Tuesday 11 June 2024 – A𝘧ternoon
A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
OCR GCE Mathematics B MEI
H640/02: Pure Mathematics and Statistics
A Level Question Paper + Mark Scheme 2024
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
Sn = 12 n^a + lh =2 1 n"2a +^n - 1hd,
Geometric series
^
Sn = a 1 -
rn 1
- r
a
S3 =1 - r 𝘧or r 1 1
Binomial series
N,
n
^a + bh = an + nC1 a n-1b + nC2 a n-2b2 +𝘧+ nCr a n-rbr +𝘧+ ne
bn JnN
n n!
where Cr = n Cr = K O =
r r!^n - rh!
L P
^ - 1h 2 n^n - 1h 𝘧^n - r + 1h
R
n n n
^r1 + xh = 1 + nx + x +𝘧+ x 1 ne
x +𝘧 1,
2! r!
Di𝘧𝘧erentiation
𝘧x 𝘧l x
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
du dv
dy v -u
u
Quotient Rule y = , = dx dx
v dx
v2
Di𝘧𝘧erentiation 𝘧rom 𝘧irst principles
𝘧^x + hh - 𝘧^xh
𝘧 l^xh = lim
h"0 h
Integration
c 𝘧 l^xh
d dx = ln 𝘧^xh + c
e 𝘧^xh
n 1 n +1
; 𝘧 l^xha𝘧^xhk dxn=+ a𝘧^xhk + c
1 ; u dv dx = uv - ; v du dx
Integration by parts
dx dx
, 3
Small angle approximations
sin i ≈ i , cos i ≈ 1 -2 1
i 2 , tan i ≈ i where i is measured in radians
, 4
Trigonometric identities
sin
A! = sin A cos B ! cos A sin B
cos B
= cos A cos B " sin A sin B
A!
B 2
tan A ! tan
tan^A ! Bh B aA ! B ! ^k + 1hrk
= 1 " tan A tan
B
Numerical methods
Trapezium rule: ; b y dx ≈ 1 b-a
h"^y 2
+ y h + 2^y + +𝘧+ h,, where h =
0
y y
a
n 1 2 n n
-1
𝘧^xn
The Newton-Raphson iteration 𝘧or solving 𝘧^xh = 0: n +1 = xn -
x h𝘧
l^xnh
Probability
P A j B = P A +P B - P A
kB P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or
P^A Bh
= P^Bh
Sample variance
2 1 ^/ xih
2
2 2
s =
n S where Sxx = /^xi - = / x2 - = / x2 - n-x
1 xx n
--
xh i i
Standard deviation, s =
variance
The binomial distribution
I𝘧 X + B n, p then P^X = rh = nCr p r q n-r where q = 1 - p
Mean o𝘧 X is np
Hypothesis testing 𝘧or the mean o𝘧 a Normal distribution
J 2N X- n
2 v
I𝘧 X + N^n, v h then X +
O and ~ N^0, 1h
NKn, nP v
L n
Percentage points o𝘧 the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
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A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
OCR GCE Mathematics B MEI
H640/02: Pure Mathematics and Statistics
A Level Question Paper + Mark Scheme 2024
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
Sn = 12 n^a + lh =2 1 n"2a +^n - 1hd,
Geometric series
^
Sn = a 1 -
rn 1
- r
a
S3 =1 - r 𝘧or r 1 1
Binomial series
N,
n
^a + bh = an + nC1 a n-1b + nC2 a n-2b2 +𝘧+ nCr a n-rbr +𝘧+ ne
bn JnN
n n!
where Cr = n Cr = K O =
r r!^n - rh!
L P
^ - 1h 2 n^n - 1h 𝘧^n - r + 1h
R
n n n
^r1 + xh = 1 + nx + x +𝘧+ x 1 ne
x +𝘧 1,
2! r!
Di𝘧𝘧erentiation
𝘧x 𝘧l x
tan kx k sec2kx
sec x sec x tan x
cot x -cosec2x
cosec x -cosec x cot x
du dv
dy v -u
u
Quotient Rule y = , = dx dx
v dx
v2
Di𝘧𝘧erentiation 𝘧rom 𝘧irst principles
𝘧^x + hh - 𝘧^xh
𝘧 l^xh = lim
h"0 h
Integration
c 𝘧 l^xh
d dx = ln 𝘧^xh + c
e 𝘧^xh
n 1 n +1
; 𝘧 l^xha𝘧^xhk dxn=+ a𝘧^xhk + c
1 ; u dv dx = uv - ; v du dx
Integration by parts
dx dx
, 3
Small angle approximations
sin i ≈ i , cos i ≈ 1 -2 1
i 2 , tan i ≈ i where i is measured in radians
, 4
Trigonometric identities
sin
A! = sin A cos B ! cos A sin B
cos B
= cos A cos B " sin A sin B
A!
B 2
tan A ! tan
tan^A ! Bh B aA ! B ! ^k + 1hrk
= 1 " tan A tan
B
Numerical methods
Trapezium rule: ; b y dx ≈ 1 b-a
h"^y 2
+ y h + 2^y + +𝘧+ h,, where h =
0
y y
a
n 1 2 n n
-1
𝘧^xn
The Newton-Raphson iteration 𝘧or solving 𝘧^xh = 0: n +1 = xn -
x h𝘧
l^xnh
Probability
P A j B = P A +P B - P A
kB P^A k Bh
P^A k Bh = P^AhP^B Ah = P^BhP^A Bh or
P^A Bh
= P^Bh
Sample variance
2 1 ^/ xih
2
2 2
s =
n S where Sxx = /^xi - = / x2 - = / x2 - n-x
1 xx n
--
xh i i
Standard deviation, s =
variance
The binomial distribution
I𝘧 X + B n, p then P^X = rh = nCr p r q n-r where q = 1 - p
Mean o𝘧 X is np
Hypothesis testing 𝘧or the mean o𝘧 a Normal distribution
J 2N X- n
2 v
I𝘧 X + N^n, v h then X +
O and ~ N^0, 1h
NKn, nP v
L n
Percentage points o𝘧 the Normal distribution
p 10 5 2 1
1 p% 1 p%
z 1.645 1.960 2.326 2.576 2 2
z
Kinematics
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