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6. A curve has parametric equations

(a) Find an expression for in terms of the parameter t.

(4)

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

(4)

(c) Find a cartesian equation of the curve in the form y = f(x). State the domain on which

the curve is defined.

(4)

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.4

tπ=

d

d

y

x

22cot , 2sin , 0 .2

x t y t tπ= = <

*N20232B01224*

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Question 6 continued

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8. Figure 2

The curve shown in Figure 2 has parametric equations

x = t – 2 sin t, y = 1 – 2 cos t, 0 t 2 .

(a) Show that the curve crosses the x-axis where and .

(2)

The finite region R is enclosed by the curve and the x-axis, as shown shaded in Figure 2.

(b) Show that the area of R is given by the integral

(3)

(c) Use this integral to find the exact value of the shaded area.(7)

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53

3

2(1 2cos ) d .t t

53

t3

t

*N23553A01820*

y

xO

R

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Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

END

*N23553A02020*

Q8

(Total 12 marks)

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4. Figure 2

The curve shown in Figure 2 has parametric equations

x = sin t, y = sin (t + ), – < t < .

(a) Find an equation of the tangent to the curve at the point where t = .(6)

(b) Show that a cartesian equation of the curve is

(3)

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23 1 (1 ), 1 < < 1.2 2

y x x x√= + √ − − �

6π

2π

2π

6π

*N23563A0820*

–0.5–1

0.5

0.5 1

y

xO

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Question 4 continued

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Q4

(Total 9 marks)

*N23563A0920*

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*N26110A01224*

6. A curve has parametric equations

(a) Find an expression for in terms of t. You need not simplify your answer.(3)

(b) Find an equation of the tangent to the curve at the point where .

Give your answer in the form , where a and b are constants to be determined.

(5)

(c) Find a cartesian equation of the curve in the form . (4)

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2π

dd

yx

t = 4π

y = ax + b

2 f ( )y x=

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Turn over*N26110A01324*

Question 6 continued

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*N26282A01624*

7.

Figure 3

The curve C has parametric equations

.

The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3.

(a) Show that the area of R is given by the integral

(4)

(b) Hence find an exact value for this area.(6)

(c) Find a cartesian equation of the curve C, in the form y = f(x).(4)

(d) State the domain of values for x for this curve.(1)

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C

11 20

2

( )( ).

t ttd

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Question 7 continued

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*H30427A02428*

8.

Figure 3

Figure 3 shows the curve C with parametric equations

The point P lies on C and has coordinates (4, 2 3).

(a) Find the value of t at the point P.(2)

The line l is a normal to C at P.

(b) Show that an equation for l is y = –x 3 + 6 3.(6)

The finite region R is enclosed by the curve C, the x-axis and the line x = 4, as shown shaded in Figure 3.

(c) Show that the area of R is given by the integral .(4)

(d) Use this integral to find the area of R, giving your answer in the form a + b 3, where a and b are constants to be determined.

(4)

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xO

y

P

l

C

4

R

x t= 8cos , y t= 4 2sin , 0 2t π .

64 22

sin cost t td

π

3π∫

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*H30427A02728*

Question 8 continued

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TOTAL FOR PAPER: 75 MARKS

END

Q8

(Total 16 marks)

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*N31013A02428*

7.

Figure 3

The curve C shown in Figure 3 has parametric equations

x = t3 – 8t, y = t2

where t is a parameter. Given that the point A has parameter t = –1,

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

The line l is the tangent to C at A.

(b) Show that an equation for l is 2x – 5y – 9 = 0.(5)

The line l also intersects the curve at the point B.

(c) Find the coordinates of B.(6)

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y

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*N31013A02728*

Question 7 continued

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TOTAL FOR PAPER: 75 MARKSEND

Q7

(Total 12 marks)

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*H34265A01428*

5.

Figure 2

Figure 2 shows a sketch of the curve with parametric equations

x = 2 cos 2t , y = 6 sin t , 0 t2

(a) Find the gradient of the curve at the point where t = 3

.(4)

(b) Find a cartesian equation of the curve in the form

y = f (x), – k x k,

stating the value of the constant k.(4)

(c) Write down the range of f (x).(2)

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*H34265A01528* Turn over

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*N35382A02228*

7.

Figure 2

Figure 2 shows a sketch of the curve C with parametric equations

x t5 42 , y t t( )9 2

The curve C cuts the x-axis at the points A and B.

(a) Find the x-coordinate at the point A and the x-coordinate at the point B.(3)

The region R, as shown shaded in Figure 2, is enclosed by the loop of the curve.

