Mathematics Mpc4 Unit Pure Core 4

General Certificate of Education January 2006 Advanced Level Examination

MATHEMATICS Unit Pure Core 4 Wednes Wednesday day 25 25 Januar January y 2006 2006

MPC4

9.00 9.00 am to 10.3 10.30 0 am

For this paper you must have: *

an 8-page answer book

*

the blue AQA booklet of formulae and statistical tables

You may use a graphics calculator.

Time allowed: 1 hour 30 minutes Instructions *

*

*

*

Use blue or black black ink or ball-point ball-point pen. Pencil Pencil should only only be used for drawing. drawing. Write Write the informatio information n required required on the front front of your answer answer book. The Examining Body for this pap paper er is is AQA AQA.. The The Paper Reference is MPC4. Answer all questions. All necessary working should be shown; otherwise marks for method may be lost.

Information *

*

The maximum mark for this paper is 75. The marks for questions are shown in brackets.

Advice *

Unless stated otherwise, formulae may be quoted, without proof, from the booklet.

P80395/Jan06/M P80395/Jan06/MPC4 PC4

6/6/6/

MPC4

2

Answer all questions.

1

(a) (a)

3 x 3

The The pol polyn ynom omia iall f x f x is defined by f x (i)

2.

Find f 1 .

(1 mark)

(ii) (ii) Show Show that that f À 2 (iii) (iii)

2 x 2 À 7 x

0.

(1 mark)

Hence He nce,, or otherw otherwise ise,, show that that x À 1 x 3 x 3

2

2 x 2 À 7 x

1 ax

2

b

where a and b are integers. (b) (b)

3 x 3

The The pol polyn ynom omia iall g x is defin defined ed by g x

2 x 2 À 7 x

d . d .

3 x À 1 , the rema remaind inder er is 2. 2. Find Find the the value value of of d .

When g x is divided by

2

(3 marks)

(3 marks)

A curve is defined by the parametric equations x

(a)

Find

3 À 4t

y

1

2 t

d y in terms of t . d x

(4 marks)

(b) Find Find the equati equation on of the tangen tangentt to the the curve at the point point where where t 2, giving your answer in the form ax by c 0, where a, b and c are integers. (4 marks) (c) Ver Verify ify that that the cartesia cartesian n equation equation of the curve curve can be writte written n as x À 3 y À 1

3

It is given given that that 3 cos cos y À 2sin y (a) (a)

Find Find the the val value ue of R.

(b) (b)

Show Show that that

a

% 33:7 .

R cos

y

a

8

(3 marks)

0

, where where R

>

0 and 0  < a < 90 . (1 mark) (2 marks)

(c) Hence Hence write write dow down n the maxi maximum mum val value ue of 3 cos y À 2 sin y and find a positive value of y at which this maximum value occurs. (3 marks)

P80395/Jan06/MPC4

3

4

On 1 January 1900, a sculpture was valued at £80. When the sculpture was sold on 1 January 1956, its value was £5000. The value, £V £ V ,, of the sculpture is modelled by the formula V years since 1 January 1900 and A and k are constants. (a) (a)

Writ Writee down down the the val value ue of of A.

(b) (b)

Show Show that that k % 1:07664.

(c) (c)

Use Use thi thiss mode modell to: to:

A k t , where where t is the time in

(1 mark) (3 marks)

(i) show show that the value value of the sculptur sculpturee on 1 January January 2006 will will be greater greater than £200 000; (2 marks)

5

(a) (a)

(ii) (ii)

find find the year in which which the value value of the sculptu sculpture re will first first exceed exceed £800 £800 000. 000. (3 marks)

(i) (i)

Obta Obtain in the the bino binomi mial al expa expans nsio ion n of 1 À x À1 up to and including the term in x 2 . (2 marks)

(ii) (ii)

Hence He nce,, or otherw otherwise ise,, show show that that 1 3 À 2 x

%

1 3

2 x 9

4 2 x 27

for small values of x.

(3 marks)

(b) Obtai Obtain n the binom binomial ial expa expansi nsion on of

(c) (c)

Give Given n that that

2 x 2 À 3 3 À 2 x 1 À x

2

1 1 À x

2

up to and including the term in x 2 . (2 marks)

can be written in the form

find the values of A, B and C .

(d) Hence Hence find find the bino binomia miall expans expansion ion of of in x 2 .

A 3 À 2 x

B 1 À x

C 1 À x

2

,

(5 marks) 2 x 2 À 3 3 À 2 x 1 À x

2

up to and including the term (3 marks)

Turn over for the next question

Turn over P80395/Jan06/MPC4

s

4

6

(a)

Express cos2 x in the form a cos2 x

b , where a and b are constants.

(2 marks)

p

(b)

Hence show that

2

p

cos2 x d x

a

0

7

, where a is an integer.

The quadrilateral ABCD has vertices A 2,1,3 , B 6,5,3 , C 6, 1,À 1 and D 2, À 3, À1 .

The line l 1 has vector equation r (a)

(i)

3

2 3 23 4 5 45 6 1 À1

l

1 1 0

:

Find the vector AB .

