Leave blank

2

*P43164A0232*

1. The curve C has equation y = f (x) where

f ( ) ,x xx

x= +−

4 12

2>

(a) Show that

f (x) =−−92 2( )x

(3)

Given that P is a point on C such that f (x) = –1,

(b) find the coordinates of P.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMTC3 papers from 2014and 2013

C3 PAPER JUNE 2014

Leave blank

4

*P43164A0432*

2. Find the exact solutions, in their simplest form, to the equations

(a) 2 ln (2x + 1) – 10 = 0(2)

(b) 3xe4x = e7

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMT

Leave blank

6

*P43164A0632*

3. The curve C has equation x = 8y tan 2y

The point P has coordinates ,8ππ⎛ ⎞⎜ ⎟⎝ ⎠

(a) Verify that P lies on C.(1)

(b) Find the equation of the tangent to C at P in the form ay = x + b, where the constants a and b are to be found in terms of .

(7)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMT

Leave blank

8

*P43164A0832*

4.

Figure 1

Figure 1 shows part of the graph with equation y = f (x), x .

The graph consists of two line segments that meet at the point Q (6, –1).

The graph crosses the y-axis at the point P(0, 11).

Sketch, on separate diagrams, the graphs of

(a) y = | f (x) |(2)

(b) y = 2f (– x) + 3(3)

On each diagram, show the coordinates of the points corresponding to P and Q.

Given that f (x) = a |x – b | – 1, where a and b are constants,

(c) state the value of a and the value of b.(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

x

y

P(0, 11)

Q(6, –1)O

y = f(x)

PMT

Leave blank

12

*P43164A01232*

5.g( ) ( ) ,x x

x

x

x xx=

++ +

+ −33 2 1

632

(a) Show that g( ) ,x xx

x= +−12

3(4)

(b) Find the range of g.(2)

(c) Find the exact value of a for which g(a) = g–1(a).(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMT

Leave blank

16

*P43164A01632*

6.

Figure 2

Figure 2 shows a sketch of part of the curve with equation

y x x x= ⎛⎝⎜

⎞⎠⎟

+ − −2 12

3 22 3cos

The curve crosses the x-axis at the point Q and has a minimum turning point at R.

(a) Show that the x coordinate of Q lies between 2.1 and 2.2(2)

(b) Show that the x coordinate of R is a solution of the equation

x x x= + ⎛⎝⎜

⎞⎠⎟

1 23

12

2sin

(4)

Using the iterative formula

x x x xn n n+ = +⎛⎝⎜

⎞⎠⎟

=12

0123

12

1 3sin , .

(c) find the values of x1 and x2 to 3 decimal places.(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Q

y

x

R

O

PMT

Leave blank

20

*P43164A02032*

7. (a) Show thatcosec 2x + cot 2x = cot x, x 90n°, n

(5)

(b) Hence, or otherwise, solve, for 0 < 180°,

cosec (4 + 10°) + cot (4 + 10°) = 3

You must show your working.

(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMT

Leave blank

24

*P43164A02432*

8. A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation

P t tt

t= +8001 3

00 1

0 1ee

.

., ,� R∈

(a) Calculate the number of primroses at the start of the study.(2)

(b) Find the exact value of t when P = 250, giving your answer in the form a ln(b) where a and b are integers.

(4)

(c) Find the exact value of d

d

Pt

when t =10. Give your answer in its simplest form.(4)

(d) Explain why the population of primroses can never be 270(1)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PMT

Leave blank

28

*P43164A02832*

9. (a) Express 2 sin – 4 cos in the form R sin( ), where R and are constants, R 0

and 02π

<

Leave blank

2

*P43163A0228*

1. Express

32 3

12 3

64 92x x x+

−−

+−

as a single fraction in its simplest form.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

C3 JUNE 2014(R) PAPER

Leave blank

4

*P43163A0428*

2. A curve C has equation y = e4x + x4 + 8x + 5

(a) Show that the x coordinate of any turning point of C satisfies the equation

x3 = –2 – e4x(3)

(b) On the axes given on page 5, sketch, on a single diagram, the curves with equations

(i) y = x3,

(ii) y = –2 – e4x

On your diagram give the coordinates of the points where each curve crosses the y-axis and state the equation of any asymptotes.

(4)

(c) Explain how your diagram illustrates that the equation x3 = –2 – e4x has only one root.(1)

The iteration formula

x xnxn

+ = − − = −1 02 113( ) ,e4

can be used to find an approximate value for this root.

(d) Calculate the values of x1 and x2, giving your answers to 5 decimal places.(2)

(e) Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve C.

