OCR A Level Mathematics B (MEI) H640/02 Pure Mathematics and Statistics Sample Question Paper

A Level Mathematics B (MEI)
H640/02 Pure Mathematics and Statistics
Sample Question Paper Version 5.3

Date – Morning/Afternoon
Time allowed: 2 hours
You must have:
• Printed Answer Booklet

You may use:
• a scientific or graphical calculator




INSTRUCTIONS
• Use black ink. HB pencil may be used for graphs and diagrams only.
• Complete the boxes provided on the Printed Answer Booklet with your name, centre number
and candidate number.
• Answer all the questions.
• Write your answer to each question in the space provided in the Printed Answer Booklet.
Additional paper may be used if necessary but you must clearly show your candidate number,
centre number and question number(s).
• Do not write in the bar codes.
• You are permitted to use a scientific or graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.

INFORMATION
• The total number of marks for this paper is 100.
• The marks for each question are shown in brackets [ ].
• You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is used. You should communicate your method with
correct reasoning.
• The Printed Answer Book consists of 20 pages. The Question Paper consists of 16 pages.




© OCR 2022 H640/02 Turn over
603/1002/9 B10026/5.3

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Formulae A Level Mathematics B (MEI) H640


Arithmetic series
S n  12 n(a  l )  12 n{2a  (n  1)d }

Geometric series
a (1  r n )
Sn 
1 r
a
S  for r  1
1 r

Binomial series
(a  b)n  a n  n C1 a n1b  n C2 a n2b2  K  n Cr a nr br  K  bn (n  ¥ ) ,
n n!
where n Cr  n Cr    
 r  r !(n  r )!
n(n  1) 2 n(n  1) K (n  r  1) r
(1  x)n  1  nx 
2!
x  K 
r!
x  K x  1, n  ¡ 
Differentiation
f ( x) f ( x)
tan kx k sec2 kx
sec x sec x tan x
cot x  cosec 2 x
cosec x  cosec x cot x
du dv
u v
u dy
Quotient Rule y  ,  dx 2 dx
v dx v

Differentiation from first principles
f ( x  h)  f ( x )
f ( x)  lim
h0 h

Integration
f ( x)
 f ( x)
dx  ln f ( x)  c



 f ( x)  f ( x)  dx  n  1 f ( x) 
n 1 n 1
c



 
dv du
Integration by parts u dx  uv  v dx
dx dx

Small angle approximations
sin    , cos   1  12  2 , tan    where θ is measured in radians


© OCR 2022 H640/02

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Trigonometric identities
sin( A  B)  sin A cos B  cos A sin B
cos( A  B)  cos A cos B m sin A sin B
tan A  tan B
tan( A  B)  ( A  B  (k  12 ) )
1 m tan A tan B

Numerical methods
b ba
Trapezium rule: a y dx  12 h{( y0  yn )  2( y1  y2  …  yn1 ) }, where h  n
f( xn )
The Newton-Raphson iteration for solving f( x)  0 : xn 1  xn 
f ( xn )

Probability
P( A  B)  P( A)  P( B)  P( A  B)
P( A  B)
P( A  B)  P( A) P( B | A)  P( B) P( A | B ) or P( A | B) 
P(B )

Sample variance
  xi   x 2  nx 2
2
1
s 2
S xx where S xx   ( xi  x ) 2   xi2  i
n 1 n

Standard deviation, s  variance

The binomial distribution
If X ~ B(n, p) then P( X  r )  n Cr p r q nr where q  1  p
Mean of X is np

Hypothesis testing for the mean of a Normal distribution
  2 X 
 
If X ~ N  ,  2 then X ~ N   ,  and
/ n
~ N(0, 1)
 n 

Percentage points of the Normal distribution

p 10 5 2 1
z 1.645 1.960 2.326 2.576

Kinematics
Motion in a straight line Motion in two dimensions
v  u  at v  u  at
s  ut  12 at 2 s  ut  12 at 2
s  12  u  v  t s  12  u  v  t
v 2  u 2  2as
s  vt  12 at 2 s  vt  12 at 2

© OCR 2022 H640/02 Turn over

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Answer all the questions.

Section A (23 marks)

1 In this question you must show detailed reasoning.

Find the coordinates of the points of intersection of the curve y  x 2  x and the line 2 x  y  4 . [5]




2 Given that f ( x)  x3 and g( x)  2 x3  1 , describe a sequence of two transformations which maps
the curve y  f ( x) onto the curve y  g( x) . [4]








12
3 Evaluate cos 3x dx , giving your answer in exact form. [3]
0




4 The function f ( x) is defined by f ( x)  x3  4 for 1  x  2.
For f 1 ( x) , determine
 the domain
 the range. [5]




5 In a particular country, 8% of the population has blue eyes. A random sample of 20 people is
selected from this population.
Find the probability that exactly two of these people have blue eyes. [2]




© OCR 2022 H640/02