Tuesday 11 June 2024 – Afternoon
A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S n = 12 n ^a + lh = 12 n "2a + ^n - 1h d ,
Geometric series
a ^1 - r nh
Sn =
1-r
a
S3 = for r 1 1
1-r
Binomial series
^a + bhn = a n + n C1 a n - 1 b + n C2 a n - 2 b 2 + f + n Cr a n - r b r + f + b n ^n ! Nh,
JnN
n!
where C r = n C r = KK OO =
n
L P r! ^n - rh !
r
n ^n - 1h 2 n ^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n ! Rh
2! r!
Differentiation
f ^xh f l^xh
tan kx k sec 2 kx
sec x sec x tan x
cot x - cosec 2 x
cosec x - cosec x cot x
du dv
v -u
u dy dx dx
Quotient Rule y = , =
v dx v 2
Differentiation from first principles
f ^x + hh - f ^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
dd dx = ln f ^xh + c
e f ^xh
; f l^xhaf ^xhk dx = n + 1 af ^xhk + c
n 1 n+1
Integration by parts ; u dx = uv - ; v dx
dv du
dx dx
Small angle approximations
sin i . i , cos i . 1 - 12 i 2 , tan i . i where i is measured in radians
, 3
Trigonometric identities
sin ^A ! Bh = sin A cos B ! cos A sin B
cos ^A ! Bh = cos A cos B " sin A sin B
tan ^A ! Bh = aA ! B ! ^k + 12h rk
tan A ! tan B
1 " tan A tan B
Numerical methods
Trapezium rule: ; y dx . 12 h "^y 0 + ynh + 2 ^y 1 + y2 + f + yn - 1h, , where h =
b
b-a
n
f ^x nh
a
The Newton-Raphson iteration for solving f ^xh = 0: xn + 1 = x n -
f l^x nh
Probability
P ^A j Bh = P ^Ah + P ^Bh - P ^A k Bh
P ^A k Bh
P ^A k Bh = P ^Ah P ^B Ah = P ^Bh P ^A Bh or P ^A Bh =
P ^Bh
Sample variance
^/ xih2
S where S xx = /^xi - xh = / x i -
1 2
2
s = - 2
= / x 2i - nx- 2
n - 1 xx n
Standard deviation, s = variance
The binomial distribution
If X + B ^n, ph then P ^X = rh = n C r p r q n - r where q = 1 - p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
J N
If X + N ^n, v 2h then X + N KKn, OO and
X -n
+ N ^0, 1h
v2
L n P v 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
Motion in a straight line Motion in two dimensions
v = u + at v = u + at
1 2
s = ut + 2 at s = ut + 12 at 2
s = 12 ^u + vh t s = 12 ^u + vh t
v 2 = u 2 + 2as
s = vt - 12 at 2 s = vt - 12 at 2
Turn over
, 4
Section A (21 marks)
1 Calculate the exact distance between the points (2, -1) and (6, 1). Give your answer in the form
a b, where a and b are prime numbers. [2]
The equation of a curve is y = e x . The curve is subject to a translation e o and a stretch scale
3
2
0
factor 2 parallel to the y-axis.
Write down the equation of the new curve. [2]
3 The histogram shows the amount spent on electricity in pounds in a sample of households in
March 2023.
4
3
Frequency
2
density
1
0
50 60 70 80 90 100
Amount spent on electricity in £
(a) Describe the shape of the distribution. [1]
A total of 16 households each spent between £60 and £65 on electricity.
(b) Determine how many households were in the sample altogether. [2]
4 (a) On the axes in the Printed Answer Booklet, sketch the graph of y = sin 2i for 0 G i G 2r .
[2]
1
(b) Solve the equation sin 2i =- for 0 G i G 2r . [3]
2
A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Time allowed: 2 hours
Turn over
, 2
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
S n = 12 n ^a + lh = 12 n "2a + ^n - 1h d ,
Geometric series
a ^1 - r nh
Sn =
1-r
a
S3 = for r 1 1
1-r
Binomial series
^a + bhn = a n + n C1 a n - 1 b + n C2 a n - 2 b 2 + f + n Cr a n - r b r + f + b n ^n ! Nh,
JnN
n!
where C r = n C r = KK OO =
n
L P r! ^n - rh !
r
n ^n - 1h 2 n ^n - 1h f ^n - r + 1h r
^1 + xhn = 1 + nx + x +f+ x +f ^ x 1 1, n ! Rh
2! r!
Differentiation
f ^xh f l^xh
tan kx k sec 2 kx
sec x sec x tan x
cot x - cosec 2 x
cosec x - cosec x cot x
du dv
v -u
u dy dx dx
Quotient Rule y = , =
v dx v 2
Differentiation from first principles
f ^x + hh - f ^xh
f l^xh = lim
h"0 h
Integration
c f l^xh
dd dx = ln f ^xh + c
e f ^xh
; f l^xhaf ^xhk dx = n + 1 af ^xhk + c
n 1 n+1
Integration by parts ; u dx = uv - ; v dx
dv du
dx dx
Small angle approximations
sin i . i , cos i . 1 - 12 i 2 , tan i . i where i is measured in radians
, 3
Trigonometric identities
sin ^A ! Bh = sin A cos B ! cos A sin B
cos ^A ! Bh = cos A cos B " sin A sin B
tan ^A ! Bh = aA ! B ! ^k + 12h rk
tan A ! tan B
1 " tan A tan B
Numerical methods
Trapezium rule: ; y dx . 12 h "^y 0 + ynh + 2 ^y 1 + y2 + f + yn - 1h, , where h =
b
b-a
n
f ^x nh
a
The Newton-Raphson iteration for solving f ^xh = 0: xn + 1 = x n -
f l^x nh
Probability
P ^A j Bh = P ^Ah + P ^Bh - P ^A k Bh
P ^A k Bh
P ^A k Bh = P ^Ah P ^B Ah = P ^Bh P ^A Bh or P ^A Bh =
P ^Bh
Sample variance
^/ xih2
S where S xx = /^xi - xh = / x i -
1 2
2
s = - 2
= / x 2i - nx- 2
n - 1 xx n
Standard deviation, s = variance
The binomial distribution
If X + B ^n, ph then P ^X = rh = n C r p r q n - r where q = 1 - p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
J N
If X + N ^n, v 2h then X + N KKn, OO and
X -n
+ N ^0, 1h
v2
L n P v 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
Motion in a straight line Motion in two dimensions
v = u + at v = u + at
1 2
s = ut + 2 at s = ut + 12 at 2
s = 12 ^u + vh t s = 12 ^u + vh t
v 2 = u 2 + 2as
s = vt - 12 at 2 s = vt - 12 at 2
Turn over
, 4
Section A (21 marks)
1 Calculate the exact distance between the points (2, -1) and (6, 1). Give your answer in the form
a b, where a and b are prime numbers. [2]
The equation of a curve is y = e x . The curve is subject to a translation e o and a stretch scale
3
2
0
factor 2 parallel to the y-axis.
Write down the equation of the new curve. [2]
3 The histogram shows the amount spent on electricity in pounds in a sample of households in
March 2023.
4
3
Frequency
2
density
1
0
50 60 70 80 90 100
Amount spent on electricity in £
(a) Describe the shape of the distribution. [1]
A total of 16 households each spent between £60 and £65 on electricity.
(b) Determine how many households were in the sample altogether. [2]
4 (a) On the axes in the Printed Answer Booklet, sketch the graph of y = sin 2i for 0 G i G 2r .
[2]
1
(b) Solve the equation sin 2i =- for 0 G i G 2r . [3]
2
