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Edexcel GCECore Mathematics C4

AdvancedWednesday 18 June 2014 Afternoon

Time: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Pink) Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for symbolicalgebra manipulation or symbolic differentiation/integration, or haveretrievable mathematical formulae stored in them.

Instructions to Candidates

In the boxes above, write your centre number, candidate number, your surname, initials and signature.Check that you have the correct question paper.Answer ALL the questions.You must write your answer for each question in the space following the question.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates

A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75.There are 28 pages in this question paper. Any blank pages are indicated.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

P43165AThis publication may only be reproduced in accordance with Pearson Education Limited copyright policy.©2014 Pearson Education Limited.

1. A curve C has the equation

x3 + 2xy – x – y3 – 20 = 0

(a) Find in terms of x and y.(5)

(b) Find an equation of the tangent to C at the point (3, –2), giving your answer in the form ax + by + c = 0, where a, b and c are integers.

(2)

2. Given that the binomial expansion of (1 + kx)–4, < 1, is

1 – 6x + Ax2 + …

(a) find the value of the constant k,(2)

(b) find the value of the constant A, giving your answer in its simplest form.(3)

P43166AThis publication may only be reproduced in accordance with Pearson Education Limited copyright policy.©2014 Pearson Education Limited.

3.

Figure 1

Figure 1 shows a sketch of part of the curve with equation , x > 0.

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis, and the lines with equations x = 1 and x = 4.

The table below shows corresponding values of x and y for .

x 1 2 3 4

y 1.42857 0.90326 0.55556

(a) Complete the table above by giving the missing value of y to 5 decimal places.(1)

(b) Use the trapezium rule, with all the values of y in the completed table, to find an estimate for the area of R, giving your answer to 4 decimal places.

(3)

(c) By reference to the curve in Figure 1, state, giving a reason, whether your estimate in part (b) is an overestimate or an underestimate for the area of R.

(1)

(d) Use the substitution u = √x, or otherwise, to find the exact value of

(6)

P43166A 3

4.

Figure 2

A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase.

When the depth of the water is h cm, the volume of water V cm3 is given by

V = 4 πh(h + 4), 0 ≤ h ≤ 25

Water flows into the vase at a constant rate of 80π cm3 s–1.

Find the rate of change of the depth of the water, in cm s–1, when h = 6.(5)

P43166A 4

5.

Figure 3

Figure 3 shows a sketch of the curve C with parametric equations

, y = 2sin t, 0 ≤ t ≤ 2π

(a) Show that

x + y = 2√3 cos t(3)

(b) Show that a cartesian equation of C is

(x + y)2 + ay2 = b

where a and b are integers to be determined.(2)

P43166A 5

6. (i) Find

(3)

(ii) Find

, x > (2)

(iii) Given that at x = 0, solve the differential equation

(7)

P43166A 6

7.

Figure 4

Figure 4 shows a sketch of part of the curve C with parametric equations

x = 3tan θ, y = 4cos2 θ, 0 ≤ θ <

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

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

(a) Find the x coordinate of the point Q.(6)

The finite region S, shown shaded in Figure 4, is bounded by the curve C, the x-axis, the y-axis and the line l. This shaded region is rotated 2π radians about the x-axis to form a solid of revolution.

(b) Find the exact value of the volume of the solid of revolution, giving your answer in the form pπ + qπ2, where p and q are rational numbers to be determined.

[You may use the formula V = πr2h for the volume of a cone.](9)

P43166A 7

8. Relative to a fixed origin O, the point A has position vector

and the point B has position vector .

The line l1 passes through the points A and B.

(a) Find the vector .(2)

(b) Hence find a vector equation for the line l1.(1)

The point P has position vector .

Given that angle PBA is θ,

(c) show that .(3)

The line l2 passes through the point P and is parallel to the line l1.

(d) Find a vector equation for the line l2.(2)

The points C and D both lie on the line l2.

Given that AB = PC = DP and the x coordinate of C is positive,

(e) find the coordinates of C and the coordinates of D.(3)

(f) find the exact area of the trapezium ABCD, giving your answer as a simplified surd.(4)

TOTAL FOR PAPER: 75 MARKS

END

P43166A 8

Paper Reference(s)

6666/01REdexcel GCECore Mathematics C4 (R)

Advanced SubsidiaryWednesday 18 June 2014 Afternoon




This paper is strictly for students outside the UK.







P43166A 9

1. (a) Find the binomial expansion of

,

in ascending powers of x up to and including the term in x2.Give each coefficient as a simplified fraction.

