Exam (elaborations)
ACTUAL OCR A LEVEL 2024 WITH MARK SCHEME MATHS A H240 PAPER 3
ACTUAL OCR A LEVEL 2024 WITH MARK SCHEME MATHS A H240 PAPER 3
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Oxford Cambridge and RSAThursday 20 June 2024 – Afternoon
A Level Mathematics A
H240/03 Pure Mathematics and Mechanics
Time allowed: 2 hours
* 1 3 3 6 2 3 6 1 2 0 *
You must have:
• the Printed Answer Booklet
• a scientific or graphical calculator
QP
INSTRUCTIONS
• Use black ink. You can use an HB pencil, but only for graphs and diagrams.
• Write your answer to each question in the space provided in the Printed Answer
Booklet. If you need extra space use the lined pages at the end of the Printed Answer
Booklet. The question numbers must be clearly shown.
• Fill in the boxes on the front of the Printed Answer Booklet.
• Answer all the questions.
• Where appropriate, your answer should be supported with working. Marks might be
given for using a correct method, even if your answer is wrong.
• Give non-exact numerical answers correct to 3 significant figures unless a different
degree of accuracy is specified in the question.
• The acceleration due to gravity is denoted by g m s–2. When a numerical value is
needed use g = 9.8 unless a different value is specified in the question.
• Do not send this Question Paper for marking. Keep it in the centre or recycle it.
INFORMATION
• The total mark for this paper is 100.
• The marks for each question are shown in brackets [ ].
• This document has 12 pages.
ADVICE
• Read each question carefully before you start your answer.
© OCR 2024 [603/1038/8] OCR is an exempt Charity
DC (PQ/SW) 336086/3 Turn over
, 2
Formulae
A Level Mathematics A (H240)
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,
n
where n C r = n C r = c m =
n!
r r! ^n - rh !
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
© OCR 2024 H240/03 Jun24
, 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
y dx . 12 h "^y 0 + y nh + 2 ^y 1 + y2 + f + y n - 1h, , where h =
b b-a
Trapezium rule: ya n
f ^x nh
The Newton-Raphson iteration for solving f ^xh = 0 : x n + 1 = xn -
f l^xnh
Probability
P ^A , Bh = P ^Ah + P ^Bh - P ^A + Bh
P ^A + Bh
P ^A + Bh = P ^Ah P ^B Ah = P ^Bh P ^A Bh or P ^A Bh =
P ^Bh
Standard deviation
/^x - -xh / f ^x - -xh
2
2
/ x2 -2 / fx 2 - 2
= - x or = / f -x
n n /f
The binomial distribution
If X + B ^n, ph then P ^X = xh = c m p x ^1 - ph , mean of X is np, variance of X is np ^1 - ph
n n-x
x
Hypothesis test for the mean of a normal distribution
If X + N ^n, v 2h then X + N cn,
v 2m
+ N ^0, 1h
X-n
and
n v n
Percentage points of the normal distribution
If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the
value of z such that P ^Z G zh = p .
p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995
z 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291
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 + vh t s = 12 ^u + vh t
v 2 = u 2 + 2as
s = vt - 12 at 2 s = vt - 12 at 2
© OCR 2024 H240/03 Jun24 Turn over
, 4
Section A
Pure Mathematics
1 Simplify each of the following.
(a) ^2a 2h # a -1
3 3
[2]
4
4x 2 - 9
^2x 2 + 5x - 12h (2x + 3)
(b) [2]
2 In this question you must show detailed reasoning.
A
O 2 rad
(3x + 1) cm
B
The diagram shows a sector AOB of a circle with centre O and radius (3x + 1) cm . The angle AOB
is 2 radians. The area of sector AOB is less than (44x - 7) cm 2 .
Find the set of possible values of x. Give your answer in set notation. [5]
3 (a) Expand (3 - 2x) -2 in ascending powers of x up to and including the term in x 2 . [4]
(b) State the set of values of x for which this expansion is valid. [1]
a+x
(c) When is expanded in ascending powers of x, the coefficient of x is zero.
(3 - 2x) 2
Determine the value of the constant a. [2]
© OCR 2024 H240/03 Jun24
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