C3 Mock to June 10

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Figure 1

y

-2 1 O 1 2 3 x

(0.5,2) (0.4,4)

C3 Mock Paper

1. Express52

2

3

)23(7

)672(

3

x

x

xx

x

as a single fraction in its simplest form.

(4)

2. The function f is defined by f : x 2x, x. (a) Find f1(x) and state the domain of f1.(2)

The function g is defined by g : x 3x2 + 2, x.

(b) Find gf1(x). (c) State the range of gf1(x).(2) (1)

3. Find the exact solutions of (i) e2x+ 3 = 6, (ii) ln (3x+ 2) = 4.(3) (3)

4. Differentiate with respect to x (i) x3 e3x, (ii)x

x

cos

2, (iii) tan2x.

(3) (3) (2)

Given that x= cos y2, (iv) findx

y

d

din terms ofy.

(4)

5. (a) Using the formulae sin (A B) = sinAcos B cosAsin B,cos (A B) = cosAcos B sinAsin B, show that

(i) sin (A+ B) sin (AB) = 2 cosAsin B, (ii) cos (AB) cos (A+ B) = 2 sinAsin B.(2) (2)

(b) Use the above results to show that )cos()cos(

)sin()sin(

BABA

BABA

= cotA.

(3)Using the result of part (b) and the exact values of sin 60 and cos 60,

(c) find an exact value for cot 75in its simplest form.(4)

6. Figure1 shows a sketch of part of the curve with

equation y= f(x), x.

The curve has a minimum point at (0.5, 2) anda maximum point at (0.4, 4). The lines x= 1,the x-axis and the y-axis are asymptotes of thecurve, as shown in Fig. 1.

On a separate diagram sketch the graphs of

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

(4) (4) (4)

In each case show clearly

(i) the coordinates of any points at which the curve has a maximum or minimum point,

(ii) how the curve approaches the asymptotes of the curve.

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7. (a) Sketch the curve with equation y= ln x.(2)

(b) Show that the tangent to the curve with equation y= ln xat the point (e, 1)passes through theorigin.

(3)

(c) Use your sketch to explain why the liney

=mx

cuts the curvey

= lnx

between x= 1 and x= e if 0 < m 0,

where is an angle related to the voltage.

Given that I= Rsin (), where R> 0 and 0 < 360,

(a) find the value ofR, and the value ofto 1 decimal place.(4)

(b) Hence solve the equation 4 sin 3 cos = 3 to find the values ofbetween 0 and 360.(5)

(c) Write down the greatest value for I.(1)

(d) Find the value ofbetween 0 and 360 at which the greatest value ofIoccurs.(2)

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Core 3 June 05

1. (a) Given that sin2+ cos2 1, show that 1 + tan2 sec2.(2)

(b) Solve, for 0 < 360, the equation 2 tan2 + sec = 1, giving your answers to 1 decimal place.(6)

2. (a) Differentiate with respect to x; (i) 3 sin2x+ sec 2x, (ii) {x+ ln (2x)}3.(3) (3)

Given that y=2

2

)1(

9105

x

xx, x 1; (b) show that

x

y

d

d=

3)1(

8

x.

(6)

3. The function f is defined by f: x2

152

xx

x

2

3

x, x> 1.

(a) Show that f(x) =1

2

x, x> 1. (b) Find f1(x).

(4) (3)

The function g is defined by g: xx2 + 5, x. (b) Solve fg(x) = 4

1 .

(3)

4. f(x) = 3ex 21 ln x 2, x> 0. (a) Differentiate to find f(x).

(3)

The curve with equation y= f(x) has a turning point at P. The x-coordinate ofPis .

(b) Show that = 61 e.

(2)

The iterative formula xn+ 1 =nxe

61

, x0 = 1, is used to find an approximate value for .

(c) Calculate the values ofx1, x2, x3 and x4, giving your answers to 4 decimal places.(2)

(d) By considering the change of sign of f(x) in a suitable interval, prove that = 0.1443 correct to 4 decimalplaces.

