Core 3 Revision Pack

8/10/2019 Core 3 Revision Pack

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Mathematics A2 Level

Core 3 Revision Pack

Contents

CORE 3 FORMULA SHEET 2

JANUARY 2006 3

JUN 2006 5

JANUARY 2007 8

JUN 2007 11

JANUARY 2008 14

JUN 2008 18

JANUARY 2009 22

JUN 2009 25

JAN 2010 28

JUN 2010 31

JAN 2011 35

JUN 2011 40

JAN 2012 43


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Core 3 Formula Sheet

Core Mathematics C3

Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and

C2.

Logarithms and exponentials

xax a=lne

Trigonometric identities

BABABA sincoscossin)(sin =

BABABA sinsincoscos)(cos m=

))((tantan1

tantan)(tan

21 +

= kBA

BA

BABA

m

2cos

2sin2sinsin

BABABA

+=+

2sin

2cos2sinsin

BABABA

+=

2cos

2cos2coscos

BABABA

+=+

2sin

2sin2coscos

BABABA

+=

Differentiation

f(x) f(x)

tan kx ksec2kx

secx secx tanx

cotx cosec2x

cosecx cosecxcotx


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January 2006

Time: 1 hour 30 minutes

1. Figure 1

Figure 1 shows the graph ofy= f(x), 5 x5.

The pointM(2, 4) is the maximum turning point of the graph.

Sketch, on separate diagrams, the graphs of

(a) y= f(x) + 3, (2)

(b) y= f(x), (2)(c) y= f(x). (3)

Show on each graph the coordinates of any maximum turning points.

2. Express

)2)(32(

32 2

++xx

xx

2

62 xx

as a single fraction in its simplest form. (7)

3. The point P lies on the curve with equation y =

x

3

1ln . The x-coordinate of P is 3.

Find an equation of the normal to the curve at the point Pin the formy= ax+ b, where aand bare constants.

(5)

4. (a) Differentiate with respect to x

(i) x2e

3x+ 2, (4)

(ii)x

x

3

)2(cos3

. (4)

(b) Given thatx= 4 sin (2y+ 6), findx

y

d

din terms ofx. (5)

y

x5O

M 2

5


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5. f(x) = 2x3x 4.

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

x=

+2

12

x. (3)

The equation 2x3x 4 = 0 has a root between 1.35 and 1.4.

(b) Use the iteration formula

xn+ 1=

+

2

12

nx,

withx0= 1.35, to find, to 2 decimal places, the value ofx1,x2andx3. (3)

The only real root of f(x) = 0 is .

(c) By choosing a suitable interval, prove that = 1.392, to 3 decimal places. (3)

6. f(x) = 12 cosx 4 sinx.

Given that f(x) =Rcos (x+ ), whereR0 and 0 90,(a) find the value ofRand the value of . (4)

(b) Hence solve the equation

12 cosx 4 sinx= 7

for 0 x< 360, giving your answers to one decimal place. (5)(c) (i) Write down the minimum value of 12 cosx 4 sinx. (1)

(ii) Find, to 2 decimal places, the smallest positive value ofxfor which this minimum value occurs.

(2)

7.(a) Show that

(i)xx

x

sincos

2cos

+cosx sinx, x(n

41 ), n, (2)

(ii)21 (cos 2x sin 2x) cos2x cosxsinx

21 . (3)

(b) Hence, or otherwise, show that the equation

cos 2

1

sincos

2cos=

+

can be written as

sin 2= cos 2. (3)

(c) Solve, for 0 < 2, sin 2= cos 2,

giving your answers in terms of . (4)

8. The functions f and g are defined by

f :xa 2x+ ln 2, x,

g :xa e2x

, x.(a) Prove that the composite function gf is

gf :xa 4e4x

, x. (4)(b) Sketch the curve with equationy= gf(x), and show the coordinates of the point where the curve cuts the

y-axis. (1)

(c) Write down the range of gf. (1)

(d) Find the value ofxfor whichxdd [gf(x)] = 3, giving your answer to 3 significant figures.

