Thursday 16 May 2024 – A𝘧ternoon
AS Level Mathematics B (MEI)
H630/01 Pure Mathematics and Mechanics
Time allowed: 1 hour 30 minutes
OCR GCE Mathematics B MEI
H630/01: Pure Mathematics and Mechanics
AS Level Question Paper plus Mark scheme 2024
Turn over
, 2
Formulae AS Level Mathematics B (MEI) (H630)
Binomial series
(a + b) n = an + nC1 an–1b + nC2 an–2b2 + ... + nCr an–rbr + ... +b n ^n e Nh,
n!
n
JC
nNr = nCr = K O =
where r r!^n - rh!
L P
n^n - 1h 2 n^n - 1h...^n - r + 1h
Rh
n
^1 + xh = 1 + nx + x + ... +
^x 1 ne
xr + ... 1,
2! r!
Di𝘧𝘧erentiation 𝘧rom 𝘧irst principles
𝘧^x + hh - 𝘧(x)
𝘧 l(x) = lim
h"0 h
Sample variance
2 1 2
2 2 ^/ xih 2 2
s = Sxx where Sxx = /(xi - = / xi - = / xi - n-x
n- n
-
x) 1
Standard deviation, s =
variance
The binomial distribution
I𝘧 X ~ B^n, ph then P (X = r) = nCr p r q n-r where q = 1 - p
Mean o𝘧X is np
Kinematics
Motion in a straight line
v = u + at
1
s = ut + at2
2
1
s= (u + v)
2
t v2 = u2 +
2as s = vt -
2
1 2
at
, 3
1 The triangle ABC has an obtuse angle at A. The angle at B is 15°. The length o𝘧 AC is 10 cm
and the length o𝘧 BC is 13 cm.
Calculate the size o𝘧 the angle at A. [2]
2 Two 𝘧orces F1 N and F2 N are given by F1 =-6i + 2j and F2 =-8i + j.
Show that the magnitude o𝘧 the resultant o𝘧 these two 𝘧orces is 205 N. [2]
3 Prove that, when n is an even number, n3 + 4 is a multiple o𝘧 4 but not a multiple o𝘧 8. [3]
4 The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the
midpoint o𝘧 AC. The angles BAC and BDC are each equal to x°. The lengths o𝘧 AB and CD
are 4 cm and 7 cm respectively.
B
4
cm
E C
A x°
7 cm
x°
D
Determine the value o𝘧 x. [4]
Turn over
, 4
5 In this question you must show detailed reasoning.
1 31
(a) Show that the gradient o𝘧 the curve y = 1 - 2xm at the point a , k is - 99
. [4]
xc
2 4 4 2
x
1 31
(b) Find the equation o𝘧 the tangent to the curve at a4 , 4 k giving your answer in the 𝘧orm
ax + by + c = 0, where a, b and c are integers. [2]
6 The polynomial x3 - 4x2 + 10x - 21 is denoted by 𝘧(x).
(a) Use the 𝘧actor theorem to show that (x - 3) is a 𝘧actor o𝘧 𝘧(x). [2]
(b) The polynomial 𝘧 (x) can be written as (x - 3)(x2 + bx + c) where b and c are constants.
Find the values o𝘧 b and c. [2]
(c) Show that x = 3 is the only real root o𝘧 the equation 𝘧(x) = 0. [2]
7 The velocity o𝘧 a particle moving in a straight line is modelled by v = 0.6t2 - 2.1t + 1.5 where v
is the velocity in metres per second and t is the time in seconds.
(a) Determine the times at which the particle is stationary. [2]
(b) Find the acceleration o𝘧 the particle at the 𝘧irst o𝘧 the times at which it is stationary. [2]
(c) Find the distance travelled by the particle between the times at which it is stationary. [2]
8 A circle with centre C has equation x2 + y2 - 6x - 16y + 48 = 0.
(a) Find the coordinates o𝘧 C. [2]
A line has equation y = x - 2 and intersects the circle at the points A and B. The midpoints o𝘧 AC
and BC are Al and Bl respectively.