(2)

5. (a) Using the identity cos (A+ B) cosAcos B sinAsin B, prove that cos 2A 1 2 sin2A.(2)

(b) Show that 2 sin 2 3 cos 2 3 sin + 3 sin (4 cos + 6 sin 3).(4)

(c) Express 4 cos + 6 sin in the form Rsin ( + ), where R> 0 and 0 < < 21 .

(4)

(d) Hence, for 0 < , solve 2 sin 2 = 3(cos 2 + sin 1),

giving your answers in radians to 3 significant figures, where appropriate.

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H31123A 4

(5)6. Figure 1

Figure 1 shows part of the graph ofy= f(x), x. The graph consists of two line segments that meet at the point(1, a), a< 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at (1, 0) and the y-axis at (0, b),b< 0.

In separate diagrams, sketch the graph with equation

(a) y= f(x+ 1),(2)

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

Indicate clearly on each sketch the coordinates of any points of intersection with the axes.

Given that f(x) = x 1 2, find

(c) the value ofaand the value ofb,

(2)

(d) the value ofxfor which f(x) = 5x.(4)

7. A particular species of orchid is being studied. The population pat time tyears after the study started is assumedto be

p=t

t

a

a2.0

0.2

e1

e2800

, where ais a constant.

Given that there were 300 orchids when the study started,

(a) show that a= 0.12,(3)

(b) use the equation with a= 0.12 to predict the number of years before the population of orchids reaches 1850.(4)

(c) Show that p=t2.0e12.0

336

.

(1)

(d) Hence show that the population cannot exceed 2800.(2)

O

y

x1 3

(1, a)

b

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H31123A 5

y

x5O

M(2, 4)

5

Core 3 Jan 06

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

The point M(2, 4) is the maximum turning point of thegraph.

Sketch, on separate diagrams, the graphs of

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

(c) y= f(x).(3)

Show on each graph the coordinates of any maximum turning points.Figure 1

2. Express)2)(32(

32 2

xx

xx

2

62 xx


(7)

3. The point Plies on the curve with equation y= ln

x

3

1.The x-coordinate ofPis 3.

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

(5)

4. (a) Differentiate with respect to x (i) x2e3x+ 2, (ii)x

x

3

)2(cos 3

.

(4) (4)

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

y

d

din terms ofx.

(5)

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,

with x0 = 1.35, to find, to 2 decimal places, the value ofx1, x2 and x3.(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)

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H31123A 6

6. f(x) = 12 cos x 4 sin x. Given that f(x) = Rcos (x+ ), where R 0 and 0 90,

(a) find the value ofRand the value of.(4)

(b) Hence solve the equation 12 cos x 4 sin x= 7

for 0 x< 360, giving your answers to one decimal place.

(5)

(c) (i) Write down the minimum value of 12 cos x 4 sin x.(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

cos x sin x, x (n 4

1 ), n,

(2)

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

1 .

(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 : x

2x+ ln 2, x

, g : x

e

2x

, x

.

(a) Prove that the composite function gf is gf : x4e4x, x.

(4)

(b) Sketch the curve with equation y= 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 which

xd

d[gf(x)] = 3, giving your answer to 3 significant figures.

(4)

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H31123A 7

Core 3June 06

1. (a) Simplify1

232

2

x

xx.

(3)

(b) Hence, or otherwise, express1

232

2

x

xx )1(

1

xx as a single fraction in its simplest form.

(3)

2. Differentiate, with respect tox,

(a) e3x

+ ln 2x,

(3)

(b) 23

)5( 2x .

(3)

3. Figure 1 shows part of the curve with equation

y = f(x), x , where f is an increasingfunction ofx. The curve passes through thepointsP(0,2) and Q(3, 0) as shown.

In separate diagrams, sketch the curve withequation

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

(b) y = f1

(x),

(3)

(c) y = 21 f(3x).