(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

5

*P43163A0528* Turn over

Question 2 continued

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

y

O x

Leave blank

8

*P43163A0828*

3. (i) (a) Show that 2 tan x – cot x = 5 cosec x may be written in the form

a cos2 x + b cos x + c = 0

stating the values of the constants a, b and c.(4)

(b) Hence solve, for 0 x 2 , the equation

2 tan x – cot x = 5 cosec x

giving your answers to 3 significant figures.(4)

(ii) Show that

tan + cot cosec 2 , ,2nπθ n≠ ∈Z

stating the value of the constant .(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

12

*P43163A01228*

4. (i) Given that

x = sec2 2y, 0 < y < 4π

show that

ddyx x x

=−

14 1( )

(4) (ii) Given that

y = (x2 + x3) ln 2x

find the exact value of dd

at eyx

x =2

, giving your answer in its simplest form.(5)

(iii) Given that

f ( ) cos( )

,x xx

x=+

≠ −31

113

show that

f g'( ) ( )( )

,x xx

x=+

≠ −1

143

where g(x) is an expression to be found.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

16

*P43163A01628*

5. (a) Sketch the graph with equation

y = | 4x – 3|

stating the coordinates of any points where the graph cuts or meets the axes.(2)

Find the complete set of values of x for which

(b)| 4x – 3| 2 – 2x

(4)

(c)| 4x – 3| 3

2 – 2x

(2)

Leave blank

20

*P43163A02028*

6. The function f is defined by

f : x e2x + k2, x , k is a positive constant.

(a) State the range of f.(1)

(b) Find f –1 and state its domain.(3)

The function g is defined by

g : x ln(2x), x 0

(c) Solve the equation g(x) + g(x2) + g(x3) = 6

giving your answer in its simplest form.(4)

(d) Find fg(x), giving your answer in its simplest form.(2)

(e) Find, in terms of the constant k, the solution of the equation

fg(x) = 2k2(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

24

*P43163A02428*

7.

Figure 1

Figure 1 shows the curve C, with equation y = 6 cos x + 2.5 sin x for 0 x 2

(a) Express 6 cos x + 2.5 sin x in the form R cos(x – ), where R and are constants

with R 0 and 0 2π . Give your value of to 3 decimal places.

(3)

(b) Find the coordinates of the points on the graph where the curve C crosses the coordinate axes.

(3) A student records the number of hours of daylight each Sunday throughout the year. She

starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later.

She models her results with the continuous function given by

2 212 6cos 2.5sin , 0 5252 52πt πtH t⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ � �

where H is the number of hours of daylight and t is the number of weeks since her first recording.

Use this function to find

(c) the maximum and minimum values of H predicted by the model,(3)

(d) the values for t when H = 16, giving your answers to the nearest whole number.

[You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.]

(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

x

y

O 2

C

Leave blank

2

*P43016A0232*

1. Given that

3 2 5 44 4

24 3 2

22

2x x x

xax bx c dx e

xx− − −

−≡ + + + +

−≠ ±,

find the values of the constants a, b, c, d and e.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2013C3 JUNE 2013 PAPER

Leave blank

4

*P43016A0432*

2. Given thatf(x) = ln x, x > 0

sketch on separate axes the graphs of

(i) y = f(x),

(ii) y = | f(x) |,

(iii) y = –f(x – 4).

Show, on each diagram, the point where the graph meets or crosses the x-axis. In each case, state the equation of the asymptote.

(7)

physicsandmathstutor.com June 2013

Leave blank

8

*P43016A0832*

3. Given that

2cos(x + 50)� = sin(x + 40)�

(a) Show, without using a calculator, that

tan x� = 13

tan ���(4)

(b) Hence solve, for 0 � � < 360,

2cos(2� + 50)� = sin(2� + 40)�

giving your answers to 1 decimal place.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2013

Leave blank

12

*P43016A01232*

4. f(x) = 25x2e2x – 16, x ���

(a) Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x).

(5)

(b) Show that the equation f(x) = 0 can be written as x = ±45

e–x(1)

The equation f(x) = 0 has a root �, where � = 0.5 to 1 decimal place.

(c) Starting with x0 = 0.5, use the iteration formula

xn+1 = 45

e–xn

to calculate the values of x1, x2 and x3, giving your answers to 3 decimal places. (3)

(d) Give an accurate estimate for � to 2 decimal places, and justify your answer.(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2013

Leave blank

16

*P43016A01632*

5. Given that

x = sec2 3y, 0 < y < 6π

(a) find dd

xy

in terms of y.(2)

(b) Hence show that

ddyx x x

=−

1

6 112( )

(4)

(c) Find an expression for dd

2

2y

x in terms of x. Give your answer in its simplest form.