(5)

(b) Hence, or otherwise, find the expansion of

,

in ascending powers of x, up to and including the term in x2.Give each coefficient as a simplified fraction.

(3)

P43137AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy.©2013 Edexcel Limited.

2.

Figure 1

Figure 1 shows a sketch of part of the curve with equation

y = (2 – x)e2x,

The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the y-axis.

The table below shows corresponding values of x and y for y = (2 – x)e2x.

x 0 0.5 1 1.5 2

y 2 4.077 7.389 10.043 0

(a) Use the trapezium rule with all the values of y in the table, to obtain an approximation for the area of R, giving your answer to 2 decimal places.

(3)

(b) Explain how the trapezium rule can be used to give a more accurate approximation for the area of R.

(1)

(c) Use calculus, showing each step in your working, to obtain an exact value for the area of R. Give your answer in its simplest form.

(5)

P43137A 11

3. x2 + y2 + 10x + 2y – 4xy = 10

(a) Find in terms of x and y, fully simplifying your answer.(5)

(b) Find the values of y for which .(5)

4. (a) Express in partial fractions.(4)

Figure 2

Figure 2 shows a sketch of part of the curve C with equation , x > 0.

The finite region R is bounded by the curve C, the x-axis, the line with equation x = 1 and the line with equation x = 4.

This region is shown shaded in Figure 2.

The region R is rotated through 360° about the x-axis.

(b) Use calculus to find the exact volume of the solid of revolution generated, giving your answer in the form a + bln c, where a, b and c are constants.

(6)

P43137A 12

5. At time t seconds the radius of a sphere is r cm, its volume is V cm3 and its surface area is S cm2.

[You are given that V = πr3 and that S = 4πr2]

The volume of the sphere is increasing uniformly at a constant rate of 3 cm3 s–1.

(a) Find when the radius of the sphere is 4 cm, giving your answer to 3 significant figures.

(4)

(b) Find the rate at which the surface area of the sphere is increasing when the radius is 4 cm.

(2)

6. With respect to a fixed origin, the point A with position vector i + 2j + 3k lies on the line l1

with equation

, where λ is a scalar parameter,

and the point B with position vector 4i + pj + 3k, where p is a constant, lies on the line l2 with equation

, where μ is a scalar parameter.

(a) Find the value of the constant p.(1)

(b) Show that l1 and l2 intersect and find the position vector of their point of intersection, C.

(4)

(c) Find the size of the angle ACB, giving your answer in degrees to 3 significant figures.(3)

(d) Find the area of the triangle ABC, giving your answer to 3 significant figures.(2)

P43137A 13

7. The rate of increase of the number, N, of fish in a lake is modelled by the differential equation

, t > 0, 0 < N < 5000

In the given equation, the time t is measured in years from the start of January 2000 and k is a positive constant.

(a) By solving the differential equation, show that

N = 5000 – Ate–kt

where A is a positive constant.(5)

After one year, at the start of January 2001, there are 1200 fish in the lake.

After two years, at the start of January 2002, there are 1800 fish in the lake.

(b) Find the exact value of the constant A and the exact value of the constant k.(4)

(c) Hence find the number of fish in the lake after five years. Give your answer to the nearest hundred fish.

(1)

P43137A 14

8.

Figure 3

The curve shown in Figure 3 has parametric equations

x = t – 4 sin t, y = 1 – 2 cos t,

The point A, with coordinates (k, 1), lies on the curve.

Given that k > 0

(a) find the exact value of k,(2)

(b) find the gradient of the curve at the point A.(4)

There is one point on the curve where the gradient is equal to .

(c) Find the value of t at this point, showing each step in your working and giving your answer to 4 decimal places.

[Solutions based entirely on graphical or numerical methods are not acceptable.](6)


END

P43137A 15

Paper Reference(s)

6666/01Edexcel GCECore Mathematics C4

Advanced SubsidiaryTuesday 18 June 2013 Morning










P43137A 16

1. (a) Find .(5)

(b) Hence find the exact value of .(2)

2. (a) Use the binomial expansion to show that

, |x| < 1(6)

(b) Substitute into

to obtain an approximation to √3.

Give your answer in the form where a and b are integers.(3)

P42954AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy.©2013 Edexcel Limited.

3.