(b) Use integration to find the area of R.(6)

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AR

C

B x

y

O

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*N35382A02328* Turn over

Question 7 continued

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*H35386A01232*

4. A curve C has parametric equations

2sin , 2 tan , 02

x t y t t π= =

(a) Find ddyx

in terms of t.

(4)

The tangent to C at the point where 3

t π= cuts the x-axis at the point P.

(b) Find the x-coordinate of P.(6)

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Question 4 continued

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*H35405A01824*

6. The curve C has parametric equations

lnx t= , 2 2y t= − , 0t

Find

(a) an equation of the normal to C at the point where 3t = ,(6)

(b) a cartesian equation of C.(3)

O

y

x

C

R

ln 2 ln 4

Figure 1

The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line ln 2x = and the line ln 4x = . The area R is rotated through360° about the x-axis.

(c) Use calculus to find the exact volume of the solid generated.(6)

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Question 6 continued

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*P38160A01824*

7.y

xO

l C

P

S

Q√3

Figure 3

Figure 3 shows part of the curve C with parametric equations

x = tan , y = sin , 0 < 2

The point P lies on C and has coordinates 1

23, 3⎛ ⎞

⎜ ⎟⎝ ⎠.

(a) Find the value of at the point P.(2)

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

(b) Show that Q has coordinates ( )3, 0 ,k giving the value of the constant k.(6)

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line 3x = and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.

(c) Find the volume of the solid of revolution, giving your answer in the form p 3 + q 2, where p and q are constants.

(7)

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Question 7 continued

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*P40085A01228*

5.

C

O x

y

Figure 2

Figure 2 shows a sketch of the curve C with parametric equations

x t y t t= +⎛⎝⎜

⎞⎠⎟

=46

3 2 0 2sin , cos ,ππ� <

(a) Find an expression for ddyx

in terms of t. (3)

(b) Find the coordinates of all the points on C where ddyx

= 0(5)

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*P41484A02132* Turn over

Question 6 continued

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*P41860A01228*

5.

Figure 2

Figure 2 shows a sketch of part of the curve C with parametric equations

x t= −1 12

, y = 2t – 1

The curve crosses the y-axis at the point A and crosses the x-axis at the point B.

(a) Show that A has coordinates (0, 3).(2)

(b) Find the x coordinate of the point B.(2)

(c) Find an equation of the normal to C at the point A.(5)

The region R, as shown shaded in Figure 2, is bounded by the curve C, the line x = –1 and the x-axis.

(d) Use integration to find the exact area of R.(6)

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C

A

B x–1

R

y

O

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*P43137A01232*

4. A curve C has parametric equations

x = 2sin t, y = 1 – cos 2t, –2π

� t ��2π

(a) Find dd

yx at the point where t =

6π

(4)

(b) Find a cartesian equation for C in the form

y = f(x), – k � x � k,

stating the value of the constant k.(3)

(c) Write down the range of f(x).(2)

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physicsandmathstutor.com June 2013

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2009 7

Core Mathematics C4

Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.

Integration (+ constant)

f(x) xx d)f(

sec2 kx k1 tan kx

xtan xsecln

xcot xsinln

xcosec )tan(ln,cotcosecln 21 xxx +−

xsec )tan(ln,tansecln 41

21 π++ xxx

−= xxuvuvx

xvu d

ddd

dd

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2009

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 =±

))(( tantan1tantan)(tan 2

1 π+≠±±=± kBABABABA

2cos

2sin2sinsin BABABA −+=+

2sin

2cos2sinsin BABABA −+=−

2cos

2cos2coscos BABABA −+=+

2sin

2sin2coscos BABABA −+−=−

Differentiation

f(x) f ′(x)

tan kx k sec2 kx

sec x sec x tan x

cot x –cosec2 x

cosec x –cosec x cot x

)g()f(

xx

))(g(

)(g)f( )g()(f2x

xxxx ′−′

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2009 5

Core Mathematics C2

Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2 = b2 + c2 – 2bc cos A

Binomial series

21

)( 221 nrrnnnnn bbarn

ban

ban

aba ++++++=+ −−− (n ∈ )

where)!(!

!C rnr

nrn

rn

−==

∈<+×××

+−−++×−++=+ nxx

rrnnnxnnnxx rn ,1(

21)1()1(

21)1(1)1( 2 )

Logarithms and exponentials

ax

xb

ba log

loglog =

Geometric series

un = arn − 1

Sn = r ra n

−−

1)1(

S∞ = r

a−1

for ⏐r⏐ < 1

Numerical integration

The trapezium rule: b

a

xy d ≈ 21 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where

nabh −=

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4π r 2

Area of curved surface of cone = π r × slant height

Arithmetic series

un = a + (n – 1)d

Sn = 21 n(a + l) =

21 n[2a + (n − 1)d]