(2 marks)

(ii) Show that the line AB is parallel to l 1 . (iii) (b)

(1 mark)

Verify that D lies on l 1 .

(2 marks)

The line l 2 passes through D 2, À 3, À1 and M 4,1,1 . (i) Find the vector equation of l 2 . (ii)

8

(5 marks)

(a)

(2 marks)

Find the angle between l 2 and AC .

Solve the differential equation d x À 2 x À 6 dt to find t in terms of x , given that x

(b)

(3 marks)

1 2

70 when t

(i) Explain what happens when x (ii)

1 2

6.

2006 AQA and its licensors. All rights reserved.

P80395/Jan06/MPC4

(1 mark)

Find how long it will take for the depth of fuel to fall from 70 cm to 22 cm. (2 marks)

END OF QUESTIONS

Ó

(6 marks)

Liquid fuel is stored in a tank. At time t minutes, the depth of fuel in the tank is x cm. Initially there is a depth of 70 cm of fuel in the tank. There is a tap 6 cm above the bottom of the tank. The flow of fuel out of the tank is modelled by the differential equation d x À 2 x À 6 dt

Copyright

0.


MATHEMATICS Unit Pure Core 4 Thursday 25 January 2007

MPC4

9.00 am to 10.30 am



*

the blue AQA booklet of formulae and statistical tables.



*

*

*

Use blue or black ink or ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MPC4. Answer all questions. Show all necessary working; otherwise marks for method may be lost.

Information *

*


Advice *

Unless stated otherwise, you may quote formulae, without proof, from the booklet.

P90454/Jan07/MPC4

6/6/6/

MPC4

2


1

A curve is defined by the parametric equations x (a)

(i)

Find

d x dt

and

(ii) Hence find

d y dt

d y d x

¼ 1 þ 2t ,

y

¼ 1 À 4t 2

.

(2 marks)

in terms of t .

(2 marks)

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

¼ 1.

(c) Find a cartesian equation of the curve.

2

(3 marks)

ð Þ ¼ 2 x 3 À 7 x 2 þ 13 .

ðÞ

The polynomial f x is defined by f x

ðÞ

(a) Use the Remainder Theorem to find the remainder when f x is divided by

(b)

(c)

3

(4 marks)

(a) (b)

ð2 x À 3Þ .

(2 marks)

ðÞ ð Þ ¼ 2 x 3 À 7 x 2 þ 13 þ d , where d is a constant. Given that ð2 x À 3Þ is a factor of g ð xÞ , show that d ¼ À4 . (2 marks) Express gð xÞ in the form ð2 x À 3Þð x 2 þ ax þ bÞ . (2 marks)

The polynomial g x is defined by g x

Express cos2 x in terms of sin x . (i)

Hence show that 3 sin x

À cos2 x ¼ 2sin2 x þ 3sin x À 1

(ii) Solve the equation 3 sin x

for all values of x . (2 marks)

À cos2 x ¼ 1 for 0° < x < 360° .

(c) Use your answer from part (a) to find

P90454/Jan07/MPC4

(1 mark)

ð

sin2 x d x .

(4 marks) (2 marks)

3

4

(a)

(i)

(ii)

(b)

(i)

À 5 in the form A þ B , where A and B are integers. (2 marks) x À 3 À3 ð 3 x À 5 Hence find d x . (2 marks) x À 3 6 x À 5 P Q Express in the form , where P and Q are integers. þ 2 x þ 5 2 x À 5 4 x2 À 25 Express

3 x x

(3 marks)

(ii) Hence find

5

À 5 d x . 4 x 2 À 25 6 x

(3 marks)

ð þ xÞ

(a) Find the binomial expansion of 1 (b)

(i) (ii)

6

ð

(i) (ii)

(b)

ð þ 3 xÞ % 2 þ 14 x À 321 x 2 p ffi9ffi % 599 . Hence show that Show that 8

for small values of x .

3

!

(2 marks) (3 marks) (2 marks)

288

À2, 4), (5, 4,0) and (11, 6, À4) respectively.

Find the vector BA .

(2 marks)

Show that the size of angle ABC is

The line l has equation r

(i)

up to the term in x 2 .

1 3

The points A , B and C have coordinates (3, (a)

1 3

cosÀ1

2 3 2 3 ¼ 4 À 5þ 4 5 8 3 2

l

1 3 2

À  5 7

.

(5 marks)

.

À

Verify that C lies on l .

(2 marks)

(ii) Show that AB is parallel to l .

(1 mark)

(c) The quadrilateral ABCD is a parallelogram. Find the coordinates of D .

(3 marks)



s

4

7

(a)

Use the identity

þ tan B ð þ BÞ ¼ 1tan A À tan A tan B

tan A

to express tan 2 x in terms of tan x . (b)

Show that 2

1 À tan xÞ2 À 2 tan x À 2tantan x ¼ ð 2 x

for all values of x , tan2 x

8

(a)

(2 marks)

(i)

6¼ 0 .

(4 marks)

d y

Solve the differential equation

(ii) Given that y

¼ 50 when t ¼

dt

p

¼ y sin t to obtain y in terms of t .

, show that y

¼ 50eÀð1þ cos t Þ .