Figure 1

Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line x = and the curve with equation

y = , 0 ≤ x ≤

The table shows corresponding values of x and y for y = .

x 0

y 1 1.035276 1.414214

(a) Complete the table above giving the missing value of y to 6 decimal places.(1)

(b) Using the trapezium rule, with all of the values of y from the completed table, find an approximation for the area of R, giving your answer to 4 decimal places.

(3)

Region R is rotated through 2π radians about the x-axis.

(c) Use calculus to find the exact volume of the solid formed.(4)

P42954A 18

4. A curve C has parametric equations

x = 2sin t, y = 1 – cos 2t, ≤ t ≤

(a) Find at the point where t = .(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)

5. (a) Use the substitution x = u2, u > 0, to show that

dx = du(3)

(b) Hence show that

dx = 2ln

where a and b are integers to be determined.(7)

P42954A 19

6. Water is being heated in a kettle. At time t seconds, the temperature of the water is θ°C.

The rate of increase of the temperature of the water at any time t is modelled by the differential equation

= λ(120 – θ), θ ≤ 100

where λ is a positive constant.

Given that θ = 20 when t = 0,

(a) solve this differential equation to show that

θ = 120 – 100e–λt

(8)

When the temperature of the water reaches 100°C, the kettle switches off.

(b) Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off.(3)

7. A curve is described by the equation

x2 + 4xy + y2 + 27 = 0

(a) Find in terms of x and y.(5)

A point Q lies on the curve.

The tangent to the curve at Q is parallel to the y-axis.

Given that the x-coordinate of Q is negative,

(b) use your answer to part (a) to find the coordinates of Q.(7)

P42954A 20

8. With respect to a fixed origin O, the line l has equation

, where λ is a scalar parameter.

The point A lies on l and has coordinates (3, – 2, 6).

The point P has position vector (–pi + 2pk) relative to O, where p is a constant.

Given that vector is perpendicular to l,

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

Given also that B is a point on l such that <BPA = 45°,

(b) find the coordinates of the two possible positions of B.(5)


END

P42954A 21

Paper Reference(s)

6666/01REdexcel GCECore Mathematics C4 (R)

Advanced SubsidiaryTuesday 18 June 2013 Morning




This paper is strictly for students outside the UK.







P42954A 22

1. Express in partial fractions

(4)

2. The curve C has equation

3x–1 + xy –y2 +5 = 0

Show that at the point (1, 3) on the curve C can be written in the form , where λ and μ are integers to be found.

(7)

3. Using the substitution u = 2 + √(2x + 1), or other suitable substitutions, find the exact value of

dx

giving your answer in the form A + 2ln B, where A is an integer and B is a positive constant.(8)


, |x| <

in ascending powers of x, up to and including the term in x3. Give each coefficient as a simplified fraction.

(6)

(b) Use your expansion to estimate an approximate value for , giving your answer to 4 decimal places. State the value of x, which you use in your expansion, and show all your working.

(3)

P41860A 23

5.

Figure 1

Figure 1 shows part of the curve with equation . The finite region R shown shaded in Figure 1 is bounded by the curve, the x-axis, the t-axis and the line t = 8.

(a) Complete the table with the value of x corresponding to t = 6, giving your answer to 3 decimal places.

t 0 2 4 6 8

x 3 7.107 7.218 5.223

(1)

(b) Use the trapezium rule with all the values of x in the completed table to obtain an estimate for the area of the region R, giving your answer to 2 decimal places.

(3)

(c) Use calculus to find the exact value for the area of R.(6)

(d) Find the difference between the values obtained in part (b) and part (c), giving your answer to 2 decimal places.

(1)

P41860A 24

6. Relative to a fixed origin O, the point A has position vector 21i – 17j + 6k and the point B has position vector 25i – 14j + 18k.

The line l has vector equation

where a, b and c are constants and λ is a parameter.

Given that the point A lies on the line l,

(a) find the value of a.(3)

Given also that the vector is perpendicular to l,

(b) find the values of b and c,(5)

(c) find the distance AB.(2)

The image of the point B after reflection in the line l is the point B´.

(d) Find the position vector of the point B´.(2)

P41860A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.©2013 Edexcel Limited.

7.

Figure 2


, , 0 ≤ t ≤

(a) Find the gradient of the curve C at the point where t = .(4)

(b) Show that the cartesian equation of C may be written in the form

, a ≤ x ≤ b

stating values of a and b.(3)

Figure 3

The finite region R which is bounded by the curve C, the x-axis and the line x = 125 is shown shaded in Figure 3. This region is rotated through 2π radians about the x-axis to form a solid of revolution.

