][4]

[3]

x

y

A

B C

4− x

2 + x

30◦

A B

y =12

(x+ 3)2

1 20

1. Find the coordinates of the stationary points on ! ! and determine their nature. [5]

State the range of values of k for which y = k intersects the curve in three distinct points [2]

2. (a) The points A and B lie on the curve ! ! and have coordinates (0,5) and (3,2) respectively. Find the area of the region enclosed by the curve and the line AB. ! ! ! [5]

(b) Find the volume of the solid formed when the finite region bounded by the curve and the x-axis is rotated through around the x-axis. ! ! ! ! ! [4]

3. A piece of wire 10m long is to be cut into three pieces which are made into two squares and a circle. The length of the side of one of the squares, is twice the length of the side of the other square.

Given that the side of the smaller square is x,

(a) Find the circumference of the circle in terms of x [1](b) Find the radius of the circle in terms of x [1](c) Show that the total area enclosed by the three shapes is given by

! ! ! ! ! ! ! ! ! ! [3]

(d) Show that for a maximum area ! ! ! ! ! [4]

(e) Hence find the maximum area correct to 3 decimal places. [2]

P1.C8_C9.E3.Differentiation and Integration

y = 2x2(x− 3)

y = 5 + 2x− x2

y = 5x− x2

360◦

A =

�5 +

36

π

�x2 − 60

πx+

25

π

x =30

5π + 36

1. Find the equation of the normal to the curve ! ! ! at the point (-1,8).! ! [4]

2. Show that the curves ! and ! ! ! have exactly one point in common, and find the gradient of each curve at this point. ! ! ! ! ! ! ! ! ! ! ! ! [6]

3. Find the coordinates of the turning points on the curve stating the nature of each turning point.! ! ! ! ! ! ! ! ! ! ! ! [5]

4. Find the coordinates of the turning points of curve C, defined by

Draw a sketch graph of C.! ! ! ! ! ! ! ! ! [9]

P1.C8_C9.E4.Differentiation and Integration

y = −3x2 + 5

y = x3 y = (x+ 1)(x2 + 4)

y = x3 − 2x2 − 4x+ 5,

y = x3 − x.

1. The diagram shows a closed cardboard box in the shape of a cuboid with base 3x cm by x cm and height h cm. The total surface area of the cardboard (i.e. the outside of the box) is

(a) By first expressing h in terms of x, show that the volume of the box, V , is given by

! ! ! ! ! ! ! ! ! ! ! [4]

(b) Given that x can vary, find the maximum value of V, explaining why it is a maximum and not a minimum. ! ! ! ! ! ! ! ! ! ! ! [6]

2. Let ! ! Write down ! and hence find an expression for the approximate small change

in y which x changes by a small amount . Use your result to estimate the cube roots of 1001. [6]

3. The formulae for the volume of a sphere of radius r and for its surface area are ! ! and

! ! respectively. Given that, when r = 5m, V is increasing at rate ! ! find the rate of

increase if A of this instant. ! ! ! ! ! ! ! ! ! ! ! ! [6]

!

P1.C8_C9.E5.Differentiation and Integration

1152 cm2

cm2

V = 9x

�48− 1

4x2

�.

y = 3√x.

dy

dx,

δx

V =4

3πr3

A = 4πr2 10m3s−1,

h cmx cm

3x cm