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2013

Leave blank

20

*P43016A02032*

6. Find algebraically the exact solutions to the equations

(a) ln(4 – 2x) + ln(9 – 3x) = 2ln(x + 1), –1 < x < 2(5)

(b) 2x e3x+1 = 10

Give your answer to (b) in the form a bc d

++

lnln

where a, b, c and d are integers.(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

physicsandmathstutor.com June 2013

Leave blank

24

*P43016A02432*

7. The function f has domain –2 � x � 6 and is linear from (–2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1.

Figure 1

(a) Write down the range of f.(1)

(b) Find ff(0).(2)

The function g is defined by

g : x � 4 35

+−

xx

, x ���, x����

(c) Find g–1(x)(3)

(d) Solve the equation gf(x) = 16(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

y

10

2–2 6O x

physicsandmathstutor.com June 2013

Leave blank

28

*P43016A02832*

8.

Figure 2

Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s–1.

Kate is 24 m ahead of John when she starts to cross the road from the fixed point A. John passes her as she reaches the other side of the road at a variable point B, as shown in

Figure 2. Kate’s speed is V m s–1 and she moves in a straight line, which makes an angle �,

0 < � < 150�, with the edge of the road, as shown in Figure 2.

You may assume that V is given by the formula

V = 21

24sin 7cos+θ θ, 0 < � < 150�

(a) Express 24sin� + 7cos� in the form Rcos(� – �), where R and � are constants and where R > 0 and 0 < � < 90�, giving the value of � to 2 decimal places.

(3)

Given that ��varies,

(b) find the minimum value of V.(2)

Given that Kate’s speed has the value found in part (b),

(c) find the distance AB.(3)

Given instead that Kate’s speed is 1.68 m s–1,

(d) find the two possible values of the angle �, given that 0 < ��< 150�.(6)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

3 m s–1

V m s–1

B

�A

24 m

7 m

physicsandmathstutor.com June 2013

physicsandmathstutor.com June 2013 (R)

C3 JUNE 2013 (R) PAPER

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

physicsandmathstutor.com June 2013 (R)

Leaveblank

2

*P41826A0232*

1. g( ) ,x xx x

x= ++ +

−6 123 2

2 02

(a) Show that g( ) ,x xx

x= −+

4 21

0(3)

(b)

Figure 1

Figure 1 shows a sketch of the curve with equation y = g(x), x 0

The curve meets the y-axis at (0, 4) and crosses the x-axis at (2, 0).

On separate diagrams sketch the graph with equation

(i) y = 2g(2x),

(ii) y = g–1(x).

Show on each sketch the coordinates of each point at which the graph meets or crosses theaxes.

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

(0, 4)

O (2, 0) x

y

PhysicsAndMathsTutor.com June 2013 (Withdrawn)C3 JUNE 2013 (W/D) PAPER

Leaveblank

6

*P41826A0632*

2. Given that tan 40° = p, find in terms of p

(a) cot 40°(1)

(b) sec 40°(2)

(c) tan 85°(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

8

*P41826A0832*

3.

Figure 2

Figure 2 shows a sketch of the graph with equation y = 2|x | – 5.

The graph intersects the positive x-axis at the point P and the negative y-axis at thepoint Q.

(a) State the coordinates of P and the coordinates of Q.(2)

(b) Solve the equation

2|x | – 5 = 3 – x(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

y

xP

Q

O

y = 2|x | – 5

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

10

*P41826A01032*

4. (a) On the same diagram, sketch and clearly label the graphs with equations

y = e x and y = 10 – x

Show on your sketch the coordinates of each point at which the graphs cut the axes.(3)

(b) Explain why the equation e x – 10 + x = 0 has only one solution.(1)

(c) Show that the solution of the equation

e x – 10 + x = 0

lies between x = 2 and x = 3(2)

(d) Use the iterative formula

xn + 1= ln(10 – xn), x1 = 2

to calculate the values of x2, x3 and x4.

Give your answers to 4 decimal places.(3)

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

14

*P41826A01432*

5. (i) (a) Show thatddx x x

x122

1ln ln

= +

√ √x x

(3)

The curve with equation y x x x= >12 0ln , has one turning point at the point P.

(b) Find the exact coordinates of P. Give your answer in its simplest form.(4)

(ii) A curve C has equation y x kx k

= −+

, where k is a positive constant.

Findddyx, and show that C has no turning points.

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

18

*P41826A01832*

6.

Figure 3

Figure 3 shows a sketch of the graph of y = f (x) where

fe

( ),

,x

x xxx

=−

− >

−

5 2

4

442 8

(a) State the range of f (x).

(1)

(b) Determine the exact value of ff (0).