(c) Use calculus to find the exact value of the volume of the solid of revolution.(5)

P41860A 26


8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant.

Let x be the mass of waste products, in kg, at time t minutes after the start of the experiment. It is known that at time t minutes, the rate of increase of the mass of waste products, in kg per minute, is k times the mass of unburned fuel remaining, where k is a positive constant.

The differential equation connecting x and t may be written in the form

, where M is a constant.

(a) Explain, in the context of the problem, what and M represent.(2)

Given that initially the mass of waste products is zero,

(b) solve the differential equation, expressing x in terms of k, M and t.(6)

Given also that x = when t = ln 4,

(c) find the value of x when t = ln 9, expressing x in terms of M, in its simplest form.(4)


END

P41860A 28

Paper Reference(s)


Advanced LevelMonday 28 January 2013 Morning



Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation or integration, or have retrievable mathematical formulae stored in them.


Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C4), the paper reference (6666), your surname, initials and signature.


A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for the parts of questions are shown in round brackets, e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75.




1. Given

f(x) = (2 + 3x)–3, |x| <

23 ,

find the binomial expansion of f(x), in ascending powers of x, up to and including the term in x3.

Give each coefficient as a simplified fraction.(5)

2. (a) Use integration to find

∫ 1

x3ln x dx

. (5)

(b) Hence calculate

∫1

2 1x3

ln x dx.

(2)

3. Express

9 x2+20x−10( x+2)(3 x−1) in partial fractions.

(4)

P41860A 30

4.

Figure 1

Figure 1 shows a sketch of part of the curve with equation y =

x1+√ x . The finite region R,

shown shaded in Figure 1, is bounded by the curve, the x-axis, the line with equation x = 1 and the line with equation x = 4.

(a) Copy and complete the table with the value of y corresponding to x = 3, giving your answer to 4 decimal places.

(1)

x 1 2 3 4

y 0.5 0.8284 1.3333

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate of the area of the region R, giving your answer to 3 decimal places.

(3)

(c) Use the substitution u = 1 + x, to find, by integrating, the exact area of R.(8)

P41860A 31

5.

Figure 2

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

x = 1 –

12 t, 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)

P41860A 32

6.

Figure 3

Figure 3 shows a sketch of part of the curve with equation y = 1 – 2 cos x, where x is measured in radians. The curve crosses the x-axis at the point A and at the point B.

(a) Find, in terms of , the x coordinate of the point A and the x coordinate of the point B.(3)

The finite region S enclosed by the curve and the x-axis is shown shaded in Figure 3. The region S is rotated through 2 radians about the x-axis.

(b) Find, by integration, the exact value of the volume of the solid generated.(6)

P41860A 33 Turn over

7. With respect to a fixed origin O, the lines l1 and l2 are given by the equations

l1 : r = (9i + 13j – 3k) + (i + 4j – 2k)

l2 : r = (2i – j + k) + (2i + j + k)

where and are scalar parameters.

(a) Given that l1 and l2 meet, find the position vector of their point of intersection.(5)

(b) Find the acute angle between l1 and l2, giving your answer in degrees to 1 decimal place.(3)

Given that the point A has position vector 4i + 16j – 3k and that the point P lies on l1 such that AP is perpendicular to l1,

(c) find the exact coordinates of P.(5)

P41860A 34

8. A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at 3 °C and t minutes after the bottle is placed in the refrigerator the temperature of the water in the bottle is °C.

The rate of change of the temperature of the water in the bottle is modelled by the differential equation

dθdt =

(3−θ)125 .

(a) By solving the differential equation, show that

= Ae–0.008t + 3,

where A is a constant.(4)

Given that the temperature of the water in the bottle when it was put in the refrigerator was 16 °C,

(b) find the time taken for the temperature of the water in the bottle to fall to 10 °C, giving your answer to the nearest minute.

(5)


END

N26109A 35

Paper Reference(s)


Advanced LevelThursday 21 June 2012 Afternoon



Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for

symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulae stored in them.




A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.There are 8 questions in this question paper. The total mark for this paper is 75.



P41484A 36

N26109A 37

1. f(x) =

1x(3 x−1 )2

=

Ax +

B(3 x−1) +

C(3 x−1)2

.

(a) Find the values of the constants A, B and C.(4)

(b) (i) Hence find ∫ f ( x )

¿dx¿.

(ii) Find ∫1

2f ( x )

¿dx¿, leaving your answer in the form a + ln b, where a and b are

constants.(6)

2.