(3)

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

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H31123A 8

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

T= 400e0.05t

+ 25, t 0.

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

(1)

(b) Find the value oftfor 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 C per 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 20 C.(1)

5.

Figure 2 shows part of the curve with equation

y = (2x1) tan 2x, 0 x 1, the functions f and g are defined by

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

g:x 2xk, x.

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

On each sketch state, in terms ofk, 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 ofk, giving your answer in its simplest form.

(2)

The curve Chas equationy = f(x). The tangent to Cat the point withx-coordinate 3 is parallel to the line

with equation 9y = 2x + 1.

(d) Find the value ofk.

(4)

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H31123A 10

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

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H31123A 11

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) = 12

3

x+

2)2(

3

x, x2.


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 pointP

4,2

lies on C.

(1)

(b) Show thatx

y

d

d=

2

1

atP.

(4)

(c) Find an equation of the normal to CatP. Give your answer in the form y = mx + c, where m and c areexact constants.

(4)

4. (i) The curve Chas equationy =2

9 x

x

.

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

(6)

(ii) Given thaty = 23

)1( 2xe , find the value ofx

y

d

datx = 2

1 ln 3.

(5)

5. Figure 1

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H31123A 12

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 =R sin (x + ), whereR and are constants,R > 0

and 0 < 2

1 .

(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 Chas equationy =x2ex.

(a) Findx

y

d

d, using the product rule for differentiation.

(3)

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

(3)

(c) Find2

2

d

d

x

y.

(2)

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

(2)

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H31123A 16

4. f(x) =x3+ 3x21.

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

x =

x3

1.

(2)

(b) Starting withx1= 0.6, use the iteration

xn + 1 =

nx3

1

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)

5. The functions f and g are defined by

f: ln (2x1), x, x > 21 ,

g: 3

2

x, x, x 3.

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

(2)

(b) Find the inverse function f1

(x), stating its domain.

(4)

(c) Sketch the graph ofy = |g(x)|. Indicate clearly the equation of the vertical asymptote and the

coordinates of the point at which the graph crosses they-axis.

(3)

(d) Find the exact values ofx for which3

2

x= 3.

(3)

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H31123A 17

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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H31123A 20

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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H31123A 21

5. The radioactive decay of a substance is given by

R = 1000ect

, t 0.

whereR is the number of atoms at time tyears 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 ofc 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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H31123A 22

7. A curve Chas 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 CatA.

(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 :x3

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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H31123A 23

C3 JUNE 08

1. The pointPlies on the curve with equation

y = 4e2x + 1

.

They-coordinate ofPis 8.

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

(2)

(b) Find the equation of the tangent to the curve at the pointPin 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) =R cos (x), whereR > 0 and 0 < 3.


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:x 2x23, x.

(d) Solve fg(x) =81 .

(3)

5. (a) Given that sin2+ cos

2 1, show that 1 + cot2 cosec2 .

(2)

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

2 cot29 cosec = 3,

giving your answers to 1 decimal place.

(6)

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H31123A 26

6. (a) Differentiate with respect tox,

(i) e3x

(sinx + 2 cosx),

(3)

(ii) x3

ln (5x + 2).

(3)

Given thaty =2

2

)1(763

xxx , x 1,

(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)

7. f(x) = 3x32x6.

(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, x 0.

(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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H31123A 27

C3 Jan 09

1. (a) Find the value ofx

y

d

dat the point wherex = 2 on the curve with equation

y =x2(5x1).

(6)

(b) Differentiate 22sin

x

x

with 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)

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H31123A 28

3.

Figure 1

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

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H31123A 29

5. The functions f and g are defined by

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

2

ex , x.

(a) Write down the range of g.

(1)

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

fg :x x2

+2

3ex , x.(2)

(c) Write down the range of fg.

(1)

(d) Solve the equation )(fgdd xx

=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 < 41

, y > 0.

