AQA A2 Mathematics MPC3 Core 3 - Friday 5th June 2015 [Exam Discussion Thread]

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*** OFFICIAL AQA CORE 3 MPC3 JUNE 2015 EXAM DISCUSSION THREAD ***
UNOFFICIAL MARK SCHEME
1.
A) Use the mid ordinate rule to find an estimate for the integral of \displaystyle \int_{1.5}^{5.5} e^{2-x} \ln (3x-2):

Mid-ordinate rule: = 2.541 (4 marks)

B) Find the gradient of the line y= e^{2-x} \ln (3x-2) at the point where x = 2:

\dfrac{3}{4} - \ln (4) (4 marks)

2.
A) Sketch y=4 - |2x+1|:

Intercepts are at (-2.5, 0), (1.5, 0), (0, 3). (4 marks)

B) Solve the equation x = 4 - |2x+1|:

x=-5, x=1 (3 marks)

C) Solve inequality x < 4 - |2x+1| :

-5 < x < 1 (2 marks)

D) Translation that maps y=|2x+1| onto y=4-|2x+1|:

EITHER
1) Reflection in x-axis
2) Translation by vector \begin{bmatrix} 0 \\ 4 \end{bmatrix}

OR

1) Translation by vector \begin{bmatrix} 0 \\ -4 \end{bmatrix}
2) Reflection in x-axis (4 marks)

3.
A) Show that 5 < \alpha < 6 (2 marks)

B) Prove x = 4 + \sqrt{13 - 6 \ln (x)} can be written as 6 \ln (x) +x^2 - 8x + 3 = 0 (3 marks)

C) Use iteration formula x_{n+1} = 4 + \sqrt{13 - 6 \ln (x_n)} with x_1=5:

x_2 = 5.828, x_3 = 5.557 (2 marks)

D) Stationary points of curve f(x) = 6 \ln x +x^2 -8x +3 are at:

(1,-4), (3, 6 \ln (3) - 12) (5 marks)

E) y = 2f(x-4), find new stationary points:

(5,-8), (7, 12 \ln (3) - 24) (2 marks)

4.
 f(x) = 5-e^{3x}

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

A) Range of f(x):

f(x) < 5 (2 marks)

B) Find the inverse function of f:

f^{-1}(x) = \dfrac{1}{3} \ln (5-x) (3 marks)

C) Find x when f^{-1}(x) = 0:

x=4 (1 mark)

D) Find gg(x) in the form \dfrac{ax+b}{cx+d}:

gg(x) = \dfrac{-(2x-3)}{6x-11} (3 marks)

5.
A) Prove \dfrac{d}{dx} (\tan x) = sec^{2} x (2 marks)

B) \displaystyle \int x sec^{2} x\ dx:

= x \tan x - \ln |sec x| + C (4 marks)

C) Volume of solid when curve y = (5 \sqrt{x})secx is rotated 2 \pi radians about the x axis between x=0 and x=1:

= 74 (2 sf) (3 marks)

6.
A) Sketch y = \sin^{-1}(3x):

End points = \left(\dfrac{1}{3}, \dfrac{\pi}{2}\right) and \left(\dfrac{-1}{3}, \dfrac{- \pi}{2}\right) (3 marks)

B) Given  x = \dfrac{1}{3} \sin y:

\dfrac{dx}{dy} = \dfrac{1}{3} \cos y

\therefore \dfrac{dy}{dx} = \dfrac{3}{\cos y} (2 marks)

7. Using substitution u = 6-x^2, solve \displaystyle \int_1^2 \dfrac{x^3}{\sqrt{6-x^2}}, expressing answer in the form:  p\sqrt5 + q\sqrt2 :

= \dfrac{13}{3} \sqrt{5} - \dfrac{16}{3} \sqrt{2} (7 marks)

8.
A) Prove 4 cosec^{2} \theta - cot^{2} \theta = k can be written as sec^{2} \theta = \dfrac{k-1}{k-4} where k \not= 4 (5 marks)

B) Given that 4 cosec^{2} (2x +75^{o}) - cot^{2} (2x +75^{o}) = 5, find values of x in the range 0^{o} < x < 180^{o}:

x = 22.5^{o}, 82.5^{o}, 112.5^{o}, 172.5^{o} (5 marks)
_____________________________

CORE 4 EXAM DISCUSSION THREAD:
http://www.thestudentroom.co.uk/showthread.php?t=3041479

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

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CD223
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How is everyone preparing? Does anyone have access to papers before 2005?

I've also updated links to the Core 4 and Mechanics 2 exam threads in the original post.


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Tiwa
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http://papers.xtremepapers.com/AQA/Maths/ Here you go. In my college, we're currently learning both M2 and C4 simultaneously. We finished C3 in November and we're on compound angles for C4 and Circular motion for M2, so we've almost finished covering all modules. I've just been using past papers and using the CGP A2 Maths revision book for revision. How about you?
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(Original post by Tiwa)
http://papers.xtremepapers.com/AQA/Maths/ Here you go. In my college, we're currently learning both M2 and C4 simultaneously. We finished C3 in November and we're on compound angles for C4 and Circular motion for M2, so we've almost finished covering all modules. I've just been using past papers and using the CGP A2 Maths revision book for revision. How about you?
Oh nice! We seem to be doing the same modules! I set up threads for the other two if you haven't already seen

Oh nice! We've done C3 and C4 and have just started M2 with resultant forces and moments revision. Basically we started off easy haha. Do you do Physics? I wondered how much M2 overlaps aha. I've been using the University of Cambridge published books for each module and then the CGP A2 Maths book and past papers for revision.