Figure 1

Figure 1 shows a metal cube which is expanding uniformly as it is heated.

At time t seconds, the length of each edge of the cube is x cm, and the volume of the cube is V cm3.

(a) Show that

dVdx = 3x2.

(1)

Given that the volume, V cm3, increases at a constant rate of 0.048 cm3 s–1,

(b) find

dxdt when x = 8,

(2)

(c) find the rate of increase of the total surface area of the cube, in cm2 s–1, when x = 8.(3)

P40085A 38

3. f(x) =

6√ (9−4 x ) , x <

94 .

(a) Find the binomial expansion of f(x) in ascending powers of x, up to and including the term in x3. Give each coefficient in its simplest form.

(6)

Use your answer to part (a) to find the binomial expansion in ascending powers of x, up to and including the term in x3, of

(b) g(x) =

6√ (9+4 x ) , x <

94 ,

(1)

(c) h(x) =

6√ (9−8 x ) , x <

98 .

(2)

4. Given that y = 2 at x =

π4 , solve the differential equation

dydx =

3ycos2 x .

(5)


16y3 + 9x2y − 54x = 0.

(a) Find

dydx in terms of x and y.

(5)

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

dydx = 0.

(7)


6.

Figure 2


x = 3 sin 2t, y = 4 cos2 t, 0 t .

(a) Show that

dydx = k3 tan 2t, where k is a constant to be determined.

(5)

(b) Find an equation of the tangent to C at the point where t =

π3 .

Give your answer in the form y = ax + b, where a and b are constants.(4)

(c) Find a cartesian equation of C.(3)

P40085A 40

7.

Figure 3

Figure 3 shows a sketch of part of the curve with equation y = x12

ln 2x.

The finite region R, shown shaded in Figure 3, is bounded by the curve, the x-axis and the lines x = 1 and x = 4.

(a) Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of R, giving your answer to 2 decimal places.

(4)

(b) Find ∫ x12 ln2 x

¿dx ¿

.(4)

(c) Hence find the exact area of R, giving your answer in the form a ln 2 + b, where a and b are exact constants.

(3)


8. Relative to a fixed origin O, the point A has position vector (10i + 2j + 3k), and the point B has position vector (8i + 3j + 4k).

The line l passes through the points A and B.

(a) Find the vector A⃗B .(2)

(b) Find a vector equation for the line l.(2)

The point C has position vector (3i + 12j + 3k) .

The point P lies on l. Given that the vector C⃗P is perpendicular to l,

(c) find the position vector of the point P.(6)


END

P40085A 42

Paper Reference(s)


Advanced LevelWednesday 25 January 2012 Afternoon




symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.








1. The curve C has the equation 2x + 3y2 + 3x2 y = 4x2.

The point P on the curve has coordinates (–1, 1).

(a) Find the gradient of the curve at P.(5)

(b) Hence find the equation of the normal to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

(3)

2. (a) Use integration by parts to find ∫ x sin 3 x

¿dx .¿

(3)

(b) Using your answer to part (a), find ∫ x2cos3 x

¿dx .¿

(3)

3. (a) Expand1

(2−5 x )2, x <

25 ,

in ascending powers of x, up to and including the term in x2, giving each term as a simplified fraction.

(5)

Given that the binomial expansion of

2+kx(2−5 x )2

, x <

25 , is

12 +

74

x + Ax2 + . . .,

(b) find the value of the constant k,(2)

(c) find the value of the constant A.(2)

N26109A 44

4.

Figure 1

Figure 1 shows the curve with equation

y = √( 2 x3 x2+4 )

, x 0.

The finite region S, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 2.

The region S is rotated 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated, giving your answer in the form k ln a, where k and a are constants.

(5)


5.

Figure 2


x = 4 sin (t + π

6 ), y = 3 cos 2t, 0 t < 2.

(a) Find an expression for

dydx in terms of t.

(3)

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

dydx = 0.

(5)

P38160A 46

6.

Figure 3

Figure 3 shows a sketch of the curve with equation y =

2 sin 2 x(1+cos x ) , 0 x

π2 .

The finite region R, shown shaded in Figure 3, is bounded by the curve and the x-axis.

The table below shows corresponding values of x and y for y =

2 sin 2 x(1+cos x ) .

x 0π8

π4

3 π8

π2

y 0 1.17157 1.02280 0

(a) Complete the table above giving the missing value of y to 5 decimal places.(1)

(b) Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 4 decimal places.