(4 marks) (3 marks)

(b) A wave machine at a leisure pool produces waves. The height of the water, y cm, above a fixed point at time t seconds is given by the differential equation d y dt

¼ y sin t

(i) Given that this height is 50 cm after height of the water after 6 seconds. (ii) Find p

p

seconds, find, to the nearest centimetre, the (2 marks)

d2 y

and hence verify that the water reaches a maximum height after dt 2 (4 marks) seconds.

END OF QUESTIONS

Copyright

Ó


P90454/Jan07/MPC4


MATHEMATICS Unit Pure Core 4 Thursday 24 January 2008

MPC4

9.00 am to 10.30 am



*




*

*

*


Information *

*


Advice *


P1029/Jan08/MPC4

6/6/6/

MPC4

2


1

(a)

(b)

2

(a)



3

1

1



can be expressed in the form k , find the value of þ 3 þ x 3 À x 9 À x 2 the rational number k . (2 marks) Given that

Show that

ð

2

3

1

9 À x 2

d x ¼

a 1 ln 2 b



, where a and b are integers.

(3 marks)

The polynomial f ð xÞ is defined by f ð xÞ ¼ 2 x 3 þ 3 x 2 À 18 x þ 8 . (i) Use the Factor Theorem to show that ð2 x À 1Þ is a factor of f ð xÞ.

(2 marks)

(ii) Write f ð xÞ in the form ð2 x À 1Þð x 2 þ px þ qÞ , where p and q are integers. (2 marks) (iii)

Simplify the algebraic fraction

4 x 2 þ 16 x 2 x 3 þ 3 x 2 À 18 x þ 8

.

(2 marks)

2 x 2 B þ Cx (b) Express the algebraic fraction in the form A þ , where ð x þ 5Þð x À 3Þ ð x þ 5Þð x À 3Þ A, B and C are integers. (4 marks)

3

1 2

(a) Obtain the binomial expansion of ð1 þ xÞ up to and including the term in x 2 . (2 marks) (b)

(c)

Hence obtain the binomial expansion of

r ffiffi ffi ffi ffi ffi ffi

q ffiffi ffi ffi ffi ffi ffi ffi 3

1 þ x up to and including the term in x 2 . 2 (2 marks)

2 þ 3 x % a þ bx þ cx 2 for small values of x, where a, b and c 8 are constants to be found. (2 marks) Hence show that

P1029/Jan08/MPC4

3

4

David is researching changes in the selling price of houses. One particular house was sold on 1 January 1885 for £20. Sixty years later, on 1 January 1945, it was sold for £2000. David proposes a model P ¼ Ak t for the selling price, £ P , of this house, where t is the time in years after 1 January 1885 and A and k are constants. (a)

(i) (ii) (iii)

Write down the value of A.

(1 mark)

Show that, to six decimal places, k ¼ 1:079775 .

(2 marks)

Use the model, with this value of k , to estimate the selling price of this house on 1 January 2008. Give your answer to the nearest £1000. (2 marks)

(b) For another house, which was sold for £15 on 1 January 1885, David proposes the model Q ¼ 15 Â 1:082709 t for the selling price, £Q, of this house t years after 1 January 1885. Calculate the year in which, according to these models, these two houses would have had the same selling price. (4 marks)

5

A curve is defined by the parametric equations (a)

At the point P on the curve, t ¼

x ¼ 2t þ

1 t 2

,

y ¼ 2t À

t 2

.

1 . 2

(i) Find the coordinates of P . (ii)

1

Find an equation of the tangent to the curve at P .

(2 marks) (5 marks)

(b) Show that the cartesian equation of the curve can be written as

ð x À yÞð x þ yÞ2 ¼ k where k is an integer.

(3 marks)



s

4

6

A curve has equation 3 xy À 2 y 2 ¼ 4 . Find the gradient of the curve at the point ð2, 1 Þ.

7

(a)

(i)

(5 marks)

Express 6 sin y þ 8cos y in the form R sinðy þ a Þ , where R > 0 and 0° < a < 90°. Give your value for a to the nearest 0.1 °.

(2 marks)

(ii) Hence solve the equation 6 sin 2 x þ 8cos2 x ¼ 7 , giving all solutions to the nearest 0.1° in the interval 0 ° < x < 360°. (4 marks) (b)

(i) (ii)

sin2 x 1 . ¼ 1 À cos 2 x tan x

Prove the identity

(4 marks)

Hence solve the equation sin2 x ¼ tan x 1 À cos 2 x giving all solutions in the interval 0 ° < x < 360°.

8

Solve the differential equation d y d x given that y ¼ 2 when x ¼

9

(4 marks)

p

2

¼

3cos3 x y

. Give your answer in the form y 2 ¼ f ð xÞ .

(5 marks)

The points A and B lie on the line l 1 and have coordinates ð2, 5, 1 Þ and ð4, 1, À2Þ respectively. (a)

(i) (ii)

!


(2 marks)

Find a vector equation of the line l 1 , with parameter l .

2 3 2 3 4 5 4 5 1

(b)

The line l 2 has equation r ¼

(i)

À3 À1

(1 mark)

1

þm

0 À2

.

Show that the point P ðÀ2, À3, 5 Þ lies on l 2 .