(4)


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4. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, TC, tminutes after it enters the liquid, isgiven by

T= 400e0.05t+ 25, t0.

(a) Find the temperature of the ball as it enters the liquid.

(1)

(b) Find the value of tfor which T= 300, giving your answer to 3 significant figures.

(4)

(c) Find the rate at which the temperature of the ball is decreasing at the instant when t= 50. Give your answer in Cper minute to 3 significant figures.

(3)

(d) From the equation for temperature Tin terms of t, given above, explain why the temperature of the ball can never

fall to 25 C.(1)

5. Figure 2

Figure 2 shows part of the curve with equation

y= (2x 1) tan 2x, 0 x 1, the functions f and g are defined by

f:xa ln (x+ k), x> k,

g:xa 2x k, x.

(a) On separate axes, sketch the graph of f and the graph of g.

On each sketch state, in terms of k, the coordinates of points where the graph meets the coordinate axes.

(5)

(b) Write down the range of f.

(1)

(c) Find fg

4

kin terms of k, giving your answer in its simplest form.

(2)

The curve Chas equationy= f(x). The tangent to C at the point withx-coordinate 3 is parallel to the line with equation 9y= 2x+ 1.

(d) Find the value of k.

(4)

8. (a) Given that cosA=43 , where 270


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January 2007

1. (a) By writing sin 3 as sin (2+ ), show that

sin 3 = 3 sin 4 sin3 .(5)

(b) Given that sin =4

3, find the exact value of sin 3.

(2)

2. f(x) = 1 2

3

+x+

2)2(

3

+x, x2.

(a) Show that f(x) =2

2

)2(

1

+++

x

xx, x2.

(4)

(b) Show thatx2+x+ 1 > 0 for all values ofx.

(3)

(c) Show that f(x) > 0 for all values ofx,x2.(1)

3. The curve Chas equationx= 2 siny.

(a) Show that the point P

4

,2

lies on C.

(1)

(b) Show thatx

y

d

d=

2

1

at P.

(4)

(c) Find an equation of the normal to Cat P. Give your answer in the formy= mx+ c, where mand

care exact constants.

(4)

4. (i) The curve Chas equationy=29 x

x

+.

Use calculus to find the coordinates of the turning points of C.

(6)

(ii) Given thaty= 23

)1( 2xe+ , find the value ofx

y

d

datx=

21 ln 3.

(5)


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5. Figure 1

Figure 1 shows an oscilloscope screen.

The curve on the screen satisfies the equationy= 3 cosx+ sinx.(a) Express the equation of the curve in the formy=Rsin (x+ ), whereRand are constants,R>

0 and 0 <

21 .

(a) Show that f (x) =12

64

x

x.

(7)

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

3. A curve C has equationy =x2ex.

(a) Findx

y

d

d, using the product rule for differentiation.

(3)

(b) Hence find the coordinates of the turning points of C.

(3)

(c) Find 2

2

d

d

x

y.

(2)

(d) Determine the nature of each turning point of the curve C.

(2)

4. f(x) = x3

+ 3x2

1.

(a) Show that the equation f(x) = 0 can be rewritten as

x=

x31

.

(2)

(b) Starting withx1= 0.6, use the iteration

xn+ 1=

nx31

to calculate the values ofx2,x3andx4, giving all your answers to 4 decimal places.

(2)

(c) Show thatx= 0.653 is a root of f(x) = 0 correct to 3 decimal places.

(3)


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P40084A 12


f:a ln (2x 1), x, x>21 ,

g:a 3

2

x, x, x3.

(a) Find the exact value of fg(4).

(2)

(b) Find the inverse function f1

(x), stating its domain.

(4)

(c) Sketch the graph of y= |g(x)|. Indicate clearly the equation of the vertical asymptote and thecoordinates of the point at which the graph crosses they-axis.

(3)

(d) Find the exact values ofx for which3

2x

= 3.

