ES1930
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2019
ENGINEERING MATHEMATICS
Candidates should answer the TWO COMPULSORY QUESTIONS.
Time Allowed: 2 hours.
Only calculators that conform to the list of models approved by the School of Engineering may
be used in this examination. The Engineering Data Book and standard graph paper will be
provided.
Read carefully the instructions on the answer book and make sure that the particulars required are
entered on each answer book.
USE A SEPARATE ANSWER BOOK FOR EACH SECTION
, ES1930
SECTION A: ENGINEERING MATHEMATICS
_________________________________________________________________________________________
1.
(i) Two vectors are given as:
3 2
⃗= 1 and ⃗= 5
−4 7
(a) Calculate the scalar product ⃗ ∙ ⃗ (3 marks)
(b) Calculate the cross product ⃗ × ⃗ (3 marks)
4 5 7 −3
(ii) Given: = with = (3 4)
3 −6 4 −3
Find and . (6 marks)
(iii) Find for the functions
(a) =4 + (3 + ) + + 5, where and are constants (3 marks)
(b) = (3 marks)
.
(iv) Find for the function ( , , ) = sin( ) ln (2 marks)
(v) Find the angle between the two planes given by the equations
2 + 7 − 3 = 8 and 6 − 4 + 5 = 10 (4 marks)
4 6
(vi) A plane has a normal vector = 5 and a point on the plane is 8 . Find the Cartesian
−3 1
equation of the plane. (3 marks)
2 3 2
(vii) Find the eigenvalue for the vector = with respect to the matrix = .
3 3 4
(4 marks)
Question 1 Continued Overleaf
1
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2019
ENGINEERING MATHEMATICS
Candidates should answer the TWO COMPULSORY QUESTIONS.
Time Allowed: 2 hours.
Only calculators that conform to the list of models approved by the School of Engineering may
be used in this examination. The Engineering Data Book and standard graph paper will be
provided.
Read carefully the instructions on the answer book and make sure that the particulars required are
entered on each answer book.
USE A SEPARATE ANSWER BOOK FOR EACH SECTION
, ES1930
SECTION A: ENGINEERING MATHEMATICS
_________________________________________________________________________________________
1.
(i) Two vectors are given as:
3 2
⃗= 1 and ⃗= 5
−4 7
(a) Calculate the scalar product ⃗ ∙ ⃗ (3 marks)
(b) Calculate the cross product ⃗ × ⃗ (3 marks)
4 5 7 −3
(ii) Given: = with = (3 4)
3 −6 4 −3
Find and . (6 marks)
(iii) Find for the functions
(a) =4 + (3 + ) + + 5, where and are constants (3 marks)
(b) = (3 marks)
.
(iv) Find for the function ( , , ) = sin( ) ln (2 marks)
(v) Find the angle between the two planes given by the equations
2 + 7 − 3 = 8 and 6 − 4 + 5 = 10 (4 marks)
4 6
(vi) A plane has a normal vector = 5 and a point on the plane is 8 . Find the Cartesian
−3 1
equation of the plane. (3 marks)
2 3 2
(vii) Find the eigenvalue for the vector = with respect to the matrix = .
3 3 4
(4 marks)
Question 1 Continued Overleaf
1