(2 marks)

(ii) The point Q lies on l 1 and is such that PQ is perpendicular to l 2 . Find the coordinates of Q. (6 marks)

END OF QUESTIONS

Copyright

Ó


P1029/Jan08/MPC4


MATHEMATICS Unit Pure Core 4 Wednesday 21 January 2009

MPC4

1.30 pm to 3.00 pm



*




*

*

*

Use black ink or black ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MPC4. Answer all questions. Show all necessary working; otherwise marks for method may be lost.

Information *

*


Advice *


P10952/Jan09/MPC4

6/6/

MPC4

2


1

(a)

(i)

2

(a)

ðÀ1Þ .

Find f

(1 mark)

(2 marks) þ 1 is a factor of f ð xÞ . 4 x 3 À 7 x À 3 (iii) Simplify the algebraic fraction . (3 marks) 2 x 2 þ 3 x þ 1 The polynomial g ð xÞ is defined by g ð xÞ ¼ 4 x 3 À 7 x þ d . When gð xÞ is divided by 2 x þ 1 , the remainder is 2. Find the value of d . (2 marks) (ii)

(b)

ð Þ ¼ 4 x 3 À 7 x À 3 .

ðÞ

The polynomial f x is defined by f x

Use the Factor Theorem to show that 2 x

Express sin x

À 3 cos x

p

ð À aÞ , where R > 0 and 0 < a < 2 .

in the form R sin x

your value of a in radians to two decimal places. (b)

Give (3 marks)

Hence: (i) write down the minimum value of sin x

À 3 cos x ;

(1 mark)

(ii) find the value of x in the interval 0 < x < 2p at which this minimum value occurs, giving your value of x in radians to two decimal places. (2 marks)

3

(a)

(i)

(ii)

(b)

(i)

þ 7 in the form A þ B , where A and B are integers. (2 marks) x þ 2 þ2 ð 2 x þ 7 Hence find d x . (2 marks) x þ 2 28 þ 4 x 2 P Q R Express in the form , where P , Q and þ þ 1 þ 3 x 5 À x ð5 À xÞ2 ð1 þ 3 xÞð5 À xÞ2 Express

2 x x

R are constants.

(ii) Hence find

P10952/Jan09/MPC4

ð

(5 marks)

þ 4 x 2 d x . ð1 þ 3 xÞð5 À xÞ2 28

(4 marks)

3

4

(a)

(i)

ð À xÞ

Find the binomial expansion of 1

1 2

(ii) Hence obtain the binomial expansion of in x 2 .


p ffi4ffi ffiÀffi ffi xffi up to and including the term

(b) Use your answer to part (a)(ii) to find an approximate value for to three decimal places.

5

(a)

(3 marks)

ffi3ffi . p

Give your answer (2 marks)

Express sin 2 x in terms of sin x and cos x .

(1 mark)

(b) Solve the equation 5sin2 x

þ 3cos x ¼ 0

giving all solutions in the interval 0 ° 4 x 4 360° to the nearest 0.1 °, where appropriate. (c)

Given that sin 2 x

(4 marks)

þ cos2 x ¼ 1 þ sin x and sin x 6¼ 0 , show that 2ðcos x À sin xÞ ¼ 1 .

(4 marks)

6

A curve is defined by the equation x 2 y (a) (b)

þ y 3 ¼ 2 x þ 1 . Find the gradient of the curve at the point ð2, 1Þ .

(6 marks)

Show that the x-coordinate of any stationary point on this curve satisfies the equation 1 x 3

¼ x þ 1

(4 marks)



s

4

7

(a) A differential equation is given by (i) (ii)

d x dt

¼À

1 x kt e2

, where k is a positive constant.

Solve the differential equation.

(3 marks)

¼ 6 when t ¼ 0 , show that x ¼ À

Hence, given that x



kt 2 2 ln 4

þe

 À 3

. (3 marks)

(b)

The population of a colony of insects is decreasing according to the model 1 x d x kt e2 , where x thousands is the number of insects in the colony after time dt t minutes. Initially, there were 6000 insects in the colony.

¼À

Given that k (i)

¼ 0:004 , find:

the population of the colony after 10 minutes, giving your answer to the nearest hundred; (2 marks)

(ii) the time after which there will be no insects left in the colony, giving your answer to the nearest 0.1 of a minute. (2 marks)

8

ð

The points A and B have coordinates 2, 1, The angle OBA is y , where O is the origin. (a)

(i)

À 1Þ and ð3, 1, À2Þ respectively.

!


(2 marks)

5 ffi7ffi . ¼ 2p ! ! The point C is such that OC ¼ 2 OB . (ii) Show that cos y

(b)

(c)

(4 marks)

!

The line l is parallel to AB and passes through the point C . Find a vector equation of l . (2 marks) The point D lies on l such that angle ODC

¼ 90°.

END OF QUESTIONS

Copyright

Ó


P10952/Jan09/MPC4

Find the coordinates of D. (4 marks)

General Certificate of Education Advanced Level Examination January 2010

Mathematics

MPC4

Unit Pure Core 4 Tuesday 19 January 2010

9.00 am to 10.30 am

For this paper you must have: * an 8-page answer book * the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator.