(3)

(c) Using the substitution u = 1 + cos x, or otherwise, show that

∫ 2sin 2 x(1+cos x )

dx = 4 ln (1 + cos x) – 4 cos x + k,

where k is a constant.(5)

(d) Hence calculate the error of the estimate in part (b), giving your answer to 2 significant figures.

(3)


7. Relative to a fixed origin O, the point A has position vector (2i – j + 5k),

the point B has position vector (5i + 2j + 10k),

and the point D has position vector (–i + j + 4k).

The line l passes through the points A and B.

(a) Find the vector A⃗B .(2)

(b) Find a vector equation for the line l.(2)

(c) Show that the size of the angle BAD is 109°, to the nearest degree.(4)

The points A, B and D, together with a point C, are the vertices of the parallelogram ABCD,

where A⃗B = D⃗C .

(d) Find the position vector of C.(2)

(e) Find the area of the parallelogram ABCD, giving your answer to 3 significant figures.(3)

(f) Find the shortest distance from the point D to the line l, giving your answer to 3 significant figures.

(2)

P38160A 48

8. (a) Express

1P (5−P ) in partial fractions.

(3)

A team of conservationists is studying the population of meerkats on a nature reserve. The population is modelled by the differential equation

dPdt =

115 P(5 – P), t 0,

where P, in thousands, is the population of meerkats and t is the time measured in years since the study began.

Given that when t = 0, P = 1,

(b) solve the differential equation, giving your answer in the form,

P =

a

b+ce− 1

3t

where a, b and c are integers.(8)

(c) Hence show that the population cannot exceed 5000.(1)


END


Paper Reference(s)


Advanced LevelMonday 20 June 2011 Morning










P38160A 50

1.

9 x2

( x−1)2 (2x+1 ) =

A( x−1) +

B( x−1)2

+

C(2 x+1 ) .

Find the values of the constants A, B and C.(4)

2. f (x) =

1√ (9+4 x2 ) , x <

32 .

Find the first three non-zero terms of the binomial expansion of f(x) in ascending powers of x. Give each coefficient as a simplified fraction.

(6)

3.

Figure 1

A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl.

When the depth of the water is h m, the volume V m3 is given by

V =

112 h2(3 – 4h), 0 h 0.25.

(a) Find, in terms of ,

dVdh when h = 0.1.

(4)

Water flows into the bowl at a rate of

π800 m3 s–1.

(b) Find the rate of change of h, in m s–1, when h = 0.1.(2)

H35405A 51

4.

Figure 2

Figure 2 shows a sketch of the curve with equation y = x3 ln (x2 + 2), x 0.

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

The table below shows corresponding values of x and y for y = x3 ln (x2 + 2).

x 0√ 24

√ 22

3 √ 24

2

y 0 0.3240 3.9210

(a) Complete the table above giving the missing values of y to 4 decimal places.(2)


(3)

(c) Use the substitution u = x2 + 2 to show that the area of R is

12∫2

4(u−2) ln u

¿du ¿

. (4)

(d) Hence, or otherwise, find the exact area of R.(6)

H35405A 52

5. Find the gradient of the curve with equation

ln y = 2x ln x, x > 0, y > 0,

at the point on the curve where x = 2. Give your answer as an exact value.(7)

6. With respect to a fixed origin O, the lines l1 and l2 are given by the equations

l1: r = ( 6−3−2 )

+ (−1

23 )

, l2: r = (−5153 )

+ μ ( 2−3

1 ),

where μ and are scalar parameters.

(a) Show that l1 and l2 meet and find the position vector of their point of intersection A.(6)

(b) Find, to the nearest 0.1°, the acute angle between l1 and l2.(3)

The point B has position vector ( 5−1

1 ).

(c) Show that B lies on l1.(1)

(d) Find the shortest distance from B to the line l2, giving your answer to 3 significant figures.

(4)

H35405A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.©2011 Edexcel Limited.

7.

Figure 3

Figure 3 shows part of the curve C with parametric equations

x = tan , y = sin , 0 <

π2 .

The point P lies on C and has coordinates (√ 3 , 1

2√ 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 (k3, 0), 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 x = 3 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)

H35405A 54

8. (a) Find ∫(4 y+3 )

−12

¿dy¿.

(2)

(b) Given that y =1.5 at x = – 2, solve the differential equation

dydx =

√ (4 y+3 )x2

,

giving your answer in the form y = f(x).(6)


END



Advanced LevelWednesday 26 January 2011 Afternoon










H35405A 56

1. Use integration to find the exact value of∫0

π2 x sin2 x

¿ dx¿.

