A level Mathematics B MIE H640/01 Pure Mathematics &
Mechanics 2025 Actual Exam Paper 1 With 100% Verified
Questions & Correct Answers Graded A+
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
na l n d
Sn = 12 ^+=h "2a + -^n 1h ,
Geometric series
Sn = a^11--rrnh
S3= 1-a r for r 1 1
Binomial series
^a + = +bhn an n
C1 an-1b + nC2 an-2 2b + +f n
Cr an-rbr + +f bn ^n !
n!
Nh, where nCr = = =n Cr KKLnrNOOP r!^n - rh! J
^1+ = + +xhn 1 nx n n^ 2-! 1hx2+ +f n n^ -1hfr!^n - +r 1hxr +f ^x
1 1, n !Rh
Differentiation
f^xh fl^xh
tan kx k sec2kx
sec x sec tanxx
cot x -cosec2x
cosec x -cosec cotx x
, 2
u d x dx
Quotient Rule y = v , ddyx = v d u v-2 u d v
Differentiation from first principles
^
fl xh= limh"0 f^x + -hhh f^xh
Integration
d h
c dffl^^xxh dx = ln
f^xh + c e
1
;fl^xhaf^xhkn dx = n + 1af^xhkn+1+ c
d d
Integration by parts ; u dvx dx = -uv; v dux dx
Small angle approximations sini i. , cosi. 1- i2, tani i. where i is
measured in radians 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 ! tan B
tan^A ! Bh= 1 "tan A tan B aA ! B !^k + hrk
Numerical methods
x
Trapezium rule: ;ab y dx . h"^y0+ +ynh 2^yf^1x+ + +h=y20:f y h,, where f^^xnhh h = b -n a
n+1 n-1
The Newton-Raphson iteration for solving = -xn fl xn
Probability
, 3
h
P^A j Bh= P^Ah+P^Bh-P^A k B h ^h ^^hh
PAkB
P^A k Bh= P^AhP^B Ah= P^BhP^A B or PA B=
PB
Sample variance s2 = n -1 1 Sxx where Sxx =/^xi - =-xh2 / xi2-
^/nxih2 =/ x2i - n-x2
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = =rh nCr p qr n-r
where q = -1 p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
K v O -n
If X + N^nv, 2h then X + NK Jn, n2NO P and vX n + N^0 1, h
L
p 10 5 2 1
z 1.645 1.960 2.326 2.576
z
Kinematics
Motion in a straight line Motion in two dimensions
v = +u at v = +u at
Percentage points of the Normal distribution
s = +ut at2 s = +ut at2 s = + ^u v th s = + ^u
1 p% 1 p%
vht 2 2
v = +u 2as
2 2
s = -vt 12 at2 s = -vt 12at2
© OCR 2024 H640/01 Jun24 Turn over
, 4
Section A (25 marks)
1 A student states that 1+ x2 1 (1+ x)2 for all values of x.
Using a counter example, show that the student is wrong. [2]
2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine
of the car produces a driving force of 6000 N. A resistance of 800 N acts on the car. A resistance
of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.
(a) Draw a force diagram showing all the horizontal forces on the car and the trailer. [2]
(b) Calculate the acceleration of the car and trailer. [3]
3 A particle hangs at the end of a string. A horizontal force of magnitude F N acting on the particle
holds it in equilibrium so that the string makes an angle of 20° with the vertical, as shown in the
diagram. The tension in the string is 12 N.
20°
FN
(a) Find the value of F. [2]
(b) Find the mass of the particle. [3]
4 The vectors v1 and v2 are defined by v1 = +2ai bj and v2 = bi-3j where a and b are constants.
© OCR 2024 H640/01 Jun24
Mechanics 2025 Actual Exam Paper 1 With 100% Verified
Questions & Correct Answers Graded A+
Formulae A Level Mathematics B (MEI) (H640)
Arithmetic series
na l n d
Sn = 12 ^+=h "2a + -^n 1h ,
Geometric series
Sn = a^11--rrnh
S3= 1-a r for r 1 1
Binomial series
^a + = +bhn an n
C1 an-1b + nC2 an-2 2b + +f n
Cr an-rbr + +f bn ^n !
n!
Nh, where nCr = = =n Cr KKLnrNOOP r!^n - rh! J
^1+ = + +xhn 1 nx n n^ 2-! 1hx2+ +f n n^ -1hfr!^n - +r 1hxr +f ^x
1 1, n !Rh
Differentiation
f^xh fl^xh
tan kx k sec2kx
sec x sec tanxx
cot x -cosec2x
cosec x -cosec cotx x
, 2
u d x dx
Quotient Rule y = v , ddyx = v d u v-2 u d v
Differentiation from first principles
^
fl xh= limh"0 f^x + -hhh f^xh
Integration
d h
c dffl^^xxh dx = ln
f^xh + c e
1
;fl^xhaf^xhkn dx = n + 1af^xhkn+1+ c
d d
Integration by parts ; u dvx dx = -uv; v dux dx
Small angle approximations sini i. , cosi. 1- i2, tani i. where i is
measured in radians 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 ! tan B
tan^A ! Bh= 1 "tan A tan B aA ! B !^k + hrk
Numerical methods
x
Trapezium rule: ;ab y dx . h"^y0+ +ynh 2^yf^1x+ + +h=y20:f y h,, where f^^xnhh h = b -n a
n+1 n-1
The Newton-Raphson iteration for solving = -xn fl xn
Probability
, 3
h
P^A j Bh= P^Ah+P^Bh-P^A k B h ^h ^^hh
PAkB
P^A k Bh= P^AhP^B Ah= P^BhP^A B or PA B=
PB
Sample variance s2 = n -1 1 Sxx where Sxx =/^xi - =-xh2 / xi2-
^/nxih2 =/ x2i - n-x2
Standard deviation, s = variance
The binomial distribution
If X + B^n, ph then P^X = =rh nCr p qr n-r
where q = -1 p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
K v O -n
If X + N^nv, 2h then X + NK Jn, n2NO P and vX n + N^0 1, h
L
p 10 5 2 1
z 1.645 1.960 2.326 2.576
z
Kinematics
Motion in a straight line Motion in two dimensions
v = +u at v = +u at
Percentage points of the Normal distribution
s = +ut at2 s = +ut at2 s = + ^u v th s = + ^u
1 p% 1 p%
vht 2 2
v = +u 2as
2 2
s = -vt 12 at2 s = -vt 12at2
© OCR 2024 H640/01 Jun24 Turn over
, 4
Section A (25 marks)
1 A student states that 1+ x2 1 (1+ x)2 for all values of x.
Using a counter example, show that the student is wrong. [2]
2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine
of the car produces a driving force of 6000 N. A resistance of 800 N acts on the car. A resistance
of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.
(a) Draw a force diagram showing all the horizontal forces on the car and the trailer. [2]
(b) Calculate the acceleration of the car and trailer. [3]
3 A particle hangs at the end of a string. A horizontal force of magnitude F N acting on the particle
holds it in equilibrium so that the string makes an angle of 20° with the vertical, as shown in the
diagram. The tension in the string is 12 N.
20°
FN
(a) Find the value of F. [2]
(b) Find the mass of the particle. [3]
4 The vectors v1 and v2 are defined by v1 = +2ai bj and v2 = bi-3j where a and b are constants.
© OCR 2024 H640/01 Jun24
