AQA A2 Mathematics MPC4 Core 4 - 9th June 2015 [Discussion & unofficial markscheme]

*** OFFICIAL AQA CORE 4 MPC4 JUNE 2015 EXAM DISCUSSION THREAD ***

UNOFFICIAL MARK SCHEME

1. It is given that

f(x) = \dfrac{19x - 2}{(5 - x)(1 + 6x)}

can be expressed as

\dfrac{A}{(5-x)} + \dfrac{B}{(1 + 6x)}

, where

A

and

B

are integers.

(a) Find the values of

A

and

B

. [3 marks]

A = 3

B = -1

\therefore \dfrac{3}{(5-x)} + \dfrac{-1}{(1 + 6x)}

(b) Hence show that

\displaystyle \int_0^4 f(x)\ dx\ = k \ln5

, where

k

is a rational number. [6 marks]

\displaystyle \int_0^4 f(x)\ dx\ = \frac{8}{3} \ln5

2.
(a) Express

2 \cos x - 5 \sin x

in the form

R \cos (x + \alpha)

, where

R > 0

and

0 < \alpha < \frac{\pi}{2}

, giving your value of

\alpha

, in radians, to three significant figures. [3 marks]

2 \cos x - 5 \sin x = \sqrt{29} \cos (x + 1.19)

(b) (i) Hence find the value of

x

in the interval

0 < x < 2 \pi

for which

2 \cos x - 5 \sin x

has its maximum value. Give your value of

x

to three significant figures. [2 marks]

x = 5.09

(ii) Use your answer to part (a) to solve the equation

2 \cos x - 5 \sin x + 1 = 0

in the interval

0 < x < 2 \pi

, giving your solutions to three significant figures. [3 marks]

x = 0.568, 3.34

3.
(a) The polynomial

f(x)

is defined as

f(x) = 8x^3 - 12x^2 - 2x + d

, where

d

is a constant. When

f(x)

is divided by

(2x + 1)

, the remainder is

-2

. Use the Remainder Theorem to find the value of

d

. [2 marks]

d = 1

(b) The polynomial

g(x)

is defined by

g(x) = 8x^3 - 12x^2 - 2x + 3

.
(i) Given that

x = \frac{-1}{2}

is a solution of the equation

g(x) = 0

, write

g(x)

as a product of three linear factors. [3 marks]

g(x) = (2x + 1)(2x - 3)(2x - 1)

(ii) The function

h

is defined by

h(x) = \dfrac{4x^2 - 1}{g(x)}

for

x > 2

. Simplify

h(x)

, and hence show that

h

is a decreasing function. [4 marks]

h(x) = \dfrac{1}{(2x - 3)}

EITHER

h'(x) = -2(2x - 3)^{-2}\ \therefore

for

x > 2

h'(x) < 0

therefore

h(x)

is a decreasing function.

OR

Plot a graph of

h(x)

h(x) \rightarrow 0

when

x \rightarrow \infty

.

4.
(a) Find the binomial expansion of

(1 + 5x)^{\frac{1}{5}}

up to and including the term in

x^2

. [2 marks]

1 + x - 2x^2

(b) (i) Find the binomial expansion of

(8 + 3x)^{\frac{-2}{3}}

up to and including the term in

x^2

. [3 marks]

\dfrac{1}{4} + \dfrac{1}{16} x + \dfrac{5}{256} x^2

(ii) Use your expansion from part (b)(i) to find an estimate for

\sqrt[3]{\frac{1}{81}}

, giving your answer to four decimal places. [2 marks]

= 0.2313

5. A curve is defined by the parametric equations

x = \cos 2t

y = \sin t

.
The point

P

on the curve is where

t = \frac{\pi}{6}

.

(a) Find the gradient at

P

. [3 marks]

\dfrac{dy}{dx} = \dfrac{-1}{2}

(b) Find the equation of the normal to the curve at

P

in the form

y = mx + c

. [3 marks]

y = 2x - \frac{1}{2}

P

intersects the curve again at the point

Q (\cos 2q, \sin q)

. Use the equation of the normal to form a quadratic equation in

\sin q

and hence find the

x

-coordinate of

Q

. [5 marks]

= \dfrac{-1}{8}

6. The points

A

and

B

have coordinates

(3, 2, 10)

and

(5, -2, 4)

respectively.
The line

l

passes through

A

and has equation

r = \begin{bmatrix} 3 \\ 2 \\ 10 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ 1 \\ -2 \end{bmatrix}

(a) Find the acute angle between the line

l

and the line

AB

. [4 marks]

60^{o}

(b) The point

C

lies on

l

such that angle

ABC

90^{o}

. Find the coordinates of

C

. [4 marks]

C = (15, 6, 2)

D

is such that

BD

is parallel to

AC

and angle

BCD

90^{o}

. The point

E

lies on the line through

B

and

D

and is such that the length of

DE

is half that of

AC

. Find the coordinates of the two possible positions of

E

. [4 marks]

E = (11, 0, 0)

E = (23, 4, -8)

7. A curve has equation

y^3 + 2e^{-3x}y - x = k

, where

k

is a constant. The point

P (\ln2, \frac{1}{2})

, lies on this curve.