(3)

6. (a) Express 3 sinx + 2 cosx in the formR sin(x + ) whereR > 0 and 0 < 2,x.

(a) Show that there is a root of f(x) = 0 in the interval 2


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P40084A 15

4.

Figure 1

Figure 1 shows a sketch of the curve with equationy = f (x).

The curve passes through the origin O and the pointsA(5, 4) andB(5, 4).

In separate diagrams, sketch the graph with equation

(a) y =f (x),

(3)

(b) y = f (x) ,(3)

(c) y = 2f(x + 1) .

(4)

On each sketch, show the coordinates of the points corresponding toA andB.


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P40084A 16

5. The radioactive decay of a substance is given by

R = 1000ect

, t 0.

whereR is the number of atoms at time t years and c is a positive constant.

(a) Find the number of atoms when the substance started to decay.

(1)

It takes 5730 years for half of the substance to decay.

(b) Find the value of c to 3 significant figures.

(4)

(c) Calculate the number of atoms that will be left when t = 22 920 .

(2)

(d) Sketch the graph ofR against t .(2)

6. (a) Use the double angle formulae and the identity

cos(A +B) cosA cosB sinA sinB

to obtain an expression for cos 3x in terms of powers of cosx only.

(4)

(b) (i) Prove that

x

x

sin1

cos

++

x

x

cos

sin1+2 secx, x(2n+ 1)

2

.

(4)

(ii) Hence find, for 0


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P40084A 17

7. A curve C has equation

y = 3 sin 2x + 4 cos 2x, x.

The pointA(0, 4) lies on C.

(a) Find an equation of the normal to the curve C atA.

(5)

(b) Expressy in the formR sin(2x + ), whereR > 0 and 0 < 0, x .

(a) Find the inverse function f1

.

(2)

(b) Show that the composite function gf is

gf :xa3

3

21

18

x

x

.

(4)

(c) Solve gf (x) = 0.

(2)

(d) Use calculus to find the coordinates of the stationary point on the graph ofy = gf(x).

(5)

TOTAL FOR PAPER: 75 MARKS


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P40084A 18

Jun 2008

1. The point P lies on the curve with equation

y= 4e2x+ 1

.

They-coordinate of P is 8.

(a) Find, in terms of ln 2, thex-coordinate of P.

(2)

(b) Find the equation of the tangent to the curve at the point P in the formy = ax + b, where a and b

are exact constants to be found.

(4)

2. f(x) = 5 cosx+ 12 sinx.

Given that f(x) =Rcos (x ), whereR > 0 and 0 < 3.

(a) Show that f(x) =1

1

+x, x> 3.

(4)

(b) Find the range of f.

(2)

(c) Find f1

(x). State the domain of this inverse function.

(3)

The function g is defined by

g:xa 2x2 3, x.

(d) Solve fg(x) =8

1.

(3)

5. (a) Given that sin2+ cos

21, show that 1 + cot2cosec2.

(2)

(b) Solve, for 0 < 180, the equation

2 cot2 9 cosec = 3,

giving your answers to 1 decimal place.

(6)

6. (a) Differentiate with respect tox,

(i) e3x

(sinx+ 2 cosx),

(3)

(ii) x3ln (5x+ 2).

(3)

Given thaty=2

2

)1(

763

++

x

xx, x1,

(b) show thatx

y

d

d=

3)1(

20

+x.

(5)

(c) Hence find2

2

d

d

x

yand the real values ofx for which

2

2

d

d

x

y=

4

15.

(3)


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P40084A 21

7. f(x) = 3x3 2x 6.

(a) Show that f (x) = 0 has a root, , betweenx = 1.4 andx = 1.45.

(2)

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

x=

+3

22

x, x0.

(3)

(c) Starting with 0x = 1.43, use the iteration

xn+ 1=

+

3

22

nx

to calculate the values of 1x , 2x and 3x , giving your answers to 4 decimal places.

