A-level Maths Pure Mathematics 2 9MA0_02 Unofficial markscheme

What did you rearrange to get the result?

What was equation of normal for the integration with xln(x) got gradient as minus a half then
when Infound where crossed I got 3e not 2e? Wondering if I got equation or basic maths wrong

Sure it crosses the x axis at 3e, but that means the length of the base of the triangle is (3e-e) = 2e.
the equation of the line is y = -1/2 x + 3/2 e

Yeah so I got area as e^2 then added this to my answer

yep, you get a final answer of 5/4e^2 + 1/4

Hmm think I might have got 9/4 oops,
Presumably will get most of the marks?

yeah if you you had a correct method, you'll get most of the marks

What did it integrate to think I got 1/2 x^2 ln(e) -1/4x^2 ?

Vote what you think the A* mark will be here: https://www.strawpoll.me/15889159

Post your answers below, these are just my answers that I remember. If you disagree with any then let me know.

1) Functions
a) (2 marks) gg(5) = 40/7? Incorrect 40/9

b) (1 mark) Range of g(x): 2 < g(x) <= 40/7 needs to be 40/9

c) (3 marks) Inverse function for g(x), and state the domain: domain of f^-1(x) = range of f(x)

2) Vectors
a) (2 marks) Position vector: 6i-7j+10k yes

b) (3 marks) Work out the value of a: 2-2root2 yes

3) Modulus & proof
a) Prove if m and n are irrational, then mn is not necessarily irrational (disprove with counter-example). Easiest method is m=root 2 and n=root 8

4) Sigma notation

a) (4 marks) Show that q: use the addition rule, seperate each term, you had to show the geometric and arithmetic sequences for each term.

b) (2 marks) Calculate sum of sequence: First determine the period, and the sum of the period (13/6). Then do (50*13/6). Answer was 325/3. yes

5) Quadratic equation for trajectory
a) (3 marks) Form the equation: 1 mark for using -x^2, 2 marks for complete equation: y= -0.03x^2 + 1.2x yes

b) (2 marks) Determine the longest distance when H=3: Equate your eq. from part a) to 3. Then calculate x.

c) (1 mark) Reason why it is unreliable: The model assumes that air resistance has a negligible/no effect on the trajectory, hence the actual height of the ball will be lower than what the model predicts. Or, you could state that the model allowed for negative values for H, which is impossible.

6) Polynomial function
a) i) (1 mark) Calculate f(2): f(2) = 0. yes

ii) (2 marks) Hence, express f(x) has a product of 2 algebraic factors: Due to factor theorem, as f(2)=0, (x-2) is a factor of f(x). So f(x)=(x-2)(-x^2-3x-5) (can't remember the second factor).

c) (2 marks) Explain why this equation has exactly 2 solutions: The right factor had no real solutions, as (b^2-4ac)<0. The left factor had 2 solutions, as it was y = +-sqrt(2). yes

d) (1 mark) State how many solutions this equation has: 3 solutions

7) Newton Raphson
a) (3 marks) Show that x=(...): x=(x-f(x)/f'(x)). Required you to find f '(x), and then combine the x and the fraction together.

b) (1 mark) Why won't this work when x=1: Because f '(0) = 0, so the denominator is 0 - hence, x is undefined yes.

8) Trigonometric equations
a) (2 marks) Show that (1-cos2x) = sin2xtanx: 1-cos2x = (1-(1-2sin^2x)) = 2sin^2(x) = tanxsin2x

b) (5 marks) Find all solutions to 5sinx - 5cosx = 2
2 marks for converting it into the form of Rsin(x+a) = 2
3 marks for finding all solutions (I got 2 solutions - from sin(x-45) = 2/5root2 ) yes

c) (6 marks) Find all solutions to (sec^2x -5)(1-cos2x) = tan^2xsin2x
1 mark for converting the (1-cos2x) factor to sin2xtanx
1 mark for dividing through by sin2xtanx loses solutions
1 mark for using tan^2x+1=sec^2(x), and forming an equation for tanx
3 marks for finding all solutions (i got 3 solutions, one being -pi/4).

9) Sucking mint question
a) (5 marks) Form an equation for the radius with respect to time: First define variables r and t. I got r^3=60/4(t) + 125 or something.

b) (3 marks) Calculate the time for the mint to dissolve: I got 5 minutes, 6 seconds. yes

c) (1 mark) Why is this model unrealistic: It allowed for infinitely small radii, which is unrealistic (it can never actually be of 0 radius).

10) Differentiation from first principles
a) (5 marks) Show that d/dx(cosx) = -sinx: Follow through standard method for differentiation from first principles

11) Area under graph
(10 marks) Calculate the area under the graph:

1 mark for getting gradient of normal
2 mark for getting equation of normal
1 mark for getting the x coords for the equations (x=1, x=e, x=2e)
3 marks for integrating y = xlnx by parts (from x=1 to x=e)
2 marks for integrating equation of normal (from x=e to x=2e)
1 mark for final answer: A = 5/4e^2 + 1/4 yes

12) Differentiation
a) Partial fractions: A=3, B=4, C=-2 yes
b) Show that function is always decreasing - For f'(x), the denominators are always positive (xyz)^2 and numerators are negative. Negative + Negative = negative, so f'(x) is always negative.

13) Mice population
a) (1 mark) State the initial population: 90 yes

b) (4 marks) Show that dP/dT = T(300-T)/1200:
c) (3 marks) Find the time at which dP/dt is at a maximum: Find the derivative of dP/dT, then equate it to 0.
time was -4ln(3/7) yes

d) (1 mark) State the largest possible population: 300

1/2x^2ln(x) - 1/4x^2. and the limits are from 1 to e
That gets you 1/4e^2 + 1/4

Btw wasnt there 14 questions?

1/2x^2ln(x) - 1/4x^2. and the limits are from 1 to e
That gets you 1/4e^2 + 1/4

The Newton raphson rearrangement - were you meant to use the formula? I did it by just rearranging the given function (the 2x3 thing) into the iterative formula. What did you do and would I lose marks /3. Stupidly didn't read that part

Guys, for the Newton iteration; I remember the question. Stupidly, I didn't read the 'Newton-Raphson' part and just went for the algebra. This was a 3 marker - do I get the marks? I did prove it?

Question:

2x^3 + x^2 - 1 = 0 --> prove that Xn+1 = 4Xn^3 + X^2n + 1 / 6X^2 n + 2Xn

My working:

2x^3 + x^2 - 1 = 0
(6x^3 - 4x^3) + (2x^2 - x^2) - 1 = 0
--> 6x^3 + 2x^2 = 4x^3 + x^2 + 1
--> x(6x^2 + 2x) = 4x^3 + x^2 + 1
--> x = (4x^3 + x^2 + 1) / (6x^2 + 2x)

I think working they wanted:

f(x) = 2x^3 + x^2 - 1
f'(x) = 6x^2 + 2x
So then you can prove it by the fractions.

Would I get 3/3 or 0/3?

I really don't know. I think they're expecting a Newton-Raphson formula use though. It really just depends how the examiners feeling, maybe they'll make a SC for something like this, who knows.