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

UNOFFICIAL MARK SCHEME

1.

A) Express f(x) = (19x-2)/(5-x)(1+6x) in the form:

f(x) = A/(5-x) + B/(1+6x):



A=3 B=-1 (3 marks)



B) Find Integral between x = 4 and x = 0 of f(x). Show it equals pln5.



= 8/3ln5 (6 marks)



2.

A) Express 2cosx - 5sinx in the form Rcos(x+alpha), R>0, 0<alpha<pi/2

√29cos(x+1.19) (3 Marks)



B) Find value of x in interval 0<x<2pi where 2cosx - 5sinx is at a maximum.

(i). x=5.09 (2 Marks)

(ii) Solve equation 2cosx - 5sinx + 1 = 0

x=0.567, 3.34 (3 Marks)



3.

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

Given when g(x) is divided by (2x+1) the remainder is -2, find d.

d=1 (2 Marks)



B) Given x = -0.5 is a solution of g(x)=0, express g(x) as 3 linear factors.

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



C) h(x) is given by 4x^2 - 1 / f(x). Simplify h(x), showing it is a decreasing function.



h(x) = 1/(2x+3), h'(x) = -2(2x+3)^-2 this is always negative, therefore it is a decreasing function (4 Marks)



4.

A) Find the binomial expansion of (1+5x)^(1.5):

= 1 + x - 2(x^2) (2 Marks)

B) Expand (8 + 3x)^(-2/3)

(i). 1/4 - (1/16)x + (5/256)x^2 (3 Marks)

(ii) Find an estimate for Cube root of 1/81.

0.2313 (2 Marks)



5. x = cos2t, y = sint

A) Find the gradient at t = pi/6.

dy/dx= -1/2 (3 Marks)



B) Equation of normal at t=pi/6

y=2x-1/2 (2 Marks)



C) Normal cuts curve at Q(cos2q, sinq). Set up a quadratic in sinq to find where the normal cuts the curve again. Then find the x co-ordinate of Q.

cos2q = -1/8 (5 Marks)



6. A = (3,2,10), B = (5,-2, 4)

Line l passes through A and has equation:

r = [3,2,10] + lambda[x,y,z]



A) Find angle between line l and AB.

Theta = 60 degrees (4 Marks)

B) C(15, 6, 2) (4 Marks)

C) E(11, 0, 0) and (23, 3, 0) (4 Marks)



7. y^3 + 2e^(-3x)y + x = k

Find k given point P (ln2, 0.5) lies on curve.

A) k=1/4 - ln2 (1 Mark)

B). Find the gradient at P:

dy/dx = 11/8 (6 Marks)



8. dx/dt = (4+5x)^0.5/5(1+t)^2

A) Given t=0 when x=0 find x in terms of t.

x=9/20-1/2(t+1)-1/20(1+t)^2

OR

x = 1/5[(-1/2(1+t)^-1 + 5/2)^2 - 4]



(7 marks)



B) Another pond models a circle of radius r. Rate of change of radius is inversely proportional to pond surface area. Write a differential equation to model this situation.

(i) dr/dt = k/(pir^2) (3 Marks)



(ii) The rate of change of radius is 4.5 when the radius is 1. Find the radius when the rate is 0.5.

r=3 metres (2 Marks)



Date: 9th June 2015

Time: 09:00am

Duration: 1h 30m

hi there, would you be able to approximate for me what 60 marks in the paper would be?.... messed up d1, so cant bank on it.... need to get pretty much and average of A in all papers now

60 is definitely an A. I can't see it being anything else.

By the way that mark scheme in the first post is ever so slightly wrong (after my last exam on Tuesday I'll fix it and polish it up so it looks as good as the C3 one).


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thanks, thats best case scenario with about 3 to 4 ecf (generous marks) added on...

did you do D1 today? or mechanics 1?

Ah I'm sure you'll be fine! I hated this exam compared to core 3. I'm just hoping they both add to give 180.

Nah I did M1 last year and have M2 on 22nd June

How about you?


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yeah i did m1, think i lost a total of 8 marks so kinda buzzing

but i pray i dont miss an A overall...

got luck on your A* with all these marksceme making i hope u get it!!!!

I'm sure you won't miss it! You seem determined not to

Ah that's very kind of you - I'm just hoping the UMS from Core 3 pull my Core 4 UMS up so that they add to 180!

Don't care how - if I get 100 in Core 3 then I'll quite happily take 80 in Core 4


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good luck is all i have to say man.. however much i prepare, i always seem to get stuck and then waste humungous amounts of time on that question, then making me not think straight and rsh the rest of the paper.. well all finihsed maths now so its ok... can i ask you 1 more thing, i did statistics 1 last year and got 79, i did m1 this year and hopefully got higher and my original aim was to swap them. I did d1, and think i got round about if not lower than S1... i cant swap d1 and s1 can i?????????????... are they not both AS units????

to summarise, my unit are : Mechanics 1, Statistics 1 and Decision 1, i need two applied modules can it be any from these??? or will it be Decsiision 1 and Mechanics?

thanks!!!!!