(2)

(c) Solve f (x) = 21

Give each answer as an exact answer.(5)

(d) Explain why the function f does not have an inverse.(1)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

y

x4

O

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

22

*P41826A02232*

7. (a) Prove that

cos 1 sin 2sec , (2 1) ,1 sin cos 2

x x πx x n nx x

−+ = ≠ + ∈−

(4)

(b) Hence find, for 0 ,4πx< < the exact solution of

cossin

sincos

sinxx

xx

x1

1 8−

+ − =

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

26

*P41826A02632*

8. (a) Express 9cos θ – 2sin θ in the form Rcos(θ + α), where R > 0 and 0 .2πα< <

Give the exact value of R and give the value of α to 4 decimal places.(3)

(b) (i) State the maximum value of 9cos θ – 2sin θ

(ii) Find the value of θ, for 0 < θ < 2π, at which this maximum occurs.(3)

Ruth models the height H above the ground of apassenger on a Ferris wheel by the equation

10 9cos 2sin5 5πt πtH = − +

where H is measured in metres and t is the timein minutes after the wheel starts turning.

(c) Calculate the maximum value of H predicted by this model, and the value of t, whenthis maximum first occurs. Give your answers to 2 decimal places.

(4)

(d) Determine the time for the Ferris wheel to complete two revolutions.(2)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

H

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leaveblank

30

*P41826A03032*

9.

Figure 4

Figure 4 shows a sketch of the curve with equation x = (9 + 16y – 2y2)12 .

The curve crosses the x-axis at the point A.

(a) State the coordinates of A.(1)

(b) Find an expression for ddxy, in terms of y.

(3)

(c) Find an equation of the tangent to the curve at A.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

y

xAO

PhysicsAndMathsTutor.com June 2013 (Withdrawn)

Leave blank

2

*P41486A0228*

1. The curve C has equation

y x= −( )2 3 5

The point P lies on C and has coordinates (w, – 32).

Find

(a) the value of w,(2)

(b) the equation of the tangent to C at the point P in the form y mx c= + , where m and c are constants.

(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

C3 JANUARY 2013 PAPER

Leave blank

4

*P41486A0428*

2. g e( )x xx= + −−1 6

(a) Show that the equation g( )x = 0 can be written as

x x= − +ln( )6 1, x < 6(2)

The root of g( )x = 0 is �.

The iterative formula

x xn n+ = − +1 6 1ln( ) , x0 2=

is used to find an approximate value for �.

(b) Calculate the values of x1 , x2 and x3 to 4 decimal places. (3)

(c) By choosing a suitable interval, show that � = 2.307 correct to 3 decimal places.(3)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

6

*P41486A0628*

3. y

x

y = f (x)

Q(0, 2)

P(–3,0) O

Figure 1

Figure 1 shows part of the curve with equation y x= f ( ) , x ∈ �.

The curve passes through the points Q( , )0 2 and P( , )−3 0 as shown.

(a) Find the value of ff ( )−3 .(2)

On separate diagrams, sketch the curve with equation

(b) y x= −f 1( ) ,(2)

(c) y x= −f ( ) 2,(2)

(d) y x= ( )2 12f .(3)

Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

10

*P41486A01028*

4. (a) Express 6 cos���+�8�sin�� in the form R�cos(�����), where R > 0 and 0 2< <πα .

Give the value of ��to 3 decimal places.(4)

(b) p( ) cos sin� � �= + +4

12 6 8 , 0 2� �θ π

Calculate

(i) the maximum value of p(�),

(ii) the value of � at which the maximum occurs.(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

14

*P41486A01428*

5. (i) Differentiate with respect to x

(a) y x x= 3 2ln

(b) y x x= +( sin )2 3(6)

Given that x y= cot ,

(ii) show that ddyx x

= −+

11 2 (5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

18

*P41486A01828*

6. (i) Without using a calculator, find the exact value of

(sin 22.5° + cos 22.5°)2

You must show each stage of your working.(5)

(ii) (a) Show that cos 2� + sin ��= 1 may be written in the form

k sin2 � – sin ��= 0, stating the value of k.(2)

(b) Hence solve, for 0 ����< 360°, the equation

cos 2� + sin ��= 1(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

22

*P41486A02228*

7. h( )( )( )

x x x x x= + + +

−+ +

22

45

185 22 2

, x� 0

(a) Show that h( )x xx

=+

252 (4)

(b) Hence, or otherwise, find ′h ( )x in its simplest form.(3)

y

O x

y = h(x)

Figure 2

Figure 2 shows a graph of the curve with equation y x= h( ) .

(c) Calculate the range of h( )x .(5)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

Leave blank

26

*P41486A02628*

8. The value of Bob’s car can be calculated from the formula

V t t= + +− −17000 2000 5000 25 0 5e e. .

where V is the value of the car in pounds (£) and t is the age in years.

(a) Find the value of the car when t = 0(1)

(b) Calculate the exact value of t when V = 9500(4)

(c) Find the rate at which the value of the car is decreasing at the instant when t = 8. Give your answer in pounds per year to the nearest pound.

(4)

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice