Monday 13 May 2024 – Afternoon
AS Level Further Mathematics B (MEI)
Y410/01 Core Pure
Time allowed: 1 hour 15 minutes
Turn over
, 2
1 The quadratic equation x 2 + ax + b = 0 , where a and b are real constants, has a root 2 - 3i.
(a) Write down the other root. [1]
(b) Hence or otherwise determine the values of a and b. [3]
J 1 aN J2 0N J- 1 0N
2 K
The matrices A, B and C are given by A = K O K
, B=K O and C = KK O, where a is
- 1 2O 1 - 1O 2 1O
a constant. L P L P L P
(a) By multiplying out the matrices on both sides of the equation, verify that A^BCh = ^ABhC .
[4]
(b) State the property of matrix multiplication illustrated by this result. [1]
2n
3 (a) Using standard summation formulae, write down an expression in terms of n for /r . 3
[1]
2n r=1
(b) Hence show that /r 3
= 14 n 2 (an + b) (cn + d ) , where a, b, c and d are integers to be
r=n+1
determined. [5]
4 In this question you must show detailed reasoning.
The roots of the cubic equation x 3 - 3x 2 + 19x - 17 = 0 are a, b and c.
(a) Find a cubic equation with integer coefficients whose roots are 12 (a - 1) , 12 ( b - 1) and
1
2 ( c - 1) . [4]
(b) Hence or otherwise solve the equation x 3 - 3x 2 + 19x - 17 = 0 . [3]
© OCR 2024 Y410/01 Jun24
, 3
J 1 2 0N
K O
5 (a) Find the volume scale factor of the transformation with associated matrix K 0 3 - 1O. [2]
K O
L- 1 0 2P
(b) The transformations S and T of the plane have associated 2 # 2 matrices P and Q respectively.
(i) Write down an expression for the associated matrix of the combined transformation S
followed by T. [1]
J k 3N
The determinant of P is 3 and Q = KK O, where k is a constant.
- 1 2O
L P
(ii) Given that this combined transformation preserves both orientation and area, determine
the value of k. [3]
J4 - 9N
6 You are given that M = KK O.
1 - 2O
L P
J1 + 3n - 9n N
(a) Prove that M n = KK O for all positive integers n. [6]
n 1 - 3nO
L P
(b) A student thinks that this formula, when n = 0 and n =- 1, gives the identity matrix and the
inverse matrix M -1 respectively.
Determine whether the student is correct. [3]
7 Three planes have equations
x + 2y - 3z = 0,
- x + 3y - 2z = 0,
x - 2y + kz = k,
where k is a constant.
(a) For the case k = 0, the origin lies on all three planes.
Use a determinant to explain whether there are any other points that lie on all three planes in
this case. [2]
(b) You are now given that k = 1.
(i) Show that there are no points that lie on all three planes. [3]
(ii) Describe the geometrical arrangement of the three planes. [1]
© OCR 2024 Y410/01 Jun24 Turn over
AS Level Further Mathematics B (MEI)
Y410/01 Core Pure
Time allowed: 1 hour 15 minutes
Turn over
, 2
1 The quadratic equation x 2 + ax + b = 0 , where a and b are real constants, has a root 2 - 3i.
(a) Write down the other root. [1]
(b) Hence or otherwise determine the values of a and b. [3]
J 1 aN J2 0N J- 1 0N
2 K
The matrices A, B and C are given by A = K O K
, B=K O and C = KK O, where a is
- 1 2O 1 - 1O 2 1O
a constant. L P L P L P
(a) By multiplying out the matrices on both sides of the equation, verify that A^BCh = ^ABhC .
[4]
(b) State the property of matrix multiplication illustrated by this result. [1]
2n
3 (a) Using standard summation formulae, write down an expression in terms of n for /r . 3
[1]
2n r=1
(b) Hence show that /r 3
= 14 n 2 (an + b) (cn + d ) , where a, b, c and d are integers to be
r=n+1
determined. [5]
4 In this question you must show detailed reasoning.
The roots of the cubic equation x 3 - 3x 2 + 19x - 17 = 0 are a, b and c.
(a) Find a cubic equation with integer coefficients whose roots are 12 (a - 1) , 12 ( b - 1) and
1
2 ( c - 1) . [4]
(b) Hence or otherwise solve the equation x 3 - 3x 2 + 19x - 17 = 0 . [3]
© OCR 2024 Y410/01 Jun24
, 3
J 1 2 0N
K O
5 (a) Find the volume scale factor of the transformation with associated matrix K 0 3 - 1O. [2]
K O
L- 1 0 2P
(b) The transformations S and T of the plane have associated 2 # 2 matrices P and Q respectively.
(i) Write down an expression for the associated matrix of the combined transformation S
followed by T. [1]
J k 3N
The determinant of P is 3 and Q = KK O, where k is a constant.
- 1 2O
L P
(ii) Given that this combined transformation preserves both orientation and area, determine
the value of k. [3]
J4 - 9N
6 You are given that M = KK O.
1 - 2O
L P
J1 + 3n - 9n N
(a) Prove that M n = KK O for all positive integers n. [6]
n 1 - 3nO
L P
(b) A student thinks that this formula, when n = 0 and n =- 1, gives the identity matrix and the
inverse matrix M -1 respectively.
Determine whether the student is correct. [3]
7 Three planes have equations
x + 2y - 3z = 0,
- x + 3y - 2z = 0,
x - 2y + kz = k,
where k is a constant.
(a) For the case k = 0, the origin lies on all three planes.
Use a determinant to explain whether there are any other points that lie on all three planes in
this case. [2]
(b) You are now given that k = 1.
(i) Show that there are no points that lie on all three planes. [3]
(ii) Describe the geometrical arrangement of the three planes. [1]
© OCR 2024 Y410/01 Jun24 Turn over