(6)

2. The current, I amps, in an electric circuit at time t seconds is given by

I = 16 – 16(0.5)t, t 0.

Use differentiation to find the value of

dIdt when t = 3 .

Give your answer in the form ln a, where a is a constant.(5)

3. (a) Express

5( x−1)(3 x+2) in partial fractions.

(3)

(b) Hence find ∫ 5

( x−1 )(3 x+2 )dx

, where x > 1.(3)

(c) Find the particular solution of the differential equation

(x – 1)(3x + 2)

dydx = 5y, x > 1,

for which y = 8 at x = 2 . Give your answer in the form y = f(x).(6)

N26109A 57

4. Relative to a fixed origin O, the point A has position vector i − 3j + 2k and the point B has position vector −2i + 2j − k. The points A and B lie on a straight line l.

(a) Find A⃗B .(2)

(b) Find a vector equation of l.(2)

The point C has position vector 2i + pj − 4k with respect to O, where p is a constant.

Given that AC is perpendicular to l, find

(c) the value of p,(4)

(d) the distance AC.(2)

5. (a) Use the binomial theorem to expand

(2 – 3x)–2, x <

23 ,

in ascending powers of x, up to and including the term in x3. Give each coefficient as a simplified fraction.

(5)

f(x) =

a+bx(2−3 x )2

, x <

23 , where a and b are constants.

In the binomial expansion of f(x), in ascending powers of x, the coefficient of x is 0 and the

coefficient of x2 is

916 .

Find

(b) the value of a and the value of b,(5)

(c) the coefficient of x3 , giving your answer as a simplified fraction.(3)

H35386A 58

6. The curve C has parametric equations

x = ln t, y = t2 −2, t > 0.

Find

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

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

Figure 1

The finite area R, shown in Figure 1, is bounded by C, the x-axis, the line x = ln 2 and the line x = ln 4. The area R is rotated through 360° about the x-axis.

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


7. I = ∫2

5 14 +√ ( x−1)

dx.

(a) Given that y =

14 +√ ( x−1) , copy and complete the table below with values of y

corresponding to x = 3 and x = 5 . Give your values to 4 decimal places.

x 2 3 4 5

y 0.2 0.1745(2)

(b) Use the trapezium rule, with all of the values of y in the completed table, to obtain an estimate of I, giving your answer to 3 decimal places.

(4)

(c) Using the substitution x = (u − 4)2 + 1, or otherwise, and integrating, find the exact value of I.

(8)


END

H35386A 60

Paper Reference(s)


AdvancedFriday 18 June 2010 Afternoon




symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.








1.

Figure 1

Figure 1 shows part of the curve with equation y = √(0.75 + cos2 x). The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-axis, the x-axis and the line with

equation x =

π3 .

(a) Copy and complete the table with values of y corresponding to x =

π6 and x =

π4 .

x 0π

12π6

π4

π3

y 1.3229 1.2973 1

(2)

(b) Use the trapezium rule

(i) with the values of y at x = 0, x =

π6 and x =

π3 to find an estimate of the area of R.

Give your answer to 3 decimal places.

(ii) with the values of y at x = 0, x =

π12 , x =

π6 , x =

π4 and x =

π3 to find a further

estimate of the area of R. Give your answer to 3 decimal places.

(6)

N35382A 62

2. Using the substitution u = cos x +1, or otherwise, show that

∫0

π2 ecos x + 1 sin x

¿dx¿ = e(e – 1).

(6)

3. A curve C has equation

2x + y2 = 2xy.

Find the exact value of

dydx at the point on C with coordinates (3, 2).

(7)

4. A curve C has parametric equations

x = sin2 t, y = 2 tan t , 0 ≤ t <

π2 .

(a) Find

dydx in terms of t.

(4)

The tangent to C at the point where t =

π3 cuts the x-axis at the point P.

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

5.

2 x2+5 x−10( x−1)( x+2 ) A +

Bx−1 +

Cx+2 .

(a) Find the values of the constants A, B and C.(4)

(b) Hence, or otherwise, expand

2 x2+5 x−10( x−1)( x+2 ) in ascending powers of x, as far as the term

in x2. Give each coefficient as a simplified fraction.

(7)

N35382A This publication may only be reproduced in accordance with Edexcel Limited copyright policy.©2010 Edexcel Limited.

6. f(θ) = 4 cos2 θ – 3sin2 θ

(a) Show that f(θ) =

12 +

72 cos 2θ.