The pointPon the curve hasx-coordinate 2. Find an equation of the tangent to CatPin the form ax +by + c = 0, where a, b and c are integers.

(6)

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N35381A 34

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 pointsA(0, 1k) andB( 21 ln k, 0), where kis 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 = f1(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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N35381A 35

6. (a) Use the identity cos (A +B) = cosA cosBsinA sinB, to show that

cos 2A = 1 2 sin2A(2)

The curves C1 and C2 have equations

C1:y = 3 sin 2x

C2:y = 4 sin2x 2 cos 2x

(b) Show that thex-coordinates of the points where C1 and C2 intersect 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 valueof 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

x

x

.

(3)

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

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N35381A 36

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

(1)

(b) Find, for 0

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N35381A 37

C3 Jan 10

1. Express33

12

x

x

13

1

x


(4)

2. f(x) =x3 + 2x2 3x 11

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

2

113

x

x, x2.

(2)

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 cosx3 sinx in the formR cos(x + ), whereR > 0 and 0 < < 21 .

(4)

(b) Hence, or otherwise, solve the equation 5 cosx3 sinx = 4

for 0 x < 2, giving your answers to 2 decimal places.(5)

4. (i) Given thaty =x

x )1ln( 2 , find

x

y

d

d.

(4)

(ii) Given thatx = tany, show that x

y

d

d= 21

1

x .

(5)

5. Sketch the graph ofy = ln x, stating the coordinates of any points of intersection with the axes.(3)

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N35381A 38

6. A sketch of the graph ofy = f (x) is shown.

The graph intersects they-axis at the point (0, 1) and the

pointA(2, 3) is the maximum turning point.

Sketch, on separate axes, the graphs of

(i) y = f(x) + 1,

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

(iii)y = 2f(2x) .

On each sketch, show the coordinates of the point at which your graph intersects the y-axis and the

coordinates of the point to whichA is transformed.

(9)

7. (a) By writing secx as xcos

1, show that

x

x

d

)(secd= secx tanx.

(3)

(b) Given thaty = e2xsec 3x, findx

y

d

d.

(4)

The curve with equationy = e2xsec 3x, 6

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N35381A 39

8. Solve cosec2 2xcot 2x = 1 for 0 x 180.

(7)

9. (i) Find the exact solutions to the equations

(a) ln (3x7) = 5, (b) 3x e7x + 2 = 15.(3) (5)

(ii) The functions f and g are defined by

f (x) = e2x+ 3, x, g(x) = ln (x1),x, x > 1.

(a) Find f1 and state its domain.

(4)

(b) Find fg and state its range.

(3)


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N35381A 40

C3 June 10

1. (a) Show that

2cos1

2sin

= tan .

(2)

(b) Hence find, for180 < 180, all the solutions of

2cos1

2sin2

= 1.

Give your answers to 1 decimal place.

(3)

2. A curve Chas equation

y =2

35

3

)( x

, x

3

5.

The pointPon Chasx-coordinate 2.

Find an equation of the normal to CatPin 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+

4

1, 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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N35381A 41

4. The function f is defined by

f :x| |2x 5|, x.

(a) Sketch the graph with equation y = f(x), showing the coordinates of the points where the graph

cuts or meets the axes.

(2)(b) Solve f(x) =15 +x.

(3)


g :x|x24x + 1, x, 0 x 5.

(c) Find fg(2).

(2)

(d) Find the range of g.(3)

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N35381A 42

5.

Figure 1

Figure 1 shows a sketch of the curve Cwith the equationy = (2x2 5x + 2)ex.

(a) Find the coordinates of the point where Ccrosses they-axis.

(1)

(b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where C

crosses thex-axis.

(3)

(c) Find x

y

d

d

.(3)

(d) Hence find the exact coordinates of the turning points ofC.

(5)

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N35381A 43

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(21 x).

(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 equation y =x2.

(c) Find f(x).

(2)

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

(1)

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

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C3 Mock to June 10