(3)

(d) By choosing a suitable interval, show that = 1.435 is correct to 3 decimal places.

(3)



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P40084A 22

January 2009

1. (a) Find the value ofx

y

d

dat the point wherex = 2 on the curve with equation

y =x2(5x 1).

(6)

(b) Differentiate2

2sin

x

xwith respect tox.

(4)

2. f(x) =32

222

+xx

x

3

1

+

x

x.

(a) Express f(x) as a single fraction in its simplest form.

(4)

(b) Hence show that f(x) =2)3(

2

x.

(3)

3.

Figure 1

Figure 1 shows the graph of y = f (x), 1


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P40084A 23

4. Find the equation of the tangent to the curvex = cos (2y+) at

4,0

.

Give your answer in the formy = ax + b, where a and b are constants to be found.

(6)


f :xa 3x+ lnx, x> 0, x,

g :xa 2

ex , x.

(a) Write down the range of g.

(1)

(b) Show that the composite function fg is defined by

fg :xa x2+

2

3ex , x.(2)

(c) Write down the range of fg.

(1)

(d) Solve the equation [ ])(fgd

dx

x

=x(2

exx + 2).

(6)

6. (a) (i) By writing 3= (2+ ), show that

sin 3= 3 sin 4 sin3.

(4)

(ii) Hence, or otherwise, for 0 < 0 and

0 < < 90.

(4)

(b) Hence find the maximum value of 3 cos + 4 sin and the smallest positive value of for

which this maximum occurs.

(3)

The temperature, f(t), of a warehouse is modelled using the equation

f (t) = 10 + 3 cos (15t) + 4 sin (15t),

where t is the time in hours from midday and 0 t < 24.

(c) Calculate the minimum temperature of the warehouse as given by this model.

(2)

(d) Find the value of t when this minimum temperature occurs.(3)



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P40084A 25

Jun 2009

1.

Figure 1

Figure 1 shows part of the curve with equation y = x3+ 2x

2+ 2, which intersects the x-axis at the

pointA wherex = .

To find an approximation to , the iterative formula

xn+ 1= 2)(

2

nx+ 2

is used.

(a) Takingx0= 2.5, find the values ofx

1,x

2,x

3andx

4.

Give your answers to 3 decimal places where appropriate. (3)

(b) Show that = 2.359 correct to 3 decimal places. (3)

2. (a) Use the identity cos2+ sin

2= 1 to prove that tan

2= sec

2 1. (2)

(b) Solve, for 0 < 360, the equation

2 tan2+ 4 sec + sec

2= 2.

(6)3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced

is modelled by the equation

P= t51e80 , t, t0.

(a) Write down the number of rabbits that were introduced to the island. (1)

(b) Find the number of years it would take for the number of rabbits to first exceed 1000.

(2)

(c) Findt

P

d

d. (2)

(d) Find P whent

P

d

d= 50. (3)


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P40084A 26

4. (i) Differentiate with respect tox

(a) x2cos 3x, (3)

(b)1

)1ln(2

2

++

x

x. (4)

(ii) A curve C has the equation

y = (4x+1), x > 41 , y > 0.

The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the

form ax + by + c = 0, where a, b and c are integers. (6)

5.

Figure 2

Figure 2 shows a sketch of part of the curve with equationy = f(x),x .

The curve meets the coordinate axes at the points A(0, 1 k) andB(21 ln k, 0), where k is a constant

and k >1, as shown in Figure 2.

On separate diagrams, sketch the curve with equation

(a) y = f(x), (3)(b) y = f

1(x). (2)

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the

axes.

Given that f(x) = e2x

k,

(c) state the range of f, (1)

(d) find f1

(x), (3)

(e) write down the domain of f1

. (1)


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P40084A 27

6. (a) Use the identity cos (A+B) = cosAcosB sinAsinB, to show that

cos 2A = 1 2 sin2A

(2)

The curves C1and C2have equations

C1:y = 3 sin 2x

C2:y = 4 sin2x 2 cos 2x

(b) Show that thex-coordinates of the points where C1and C2intersect satisfy the equation

4 cos 2x + 3 sin 2x = 2

(3)

(c) Express 4cos 2x + 3 sin 2x in the formR cos (2x ), whereR > 0 and 0 < < 90, giving the

value of to 2 decimal places.