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(Original post by CD223)
Oh nice! We seem to be doing the same modules! I set up threads for the other two if you haven't already seen

Oh nice! We've done C3 and C4 and have just started M2 with resultant forces and moments revision. Basically we started off easy haha. Do you do Physics? I wondered how much M2 overlaps aha. I've been using the University of Cambridge published books for each module and then the CGP A2 Maths book and past papers for revision.


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Yeah, I do Physics. Physics and M2 overlap a lot. For one circular motion shows up in both Physics and M2. Moments, centres of gravity, Power, kinetic energy. Basically, everything except from i and j vectors shows us in Physics.
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(Original post by Tiwa)
Yeah, I do Physics. Physics and M2 overlap a lot. For one circular motion shows up in both Physics and M2. Moments, centres of gravity, Power, kinetic energy. Basically, everything except from i and j vectors shows us in Physics.
Ah that's good news! Will hopefully help me understand both a bit better! Which board are you on for physics?


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Tiwa
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I do OCR A Physic's. How about you?
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Actually, what I said before, scratch that. I think everything except i and j vectors, differentiation and integration show up in Physics.
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(Original post by Tiwa)
I do OCR A Physic's. How about you?
Ah I do AQA A good to know so much of M2 comes up!


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Pumuki63
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Doing AQA as well!!
Wanted to know which identities and rules we should be able to prove and know the proof?
Do we have to know how to show the integral of cotx dx = ln mod(sinx) +c
Thank you!
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(Original post by Pumuki63)
Doing AQA as well!!
Wanted to know which identities and rules we should be able to prove and know the proof?
Do we have to know how to show the integral of cotx dx = ln mod(sinx) +c
Thank you!
AQA Maths? Nice!
The only identities you need to know are:
sinx / cosx = tanx
sin^2x+cos^2x = 1
Cosec^2x = cot^2x + 1
Sec^2x = tan^2x + 1

Then derivatives, just that sinx -> cosx -> -sinx -> -cosx

and

tanx -> sec^2x

Any others will be broken down (ie: they won't expect you to already know it). Even the derivative of tanx is in the formulae booklet.


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(Original post by CD223)
AQA Maths? Nice!
The only identities you need to know are:
sinx / cosx = tanx
sin^2x+cos^2x = 1
Cosec^2x = cot^2x + 1
Sec^2x = tan^2x + 1

Then derivatives, just that sinx -> cosx -> -sinx -> -cosx

and

tanx -> sec^2x

Any others will be broken down (ie: they won't expect you to already know it). Even the derivative of tanx is in the formulae booklet.


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Okey dokey, that's good!! Thank you Think I'm going to make a little formula sheet of things I have to know that won't be in the formula booklet
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(Original post by Pumuki63)
Okey dokey, that's good!! Thank you Think I'm going to make a little formula sheet of things I have to know that won't be in the formula booklet
Ah that's what I'm planning on doing! It'll make things so much easier. Even trying to learn things on the formula booklet will help you tackle questions quicker.


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(Original post by CD223)
Ah that's what I'm planning on doing! It'll make things so much easier. Even trying to learn things on the formula booklet will help you tackle questions quicker.


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100% agree, not just time but I seem to find that if I don't remember a formula I don't always think to look in the booklet and so tackle the question without a formula and ...fail... only to find out that if I'd learnt the formula I would have got the question right and saved myself a lot of stress - stress which only makes you under-perform as you go through the paper...
Do you know of any resources other than past papers that have useful and challenging questions?
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(Original post by Pumuki63)
100% agree, not just time but I seem to find that if I don't remember a formula I don't always think to look in the booklet and so tackle the question without a formula and ...fail... only to find out that if I'd learnt the formula I would have got the question right and saved myself a lot of stress - stress which only makes you under-perform as you go through the paper...
Do you know of any resources other than past papers that have useful and challenging questions?
True. Have you tried Solomon papers?


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(Original post by CD223)
True. Have you tried Solomon papers?


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No I haven't but I will definitely check them out, thank you!!
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(Original post by Pumuki63)
No I haven't but I will definitely check them out, thank you!!
No worries!


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Does anyone know if we will be able to use C4 knowledge in answering a C3 question?


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(Original post by CD223)
Does anyone know if we will be able to use C4 knowledge in answering a C3 question?


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Unless they tell you to work it out in a specific way in the question than you can use whichever method you like, it's best examining wise if you try and stick to C3 though


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(Original post by jjsnyder)
Unless they tell you to work it out in a specific way in the question than you can use whichever method you like, it's best examining wise if you try and stick to C3 though


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Oh right, thank you! I'll try to stick to C3 methods. I just know in C4 they synoptically test everything so I wondered if C3 was similar!


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