(a) Show that the exact value of

k

q - \ln2

, where

q

is a rational number. [1 mark]

k = \dfrac{1}{4} - \ln2

(b) Find the gradient of the curve at

P

. [6 marks]

\dfrac{dy}{dx} = \dfrac{11}{8}

8.
(a) A pond is initially empty and is then filled gradually with water. After

t

minutes, the depth of the water,

x

metres, satisfies the equation

\dfrac{dx}{dt} = \dfrac{\sqrt{4 + 5x}}{5 (1 + t)^2}

Solve this differential equation to find

x

in terms of

t

. [7 marks]

x = \frac{1}{5} [(\frac{-1}{2} (1 + t)^{-1} + \frac{5}{2})^2 - 4]

(b) Another pond is gradually filling with water. After

t

minutes, the surface of the water forms a circle of radius

r

metres. The rate of change of the radius is inversely proportional to the area of the surface of the water.

(i) Write down a differential equation, in the variables

r

and

t

and a constant of proportionality, which represents how the radius of the surface of the water is changing with time. (You are not required to solve your differential equation.) [3 marks]

\dfrac{dr}{dt} = \dfrac{k}{\pi r^2}

(ii) When the radius of the pond is

1\ \text{metre}

, the radius is increasing at a rate of

4.5

metres per second. Find the radius of the pond when the radius is increasing at a rate of

0.5

metres per second.

r = 3\ \text{metres}

__________________________________________________________________________

[TOTAL: 75 marks]
__________________________________________________________________________

CORE 3 EXAM DISCUSSION THREAD:
http://www.thestudentroom.co.uk/showthread.php?t=3041463

MECHANICS 2 EXAM DISCUSSION THREAD:
http://www.thestudentroom.co.uk/showthread.php?t=3047357
_____________________________________________________________________________

RESOURCES:

(edited 8 years ago)

AQA Core 4 Revision Notes.pdf277.7KB

C4 revision notes 2.pdf372.2KB

Core 4 Revision Sheet.docx80.5KB

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Reply 1

CD223

How is everyone preparing? Does anyone have access to papers before 2005?

I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.

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Reply 2

madmenace

Original post by CD223

How is everyone preparing? Does anyone have access to papers before 2005?

I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.

Posted from TSR Mobile

hows your preperation going for c3 and c4? have you finsihed all the content?

Reply 3

CD223

Original post by madmenace

hows your preperation going for c3 and c4? have you finsihed all the content?

Hey! Not bad thanks :smile:

yeah we've finished the contents for both and I'm about to move onto M2. How about you?
Getting around 63-65 in each paper. Desperately want to get that up to about 72 before the exam so I can get at least an A overall :smile:

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Reply 4

madmenace

Original post by CD223

Hey! Not bad thanks :smile:

Posted from TSR Mobile

Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already

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Reply 5

CD223

Original post by madmenace

Hey!!! Yeah I'm abit slower I've finished core 3 and I'm hitting around 69... just abit of c4 and D1 to go... exams are fast approaching I'm getting scared already

Posted from TSR Mobile

Ah I'm scared too! What do you have left of C4? I only finished it like 2 weeks ago, so we're starting M2 tomorrow I believe because the last few weeks has been mock period! What are you aiming for grade wise?

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Reply 6

madmenace

Vectors, but I've nearly finished.... Some of the harder parametric equations I'm still getting used to.. I've got 10 exams in total :-( I want to get AAA minimum... but need to hurry up so i can start past papaer season lol

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Reply 7

CD223

Original post by madmenace

Oh nice! Yeah parametric questions can be hard. I need AAA for my uni offer haha. I've got 8 exams :/ what other subjects do you do?

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Reply 8

madmenace

Physics and chemistry.... no further maths;-) Chemistry is the killer

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Reply 9

CD223

Original post by madmenace

Physics and chemistry.... no further maths;-) Chemistry is the killer

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Ah I've heard chemistry is hard. AQA physics?

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Reply 10

Tiwa

I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?

Reply 11

CD223

Original post by Tiwa

I'm currently learning C4 and M2 simultaneously. C4 is very challenging. I've been getting mid 55-62 on the C3 past papers. Need to raise my marks higher. Anyone have any revision tips?

Is your issue with the content or timing? Because I'd say nail the content before even worrying about the 90 minutes! Try Solomon papers for a variety of extra practice. Exam solutions and various other videos on YouTube are great. Have you seen Jack Brown's videos?

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Reply 12

Tiwa

Original post by CD223

I think it's about the content mostly. Sometimes when I do past papers, I sometimes forget the method of answering a questions. I know of Exam solutions. I haven;t heard of Solomon papers, though. I'll take it into account. Thank you so much!

Reply 13

CD223

Original post by Tiwa

Yeah the method varies much more at A2 compared to AS aha. Sure, any time!

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Reply 14

CD223

Does anyone know what standard integrals we have to know? There's some in my textbook that refer to sin and tan in a fraction but I wasn't sure if this was still relevant to today's syllabus.

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Reply 15

madmenace

Hey, yeah it's those 2 you have to be able to recognise....

Reply 16

tiny hobbit

Original post by CD223

How is everyone preparing? Does anyone have access to papers before 2005?

I've also updated links to the Core 3 and Mechanics 2 exam threads in the original post.

Posted from TSR Mobile

C3 didn't exist before 2005. Before that, C1 - 4 was in 3 modules, P1 - 3.