(3)

(b) Hence, using calculus, find the exact value of ∫0

π2 θ f (θ )

¿dθ¿.

(7)

7. The line l1 has equation r = ( 2

3−4) + λ ( 1

21 )

, where λ is a scalar parameter.

The line l2 has equation r = ( 0

9−3) + μ ( 5

02 )

, where is a scalar parameter.

Given that l1 and l2 meet at the point C, find

(a) the coordinates of C.(3)

The point A is the point on l1 where λ = 0 and the point B is the point on l2 where μ = –1.

(b) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places.(4)

(c) Hence, or otherwise, find the area of the triangle ABC.(5)

N35382A 64

8.

Figure 2

Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is 6 m. Water is flowing into the tank at a constant rate of 0.48π m3 min−1. At time t minutes, the depth of the water in the tank is h metres. There is a tap at a point T at the bottom of the tank. When the tap is open, water leaves the tank at a rate of 0.6πh m3 min−1.

(a) Show that, t minutes after the tap has been opened,

75

dhdt = (4 – 5h).

(5)

When t = 0, h = 0.2

(b) Find the value of t when h = 0.5(6)


END


Paper Reference(s)


Advanced LevelMonday 25 January 2010 Morning


Materials required for examination Items included with question papersMathematical Formulae (Pink or Green) Nil








N35382A 66


(1 – 8x), x < 18 ,

in ascending powers of x up to and including the term in x3, simplifying each term.(4)

(b) Show that, when x =

1100 , the exact value of (1 – 8x) is

√ 235 .

(2)

(c) Substitute x =

1100 into the binomial expansion in part (a) and hence obtain an

approximation to 23. Give your answer to 5 decimal places.

(3)

P43165A 67

2.

Figure 1

Figure 1 shows a sketch of the curve with equation y = x ln x, x 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axis and the line x = 4.

The table shows corresponding values of x and y for y = x ln x.

x 1 1.5 2 2.5 3 3.5 4

y 0 0.608 3.296 4.385 5.545

(a) Copy and complete the table with the values of y corresponding to x = 2 and x = 2.5, giving your answers to 3 decimal places.

(2)


(4)

(c) (i) Use integration by parts to find ∫ x ln x

¿dx¿.

(ii) Hence find the exact area of R, giving your answer in the form

14 (a ln 2 + b), where

a and b are integers.(7)

N35382A 68


cos 2x + cos 3y = 1, –

π4 x

π4 , 0 y

π6 .

(a) Find

dydx in terms of x and y.

(3)

The point P lies on C where x =

π6 .

(b) Find the value of y at P.(3)

(c) Find the equation of the tangent to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

(3)


4. The line l1 has vector equation

r = (−6

4−1 )

+ λ ( 4−1

3 )and the line l2 has vector equation

r = (−6

4−1 )

+ ( 3−4

1 )where λ and μ are parameters.

The lines l1 and l2 intersect at the point A and the acute angle between l1 and l2 is θ.

(a) Write down the coordinates of A.(1)

(b) Find the value of cos θ.(3)

The point X lies on l1 where λ = 4.

(c) Find the coordinates of X.(1)

(d) Find the vector A⃗X .(2)

(e) Hence, or otherwise, show that A⃗X = 4√26.(2)

The point Y lies on l2. Given that the vector Y⃗X is perpendicular to l1,

(f) find the length of AY, giving your answer to 3 significant figures.(3)

N35382A 70

5. (a) Find ∫ 9 x+6

xdx

, x > 0.(2)

(b) Given that y = 8 at x =1, solve the differential equation

dydx =

(9 x+6 ) y13

x

giving your answer in the form y 2 = g(x).(6)

6. The area A of a circle is increasing at a constant rate of 1.5 cm2 s–1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasing when the area of the circle is 2 cm2.

(5)


7.

Figure 2


x = 5t 2 − 4, y = t(9 − t 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)

N35382A 72

8. (a) Using the substitution x = 2 cos u, or otherwise, find the exact value of

∫1

√ 2 1x2 √ (4−x2 )

dx.

(7)

Figure 3

Figure 3 shows a sketch of part of the curve with equation y =

4

x( 4−x2)14

, 0 < x < 2.

The shaded region S, shown in Figure 3, is bounded by the curve, the x-axis and the lines with equations x = 1 and x = √2. The shaded region S is rotated through 2π radians about the x-axis to form a solid of revolution.

(b) Using your answer to part (a), find the exact volume of the solid of revolution formed.(3)


END


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