Time allowed 1 hour 30 minutes *

Instructions Use black ink or black ball-point pen. Pencil should only be used for drawing. Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MPC4. Answer all questions. Show all necessary working; otherwise marks for method may be lost. * *

* *

Information The marks for questions are shown in brackets. The maximum mark for this paper is 75. * *

Advice Unless stated otherwise, you may quote formulae, without proof, from the booklet. *

P21993/Jan10/MPC4

6/6/6/

MPC4

2


1

The polynomial f ð xÞ is defined by f ð xÞ ¼ 15 x 3 þ 19 x 2 À 4 . (a)

(i)

Find f ð À1Þ .

(1 mark)

(ii) Show that ð5 x À 2Þ is a factor of f ð xÞ . (b)

(2 marks)

Simplify 15 x 2 À 6 x f ð xÞ giving your answer in a fully factorised form.

2

(a)

(5 marks)

Express cos x þ 3 sin x in the form R cosð x À a Þ , where R > 0 and 0 < a < your value of a , in radians, to three decimal places.

(b)

(i)

Hence write down the minimum value of cos x þ 3sin x .

p

2

. Give (3 marks) (1 mark)

(ii) Find the value of x in the interval 0 4 x 4 2p at which this minimum occurs, giving your answer, in radians, to three decimal places. (2 marks) (c)

3

(a)

Solve the equation cos x þ 3 sin x ¼ 2 in the interval 0 4 x 4 2p , giving all solutions, in radians, to three decimal places. (4 marks)

(i)

(ii)

Find the binomial expansion of ð1 þ xÞ

Hence find the binomial expansion of in x 2 .

r ffiffi ffi ffi ffi ffi ffi 3

(b)

À1 3



up to and including the term in x 2 . (2 marks) 3 4

1 þ x

À1



3

up to and including the term (2 marks)

256 % a þ bx þ cx 2 for small values of x, stating the values of 4 þ 3 x the constants a, b and c. (3 marks)

Hence show that

P21993/Jan10/MPC4

3

4

5

10 x 2 þ 8 A B The expression can be written in the form 2 þ , where A þ x þ 1 5 x À 1 ð x þ 1Þð5 x À 1Þ and B are constants. (a)

Find the values of A and B.

(4 marks)

(b)

ð

(4 marks)

Hence find

10 x 2 þ 8 d x . ð x þ 1Þð5 x À 1Þ

A curve is defined by the equation x 2 þ xy ¼ e y Find the gradient at the point ðÀ1, 0Þ on this curve.

6

(a)

(i)

Express sin 2y and cos 2y in terms of sin y and cos y .

(ii) Given that 0 < y <

p

2

and cos y ¼

of cos 2y .

(5 marks)

(2 marks)

3 24 , show that sin 2 y ¼ and find the value 5 25 (2 marks)

(b) A curve has parametric equations x ¼ 3sin2y , (i)

Find

d y d x

y ¼ 4cos2y

in terms of y .

(3 marks)

(ii) At the point P on the curve, cos y ¼ tangent to the curve at the point P .

7

d y

p 3 and 0 < y < . Find an equation of the 5 2 (3 marks)

p 1 x Solve the differential equation , given that y ¼ 1 when x ¼ . ¼ cos 3 2 d x y



Write your answer in the form y 2 ¼ f ð xÞ .

(6 marks)



s

4

8

The points A, B and C have coordinates ð2, À 1, À 5Þ , ð0, 5, À 9Þ and ð9, 2, 3 Þ respectively.

2 3 2 3 4 5 4 5 2

The line l has equation r ¼

1

þ l À3 .

À1 À5

2

(a) Verify that the point B lies on the line l .

!

(b)

Find the vector BC .

(c)

The point D is such that AD ¼ 2 BC . (i)

(2 marks) (2 marks)

!

!

Show that D has coordinates ð20, À 7, 19Þ .

(2 marks)

(ii) The point P lies on l where l ¼ p . The line PD is perpendicular to l . Find the value of p. (5 marks)

9

A botanist is investigating the rate of growth of a certain species of toadstool. She observes that a particular toadstool of this type has a height of 57 millimetres at a time 12 hours after it begins to grow.



She proposes the model h ¼ A 1 À e



À 1 t 4

, where A is a constant, for the height

h millimetres of the toadstool, t hours after it begins to grow. (a)

Use this model to: (i) find the height of the toadstool when t ¼ 0 ; (ii) show that A ¼ 60 , correct to two significant figures.

(b)



Use the model h ¼ 60 1 À e (i)



À 1 t 4

(1 mark) (2 marks)

to:

show that the time T hours for the toadstool to grow to a height of 48 millimetres is given by T ¼ a ln b where a and b are integers;

(ii) show that (iii)

dh dt

h 4

¼ 15 À ;

Ó


P21993/Jan10/MPC4

(3 marks)

find the height of the toadstool when it is growing at a rate of 13 millimetres per hour. (1 mark) END OF QUESTIONS

Copyright

(3 marks)

General Certificate of Education June 2006 Advanced Level Examination

MATHEMATICS Unit Pure Core 4 Thursday 15 June 2006

MPC4

1.30 pm to 3.00 pm



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the blue AQA booklet of formulae and statistical tables



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Information *

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P85556/Jun06/MPC4

6/6/6/

MPC4

2


1

(a)

(i) (ii) (iii)

9 x

10 .