(b) Determine the exact distance AlBl. [8]
AS Level Mathematics B (MEI)
H630/01 Pure Mathematics and Mechanics
Time allowed: 1 hour 30 minutes
OCR GCE Mathematics B MEI
H630/01: Pure Mathematics and Mechanics
AS Level Question Paper plus Mark scheme 2024
Turn over
, 2
Formulae AS Level Mathematics B (MEI) (H630)
Binomial series
(a + b) n = an + nC1 an–1b + nC2 an–2b2 + ... + nCr an–rbr + ... +b n ^n e Nh,
n!
n
JC
nNr = nCr = K O =
where r r!^n - rh!
L P
n^n - 1h 2 n^n - 1h...^n - r + 1h
Rh
n
^1 + xh = 1 + nx + x + ... +
^x 1 ne
xr + ... 1,
2! r!
Di𝘧𝘧erentiation 𝘧rom 𝘧irst principles
𝘧^x + hh - 𝘧(x)
𝘧 l(x) = lim
h"0 h
Sample variance
2 1 2
2 2 ^/ xih 2 2
s = Sxx where Sxx = /(xi - = / xi - = / xi - n-x
n- n
-
x) 1
Standard deviation, s =
variance
The binomial distribution
I𝘧 X ~ B^n, ph then P (X = r) = nCr p r q n-r where q = 1 - p
Mean o𝘧X is np
Kinematics
Motion in a straight line
v = u + at
1
s = ut + at2
2
1
s= (u + v)
2
t v2 = u2 +
2as s = vt -
2
1 2
at
, 3
1 The triangle ABC has an obtuse angle at A. The angle at B is 15°. The length o𝘧 AC is 10 cm
and the length o𝘧 BC is 13 cm.
Calculate the size o𝘧 the angle at A. [2]
2 Two 𝘧orces F1 N and F2 N are given by F1 =-6i + 2j and F2 =-8i + j.
Show that the magnitude o𝘧 the resultant o𝘧 these two 𝘧orces is 205 N. [2]
3 Prove that, when n is an even number, n3 + 4 is a multiple o𝘧 4 but not a multiple o𝘧 8. [3]
4 The perpendicular lines AC and BD intersect at E as shown in the diagram. The point E is the
midpoint o𝘧 AC. The angles BAC and BDC are each equal to x°. The lengths o𝘧 AB and CD
are 4 cm and 7 cm respectively.
B
4
cm
E C
A x°
7 cm
x°
D
Determine the value o𝘧 x. [4]
Turn over
, 4
5 In this question you must show detailed reasoning.
1 31
(a) Show that the gradient o𝘧 the curve y = 1 - 2xm at the point a , k is - 99
. [4]
xc
2 4 4 2
x
1 31
(b) Find the equation o𝘧 the tangent to the curve at a4 , 4 k giving your answer in the 𝘧orm
ax + by + c = 0, where a, b and c are integers. [2]
6 The polynomial x3 - 4x2 + 10x - 21 is denoted by 𝘧(x).
(a) Use the 𝘧actor theorem to show that (x - 3) is a 𝘧actor o𝘧 𝘧(x). [2]
(b) The polynomial 𝘧 (x) can be written as (x - 3)(x2 + bx + c) where b and c are constants.
Find the values o𝘧 b and c. [2]
(c) Show that x = 3 is the only real root o𝘧 the equation 𝘧(x) = 0. [2]
7 The velocity o𝘧 a particle moving in a straight line is modelled by v = 0.6t2 - 2.1t + 1.5 where v
is the velocity in metres per second and t is the time in seconds.
(a) Determine the times at which the particle is stationary. [2]
(b) Find the acceleration o𝘧 the particle at the 𝘧irst o𝘧 the times at which it is stationary. [2]
(c) Find the distance travelled by the particle between the times at which it is stationary. [2]
8 A circle with centre C has equation x2 + y2 - 6x - 16y + 48 = 0.
(a) Find the coordinates o𝘧 C. [2]
A line has equation y = x - 2 and intersects the circle at the points A and B. The midpoints o𝘧 AC
and BC are Al and Bl respectively.
(b) Determine the exact distance AlBl. [8]