(3)

(d) Hence find, for 0 x < 180, all the solutions of

4 cos 2x + 3 sin 2x = 2,

giving your answers to 1 decimal place.

(4)

7. The function f is defined by

f(x) = 1

)4(

2

+x+

)4)(2(

8

+

xx

x, x , x 4, x 2.

(a) Show that f (x) =2

3

x

x. (5)


g(x) =2e

3e

x

x

, x , x ln 2.

(b) Differentiate g(x) to show that g(x) =2)2(e

e

xx

. (3)

(c) Find the exact values ofx for which g(x) = 1 (4)

8. (a) Write down sin 2x in terms of sinx and cosx. (1)

(b) Find, for 0


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P40084A 28

Jan 2010

1. Express

33

12 +

x

x

13

1

+x

as a single fraction in its simplest form.

(4)

2. f(x) =x3+ 2x

2 3x 11

(a) Show that f(x) = 0 can be rearranged as

(2)

x=

++

2

113

x

x, x2.

The equation f(x) = 0 has one positive root .

The iterative formula xn+ 1=

++

2

113

n

n

x

x is used to find an approximation to .

(b) Taking 1x = 0, find, to 3 decimal places, the values of 2x , 3x and 4x .

(3)

(c) Show that = 2.057 correct to 3 decimal places.

(3)

3. (a) Express 5 cosx 3 sinx in the formR cos(x + ), whereR > 0 and 0 < 1.

(a) Find f1

and state its domain.

(4)

(b) Find fg and state its range.

(3)



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P40084A 31

Jun 2010

1. (a) Show that

2cos12sin+ = tan .

(2)

(b) Hence find, for 180 < 180, all the solutions of

2cos1

2sin2

+= 1.

Give your answers to 1 decimal place.

(3)

2. A curve C has equation

y=235

3

)( x, x

3

5.

The point P on C hasx-coordinate 2.

Find an equation of the normal to C at P in the form ax + by + c = 0, where a, b and c are integers.

(7)

3. f(x) = 4 cosecx 4x +1, wherex is in radians.

(a) Show that there is a root of f(x) = 0 in the interval [1.2, 1.3].

(2)

(b) Show that the equation f(x) = 0 can be written in the form

x=xsin

1+

4

1

(2)

(c) Use the iterative formula

xn+ 1=nxsin

1 +41 , x0= 1.25,

to calculate the values ofx1,x2andx3, giving your answers to 4 decimal places.

(3)

(d) By considering the change of sign of f(x) in a suitable interval, verify that = 1.291 correct to 3

decimal places.

(2)


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P40084A 33

6.

Figure 2

Figure 2 shows a sketch of the curve with the equationy = f(x),x.

The curve has a turning point atA(3, 4) and also passes through the point (0, 5).

(a) Write down the coordinates of the point to whichA is transformed on the curve with equation

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

(ii) y = 2f(21x).

(4)

(b) Sketch the curve with equationy = f(|x|).

On your sketch show the coordinates of all turning points and the coordinates of the point at

which the curve cuts they-axis.

(3)

The curve with equationy = f(x) is a translation of the curve with equationy =x2.

(c) Find f(x).

(2)

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

(1)


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P40084A 34

7. (a) Express 2 sin 1.5 cos in the formR sin ( ), whereR > 0 and 0 < 0 and 0 < 1,

(b) show that

f(x) =12

3

x.

(2)

(c) Hence differentiate f(x) and find f(2).(3)

3. Find all the solutions of

2 cos 2 = 1 2 sin

in the interval 0


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P40084A 36

4. Joan brings a cup of hot tea into a room and places the cup on a table. At time t minutes after Joan

places the cup on the table, the temperature, C, of the tea is modelled by the equation

= 20 +Aekt

,

whereA and k are positive constants.

