A-Level Pure Maths Paper 2 Official Unofficial Mark Scheme
Question
number
1
2
Question
Answer
a) find f’’(x)
b) When is f’’(x) = 0
c) Hence find the range of values for
which f(x) is concave
ai) Prove that 𝑠2 = 40
a)
Marks
a) 2
b) 2
Total: 4
6𝑥 + 4
2
3
b) 𝑥 =−
c) 𝑥 ≤−
2
3
a) i) Input 𝑥𝑛 and 𝑛 = 1to get 40
aii) Find 𝑠3 and 𝑠4
ii) 𝑠3 = 28
bi) Given that it is periodic with order 4, find
the value of 𝑠5
bii) Find sum of first 25 terms
𝑠4 = 33
b)
i) 𝑠5 = 35
a) 2
b) 3
Total: 5
ii) 851
can we starts putting marks in
3
Given the following logarithmic equation …
2
a) Prove 3𝑥 − 13𝑥 − 30 = 0
b) Find the roots of the quadratic
c) Explain which root is invalid
2
a) 3𝑥 − 13𝑥 − 30 = 0
b) 6 𝑜𝑟 −
c) −
5
3
5
3
, as it is negative.
a) 2
b) 2
c) 1
Total: 5
4
5
6
7
a) Given the following vectors:
𝑎 = 12𝑗
𝑏 = 16𝑗
𝑐 = 50𝑖 + 136𝑗
d=22𝑖 + 24𝑗
b) ABCD is a track and a runner
completes 2 laps in 5 minutes. Find
the average speed.
Given the following model…
And that the:
- Temperature is initially 85
- The rate of cooling -7.5
a) Find the value of A
b) FInd a full equation for the model
a) Show that 𝑘 = 12
b) Find the coordinates when it
intersects y axis
Partial fractions integration with k
a) Express in partial fractions
b) Given this integral, find k
a) 5𝐴𝐷 = 𝐵𝐶, therefore parallel
−1
b) 6. 99 𝑘𝑚ℎ
a) 𝐴 = 55
−1.136𝑥
b) 𝐻 = 55𝑒
+ 30
a) Show that 𝑘 = 12
b) (0, − 28)
a) 2
b) 3
Total: 5
a) 2
b) 2
Total: 4
a) 2
b) 3
Total: 5
2𝑘+3
𝑥+4
𝑘 =
,
𝑘−3
𝑥−2
21
𝑙𝑛5
−6
a) 3
b) 5
Total: 8
8
9
Harmonic identity
a) Write .. in the form
b) Find the minimum value of the
function
c) Find the value of 𝑥
a) 2 17 𝑐𝑜𝑠(θ − 1. 326)
Given the parametric equations
a) 𝑦 = 3𝑙𝑛(𝑥 + 25)
a) 3
b) 3
c) 3
Total: 9
b) 9 17
c) 𝑥 = 1. 326
2
𝑥 = 𝑡 + 3𝑡 − 16 and
𝑦 = 6𝑙𝑛(𝑡 + 3), with 𝑡 ≥ 3
b) 3𝑥 − 25𝑦 =− 150𝑙𝑛5
a) 3
b) 3
Total: 6
a) Find the cartesian equation
b) Find the tangent at…
10
3
2
𝑥 + 2𝑥𝑦 + 3𝑦 = 47
a) Find
a) −
𝑑𝑦
𝑑𝑥
a) 3
b) 3
a) Total: 6
2
2𝑦+3𝑥
6𝑦+2𝑐
b) 13𝑥 − 11𝑦 + 81 = 0
b) Find the equation of the normal at
the point 𝑃(− 2, 5)
Find the normal when P(x,y)
12
Modulus Music companies
a) Find the difference in subscribers
b) Find the point at which the price
change is likely to have occurred,
giving a reason for your answer
c) Find the range of values where
13
Cuboid with water:
● Height 5m
a) 5000
(b) When 𝑡 = 3, because this is when the number of subscribers begins to
increase.
(c) {𝑡: 0 ≤ 𝑡 ≤
5
3
} ∪ {𝑡 >
13
3
}
(d) It is invalid beyond 𝑡 = 7 because it predicts negative subscribers
a)
𝑑ℎ
𝑑𝑡
=
λ
ℎ
(Should get a value of 1/200 at some stage)
a) 2
b) 2
c) 5
d) 1
Total: 10
a) 3
b) 5
● Length 20m
● Width 10m
● Water height h
Rate of change of water volume is inversely
proportional to √h.
a) Show that
𝑑ℎ
𝑑𝑡
=
λ
ℎ
3
b)ℎ 2 = 0. 513𝑡 + 1. 152
c) 2
Total: 10
c) 18. 4
where λ is a
constant.
b) Show that h^(3/2)=At+B where A
and B are constants to 3 decimal
places.
c) How long does it take for h to be
maximum?
13
Binomial and integration
a) Expand..
b) Approximate the integral
c) Find the real integral
a)
1
9
−
2
27
𝑥+
1
27
b) 0. 03304
17
c) 6𝑙𝑛( 16 ) −
45
136
2
𝑥
a) 4
b) 4
c) 5
Total: 13
14
a) Trig
a) …
Given that
2𝑡𝑎𝑛θ(𝑐𝑜𝑠θ − 23…) = 𝑠𝑖𝑛2θ(𝑡𝑎𝑛²θ + 1) b) 360, 49, 540
Show that
14𝑠𝑖𝑛2θ(− 23𝑐𝑜𝑠²θ + 8𝑐𝑜𝑠θ + 15) = 0
b) Hence solve …
a) 3
b) 4
Total: 7
15
Proof
2
(𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥) < 1
(cosx-sinx)^2<1
cos^2(x)+sin^2(x)-2sin(x)cos(x)<1
1-2sin(x)cos(x)<1
-2sin(x)cos(x)<0
2sin(x)cos(x)><0
sin(2x)><0
Contradiction, for value the sin(2x) contradiction needs to be explained
otherwise the proof is incomplete.
Sin(2x)>0 is impossible because by the definition of obtuse, x is between 90
and 180 and so 2x is between 180 and 360, let a=2x. So sin(2x)=sin(a)
which is negative because from the graph of sin(x), the ranges x at 180 and
360 has sin(x) below the axis.
3 marks