Find p 2 .

(1 mark)

Use the Factor Theorem to show that 2 x

1 is a factor of p x .

(3 marks)

Write p x as the product of three linear factors.

(b) Hence simplify

2

6 x 3 À 19 x 2

The polynomial p x is defined by p x

3 x 2 À 6 x 6 x 3 À 19 x 2

9 x

10

(2 marks)

.

(2 marks)

(a) Obtain the binomial expansion of 1 À x À3 up to and including the term in x 2 . (2 marks)

(b)

Hence obtain the binomial expansion of in x 2 .

(c)

3 5 À 1 À x up to and including the term 2 (2 marks)





Find the range of values of x for which the binomial expansion of

5 À3 1 À x would 2





be valid.

(d)

(2 marks)

Given that x is small, show that



4 2 À 5 x

3



%a

bx

cx 2 , where a, b and c are

integers.

3

(a)

(2 marks)

9 x 2 À 6 x 5 Given that can be written in the form 3 3 x À 1 x À 1 and B are integers, find the values of A and B.

(b) Hence, or otherwise, find

P85556/Jun06/MPC4

9 x 2 À 6 x 5 d x . 3 x À 1 x À 1

A

B

3 x À 1

x À 1

, where A (4 marks)

(4 marks)

3

4

(a)

(i)

Express sin 2 x in terms of sin x and cos x .

(1 mark)

(ii) Express cos 2 x in terms of cos x . (b)

(1 mark)

Show that sin 2 x À tan x

tan x cos2 x

for all values of x . (c)

5

(3 marks)

Solve the equation sin 2 x À tan x 0° < x < 360° .

A curve is defined by the equation y2 À xy (a) (b)

3 x 2 À 5

0

Find the y-coordinates of the two points on the curve where x (i) (ii) (iii)

6

0, giving all solutions in degrees in the interval (4 marks)

Show that

d y d x

1.

(3 marks)

y À 6 x . 2 y À x

Find the gradient of the curve at each of the points where x Show that, at the two stationary points on the curve, 33 x2 À 5

(6 marks) 1.

(2 marks) 0.

(3 marks)

The points A and B have coordinates 2, 4, 1 and 3, 2, À 1 respectively. The point C is 3 3 such that OC 2 OB , where O is the origin. (a)

Find the vectors: (i) (ii)

(b)

(i)

3

OC ;

(1 mark)

3

AB .

(2 marks)

Show that the distance between the points A and C is 5 .

(2 marks)

(ii) Find the size of angle BAC , giving your answer to the nearest degree. (c)

(4 marks)

The point P a , b , g is such that BP is perpendicular to AC . Show that 4 a À 3g

15 .

(3 marks)


Turn over P85556/Jun06/MPC4

s

4

7

Solve the differential equation d y

6 xy2

d x given that y

8

1 when x

2 . Give your answer in the form y

f x .

(6 marks)

A disease is spreading through a colony of rabbits. There are 5000 rabbits in the colony. At time t hours, x is the number of rabbits infected. The rate of increase of the number of rabbits infected is proportional to the product of the number of rabbits infected and the number not yet infected. (a)

(i)

Formulate a differential equation for

d x dt

in terms of the variables x and t and a

constant of proportionality k .

(2 marks)

(ii) Initially, 1000 rabbits are infected and the disease is spreading at a rate of 200 rabbits per hour. Find the value of the constant k . (You are not required to solve your differential equation.) (b)

(2 marks)

The solution of the differential equation in this model is t (i)



4 x 4 ln 5000 À x



Find the time after which 2500 rabbits will be infected, giving your answer in hours to one decimal place. (2 marks)

(ii) Find, according to this model, the number of rabbits infected after 30 hours. (4 marks)

END OF QUESTIONS

Copyright

Ó


P85556/Jun06/MPC4


MATHEMATICS Unit Pure Core 4 Monday 18 June 2007

MPC4

9.00 am to 10.30 am



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P94574/Jun07/MPC4

6/6/6/

MPC4

2


1

(a) (b)

2

(a)

Find the remainder when 2 x 2 þ x À 3 is divided by 2 x þ 1 . Simplify the algebraic fraction

(i)

2 x 2 þ x À 3 x 2 À 1

.

in x 3 .

(c)

Express

(i) (ii)

3

(3 marks)

Find the binomial expansion of ð1 þ xÞÀ1 up to the term in x 3 .