Given that the initial temperature of the tea was 90 C,

(a) find the value ofA.

(2)

The tea takes 5 minutes to decrease in temperature from 90 C to 55 C.

(b) Show that k=

5

1ln 2.

(3)

(c) Find the rate at which the temperature of the tea is decreasing at the instant when t = 10. Give

your answer, in C per minute, to 3 decimal places.

(3)


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P40084A 37

5.

Figure 1

Figure 1 shows a sketch of part of the curve with equationy = f(x) , where

f(x) = (8 x)lnx, x > 0.

The curve cuts the x-axis at the points A andB and has a maximum turning point at Q, as shown in

Figure 1.

(a) Write down the coordinates ofA and the coordinates ofB.

(2)

(b) Find f(x).(3)

(c) Show that thex-coordinate of Q lies between 3.5 and 3.6

(2)

(d) Show that thex-coordinate of Q is the solution of

x=xln1

8

+.

(3)

To find an approximation for thex-coordinate of Q, the iteration formula

n

nx

xln1

81 +=+

is used.

(e) Taking 0x = 3.55, find the values of 1x , 2x and 3x .

Give your answers to 3 decimal places.

(3)


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P40084A 38


f:x a 5

23

x

x, x, x 5.

(a) Find f

1

(x).(3)

Figure 2

The function g has domain 1 x 8, and is linear from (1, 9) to (2, 0) and from (2, 0) to (8, 4).Figure 2 shows a sketch of the graph ofy = g(x).

(b) Write down the range of g.

(1)

(c) Find gg(2).

(2)

(d) Find fg(8).

(2)

(e) On separate diagrams, sketch the graph with equation

(i) y = g(x),

(ii) y = g1

(x).

Show on each sketch the coordinates of each point at which the graph meets or cuts the axes.

(4)

(f) State the domain of the inverse function g1

.

(1)


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P40084A 39

7. The curve C has equation

y=x

x

2cos2

2sin3

++

.

(a) Show that

x

y

d

d=

2)2cos2(

22cos42sin6

x

xx

+++

(4)

(b) Find an equation of the tangent to C at the point on C wherex =2

.

Write your answer in the formy = ax + b, where a and b are exact constants.

(4)

8. Given that

xd

d(cosx) = sinx,

(a) show thatxd

d(secx) = secxtanx.

(3)

Given that

x= sec 2y,

(b) findy

x

d

din terms ofy.

(2)

(c) Hence findx

y

d

din terms ofx.

(4)



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P40084A 41


f:xa 4 ln (x + 2), x , x 1.

(a) Find f1

(x).

(3)

(b) Find the domain of f1

.

(1)


g:xa2

ex 2, x .

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

(3)

(d) Find the range of fg.

(1)

5. The mass, m grams, of a leaf t days after it has been picked from a tree is given by

m =pekt

,

where k andp are positive constants.

When the leaf is picked from the tree, its mass is 7.5 grams and 4 days later its mass is 2.5 grams.

(a) Write down the value ofp.

(1)

(b) Show that k =4

1ln 3.

(4)

(c) Find the value of t whent

m

d

d= 0.6 ln 3.

(6)

6. (a) Prove that

2sin

2cos

2sin

1 = tan , 90n, n.

(4)

(b) Hence, or otherwise,

(i) show that tan 15= 2 3,(3)

(ii) solve, for 0


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7. f(x) =)3)(12(

54

+xx

x

9

22 xx

, x3, x2

1.

(a) Show that

f(x) = )3)(12(

5

++ xx .

(5)

The curve C has equationy = f (x). The point P

2

5,1 lies on C.

(b) Find an equation of the normal to C at P.

(8)

8. (a) Express 2 cos 3x 3 sin 3x in the formR cos (3x + ), whereR and are constants,R > 0 and 0

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