Ooh that's an interesting one - I believe they will take your core units and the best two of your applied units to give you an overall grade


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really?????????.... so it doesnt matter which 2??? i originally thought D1 was an A2 unit and the other 2 were AS but when i got on ums converter.... they are all under AS OMG!!!

NO IT CANT BEEE....

i did mechanics at AS and decision 1 at A2

yeah, how was d1 for you/ good luck on getting the grade you want.... just found out, that i can swap d1 for s1 really happy beacuse i think i got about 60 ums in it

ahh thank you you too!

i think i did okay, a few questions threw me and i almost ran out of time, aha, im hoping for an A but i dont know how many marks i'd need for that so
how did you find it?

UNOFFICIAL MARK SCHEME

1.

A) Express f(x) = (19x-2)/(5-x)(1+6x) in the form:

f(x) = A/(5-x) + B/(1+6x):



A=3 B=-1 (3 marks)



B) Find Integral between x = 4 and x = 0 of f(x). Show it equals pln5.



= 8/3ln5 (6 marks)



2.

A) Express 2cosx - 5sinx in the form Rcos(x+alpha), R>0, 0<alpha<pi/2

√29cos(x+1.19) (3 Marks)



B) Find value of x in interval 0<x<2pi where 2cosx - 5sinx is at a maximum.

(i). x=5.09 (2 Marks)

(ii) Solve equation 2cosx - 5sinx + 1 = 0

x=0.567, 3.34 (3 Marks)



3.

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

Given when g(x) is divided by (2x+1) the remainder is -2, find d.

d=1 (2 Marks)



B) Given x = -0.5 is a solution of g(x)=0, express g(x) as 3 linear factors.

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



C) h(x) is given by 4x^2 - 1 / f(x). Simplify h(x), showing it is a decreasing function.



h(x) = 1/(2x+3), h'(x) = -2(2x+3)^-2 this is always negative, therefore it is a decreasing function (4 Marks)



4.

A) Find the binomial expansion of (1+5x)^(1.5):

= 1 + x - 2(x^2) (2 Marks)

B) Expand (8 + 3x)^(-2/3)

(i). 1/4 - (1/16)x + (5/256)x^2 (3 Marks)

(ii) Find an estimate for Cube root of 1/81.

0.2313 (2 Marks)



5. x = cos2t, y = sint

A) Find the gradient at t = pi/6.

dy/dx= -1/2 (3 Marks)



B) Equation of normal at t=pi/6

y=2x-1/2 (2 Marks)



C) Normal cuts curve at Q(cos2q, sinq). Set up a quadratic in sinq to find where the normal cuts the curve again. Then find the x co-ordinate of Q.

cos2q = -1/8 (5 Marks)



6. A = (3,2,10), B = (5,-2, 4)

Line l passes through A and has equation:

r = [3,2,10] + lambda[x,y,z]



A) Find angle between line l and AB.

Theta = 60 degrees (4 Marks)

B) C(15, 6, 2) (4 Marks)

C) E(11, 0, 0) and (23, 3, 0) (4 Marks)



7. y^3 + 2e^(-3x)y + x = k

Find k given point P (ln2, 0.5) lies on curve.

A) k=1/4 - ln2 (1 Mark)

B). Find the gradient at P:

dy/dx = 11/8 (6 Marks)



8. dx/dt = (4+5x)^0.5/5(1+t)^2

A) Given t=0 when x=0 find x in terms of t.

x=9/20-1/2(t+1)-1/20(1+t)^2

OR

x = 1/5[(-1/2(1+t)^-1 + 5/2)^2 - 4]



(7 marks)



B) Another pond models a circle of radius r. Rate of change of radius is inversely proportional to pond surface area. Write a differential equation to model this situation.

(i) dr/dt = k/(pir^2) (3 Marks)



(ii) The rate of change of radius is 4.5 when the radius is 1. Find the radius when the rate is 0.5.

r=3 metres (2 Marks)



Date: 9th June 2015

Time: 09:00am

Duration: 1h 30m

hey there, i need to clear something up in core 4, when its said show that the function is decreasing, i said as x increases, the function 1/f(x) decreases.... is this wrong? or did you explicity need to say it was negative, how many marks do you think i lost?

I'll alter the mark scheme soon. I plotted a graph of 1/2x-3 and said that as x tends towards infinity, h(x) decreases towards zero. I'm sure you'll get the marks for saying that as well because it does show it's a decreasing function. Differentiating it is just the more common method.


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oh nice so we basically did the same thing.... hope so.. whatever i did seemed 100% right at the time... i cant even remember if i differentiated..... when i look back though 4 marks for just saying as x tends to infinity it decreases seems wrong....