(ii) Hence, or otherwise, obtain the binomial expansion of

(b)

(2 marks)

1 1 þ 3 x

up to the term (2 marks)

1 þ 4 x in partial fractions. ð1 þ xÞð1 þ 3 xÞ

Find the binomial expansion of

(2 marks)

1 þ 4 x up to the term in x 3 . ð1 þ xÞð1 þ 3 xÞ

Find the range of values of x for which the binomial expansion of 1 þ 4 x is valid. ð1 þ xÞð1 þ 3 xÞ

(3 marks)

(3 marks)

(2 marks)

(a)

Express 4 cos x þ 3sin x in the form R cosð x À a Þ , where R > 0 and 0 ° < a < 360° , giving your value for a to the nearest 0.1 °. (3 marks)

(b)

Hence solve the equation 4 cos x þ 3sin x ¼ 2 in the interval 0 ° < x < 360°, giving all solutions to the nearest 0.1 °. (4 marks)

(c)

Write down the minimum value of 4 cos x þ 3sin x and find the value of x in the interval 0° < x < 360° at which this minimum value occurs. (3 marks)

P94574/Jun07/MPC4

3

4

A biologist is researching the growth of a certain species of hamster. She proposes that the length, x cm, of a hamster t days after its birth is given by À

x ¼ 15 À 12e (a)

t 14

Use this model to find: (i) the length of a hamster when it is born;

(1 mark)

(ii) the length of a hamster after 14 days, giving your answer to three significant figures. (2 marks) (b)

(i)

Show that the time for a hamster to grow to 10 cm in length is given by a t ¼ 14ln , where a and b are integers. (3 marks) b



(ii) (c)

(i)

Find this time to the nearest day. Show that d x dt

(ii)

5

(1 mark)

¼

1 14

ð15 À xÞ

(3 marks)

Find the rate of growth of the hamster, in cm per day, when its length is 8 cm. (1 mark)

The point P (1, a) , where a > 0 , lies on the curve y þ 4 x ¼ 5 x 2 y 2 . (a)

Show that a ¼ 1 .

(2 marks)

(b) Find the gradient of the curve at P . (c) Find an equation of the tangent to the curve at P .

(7 marks) (1 mark)



s

4

6

A curve is given by the parametric equations x ¼ cos y (a)

(i)

(ii)

Find

d x dy

and

d y dy

y ¼ sin 2y

.

(2 marks)

Find the gradient of the curve at the point where y ¼

p

6

.

(2 marks)

(b) Show that the cartesian equation of the curve can be written as y 2 ¼ kx2 ð1 À x 2 Þ where k is an integer.

(4 marks)

2 3 2 3 5 þ 4À 5 ¼4 8

7

The lines l 1 and l 2 have equations r respectively.

3

l

6 À9

3 À1

2À 3 2 3 5þ 4 5 ¼4 4

and r

0 11

m

2 À3

(a)

Show that l 1 and l 2 are perpendicular.

(2 marks)

(b)

Show that l 1 and l 2 intersect and find the coordinates of the point of intersection, P . (5 marks)

(c)

The point A (À4,0, 11) lies on l 2 . The point B on l 1 is such that AP ¼ BP . Find the length of AB .

8

1

(4 marks)

(a) Solve the differential equation d y d x

¼

p ffiffi ffiþffi ffi ffi ffi 1

2 y

x2

given that y ¼ 4 when x ¼ 1 .

(6 marks)



1 8 1 (b) Show that the solution can be written as y ¼ 15 À þ 2 2 x x

END OF QUESTIONS

Copyright

Ó


P94574/Jun07/MPC4



.

(2 marks)



MPC4

9.00 am to 10.30 am



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P6320/Jun08/MPC4

6/6/6/

MPC4

2


1

ð Þ ¼ 27 x 3 À 9 x þ 2 . Find the remainder when f ð xÞ is divided by 3 x þ 1 . ðÞ

The polynomial f x is defined by f x (a) (b)

(i)

(2 marks)

  Show that f À ¼ 0 .

(1 mark)

ðÞ

(4 marks)

2 3

(ii) Express f x as a product of three linear factors. (iii)

Simplify 27 x 3

À 9 x þ 2 9 x 2 þ 3 x À 2

2

A curve is defined, for t

6¼ 0 , by the parametric equations x

At the point P on the curve, t

3

¼ 4t þ 3,

y

¼ 2t 1 À 1

¼ 12 .

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

(4 marks)

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

(3 marks)

(c) Find a cartesian equation of the curve.

(3 marks)

(a)

ð þ 2 xÞ , show that

By writing sin 3 x as sin x of x.

(b) Hence, or otherwise, find

4

(2 marks)

(a)

(i)

¼ 3 sin x À 4sin3 x

sin3 x d x .

(3 marks)

ð À xÞ

1 4

1 4


8 2 x ð81 À 16 xÞ % 3 À 274 x À 729

for small values of x .

(b) Use the result from part (a)(ii) to find an approximation for to seven decimal places. P6320/Jun08/MPC4

for all values (5 marks)

ð

Obtain the binomial expansion of 1

(ii) Hence show that

sin 3 x

(3 marks)

p ffi80ffi ffi , giving your answer 4

(2 marks)

3

5

(a)

The angle a is acute and sin a (i)

¼ 45 .

Find the value of cos a .

(1 mark)

ð À bÞ in terms of sin b

(ii) Express cos a (iii)

Given also that the angle b is acute and cos b

ð À bÞ .

of cos a (b)

(i) (ii)

6

(a)

¼ 1 , show that tan2 x þ 2tan x À 1 ¼ 0 . p ffiffi 1 Hence, given that tan 45° ¼ 1 , show that tan 22 ° ¼ 2 À 1 .

Express

2

2 x 2

À1

ð

in the form

B . þ x À 1 x þ 1

Hence find

(c)

Solve the differential equation

x 2

A

(2 marks) (2 marks) (3 marks)

(3 marks)

2

(b)

(2 marks)

¼ 135 , find the exact value

Given that tan 2 x

À 1 d x .

(2 marks) d y

2 y , given that y ¼ 1 when x ¼ 3 . ¼ d x 3ð x 2 À 1Þ

Show that the solution can be written as y 3

7

and cos b .

¼ 2 xð xþÀ11Þ .

(5 marks)

ð À 2, 1Þ and ð5, 3, 0Þ respectively.

The coordinates of the points A and B are 3,

The line l has equation r

253 2 13 ¼ 4 3 5 þ l4 0 5 . 0

À3

(a) Find the distance between A and B.

(2 marks)

(b) Find the acute angle between the lines AB and l . Give your answer to the nearest degree. (5 marks) (c)

The points B and C lie on l such that the distance AC is equal to the distance AB . Find the coordinates of C . (5 marks)



s

4

8

(a) The number of fish in a lake is decreasing. After t years, there are x fish in the lake. The rate of decrease of the number of fish is proportional to the number of fish currently in the lake.

(b)

(i)

Formulate a differential equation, in the variables x and t and a constant of proportionality k , where k > 0 , to model the rate at which the number of fish in the lake is decreasing. (2 marks)

(ii)

At a certain time, there were 20 000 fish in the lake and the rate of decrease was 500 fish per year. Find the value of k . (2 marks)

The equation P

¼ 2000 À AeÀ0:05t

is proposed as a model for the number of fish, P , in another lake, where t is the time in years and A is a positive constant. On 1 January 2008, a biologist estimated that there were 700 fish in this lake. (i) Taking 1 January 2008 as t

¼ 0, find the value of A.

(1 mark)

(ii) Hence find the year during which, according to this model, the number of fish in this lake will first exceed 1900. (4 marks)

END OF QUESTIONS

Copyright

Ó


P6320/Jun08/MPC4



MPC4

9.00 am to 10.30 am



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P15570/Jun09/MPC4

6/6/6/

MPC4

2


1

(a) Use the Remainder Theorem to find the remainder when 3 x 3 by 3 x 1 .

À

(b)

Express

3 x 3

are integers.

2

þ 8 x 2 À 3 x À 5 is divided

(2 marks)

þ 8 x 2 À 3 x À 5 in the form ax 2 þ bx þ c , where a, b and c 3 x À 1 3 x À 1

(3 marks)

A curve is defined by the parametric equations x

(a)

Find

d y d x

¼ 1t ,

y

¼ t þ 2t 1

in terms of t .

(4 marks)

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

¼ 1.

(4 marks)

(c) Show that the cartesian equation of the curve can be written in the form x 2

À 2 xy þ k ¼ 0

where k is an integer.

3

(3 marks)

ð À xÞÀ1

(a) Find the binomial expansion of 1

(b)

(i)

(ii)

Express

3 x 1 A in the form x 2 3 x 1 x

Find the binomial expansion of in x 2 .

(c)

À ð1 À Þð À Þ

up to and including the term in x 2 . (2 marks) B þ À 2 À 3 x , where A and B are (3integers. marks)


3 x 1 up to and including the term x 2 3 x (6 marks)

Find the range of values of x for which the binomial expansion of is valid.

P15570/Jun09/MPC4


3 x 1 x 2 3 x (2 marks)

3

4

A car depreciates in value according to the model V

¼ Ak t

where £V is the value of the car t months from when it was new, and A and k are constants. Its value when new was £12 499 and 36 months later its value was £7000. (a)

(i) (ii)

Write down the value of A.

(1 mark)

Show that the value of k is 0.984 025, correct to six decimal places.

(2 marks)

(b) The value of this car first dropped below £5000 during the nth month from new. Find the value of n. (3 marks)

5

A curve is defined by the equation 4 x 2 Find the gradient at the point

6

(a)

(i)

(b)

(i)

ð1, 3Þ on this curve. þ

(5 marks)

þ ¼

Show that the equation 3 cos 2 x 7 cos x 5 0 can be written in the form a cos2 x b cos x c 0 , where a, b and c are integers. (3 marks)

þ

(ii)

þ y 2 ¼ 4 þ 3 xy .

þ ¼

Hence find the possible values of cos x .

þ

(2 marks)

ð þ Þ

Express 7 sin y 3cos y in the form R sin y a , where R > 0 and a is an acute angle. Give your value of a to the nearest 0.1 ° . (3 marks)

þ

¼

(ii) Hence solve the equation 7 sin y 3cos y 4 for all solutions in the interval 0° 4 y 4 360° , giving y to the nearest 0.1 ° . (3 marks) (c)

(i)

Given that b is an acute angle and that tan b

¼ 2p ffi2ffi , show that cos b ¼ 13 .

(2 marks)

(ii)

ffi ffi p Hence show that sin 2 b ¼ p 2 , where p is a rational number.

(2 marks)



s

Mathematics Mpc4 Unit Pure Core 4

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