ES1930
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2018
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) Determine the position of any extremum points of the function
𝑦 = 12𝑥 + 3𝑥 2
and ascertain if they are a maximum or a minimum. (4 marks)
y sin cos3x .
dy
(ii) Find the derivative for the function (4 marks)
dx
(iii) A quantity 𝑧 = 𝑓(𝑥, 𝑦), has 𝑓(𝑥, 𝑦) defined as
𝑓(𝑥, 𝑦) = 𝑥 2 + 3𝑥𝑦 2 − 𝑦 4 + 4 .
Find the total differential, 𝑑𝑧. (4 marks)
dy
(iv) Use implicit differentiation to find where x and y are related through
dx
x 2e y 3x y 2 2 0 . (4 marks)
(v) Three vectors are given as:
𝒂 = 3𝒊 + 4𝒋 + 2𝒌, 𝒃 = 2𝒊 + 5𝒋 + 3𝒌 and 𝒄 = 4𝒊 + 10𝒋 − 5𝒌 .
(a) Calculate the cross product 𝒂 × 𝒃 (3 marks)
(b) Calculate the triple product (𝒂 × 𝒃) ∙ 𝒄 (3 marks)
3 −4 𝑥 −6 4
(vi) Given: ( ) ( ) = 𝐴𝑇 with 𝐴 = (2 1) ( ).
2 −1 𝑦 3 −9
Find 𝑥 and 𝑦. (6 marks)
(vii) Two complex numbers are given by 𝒛𝟏 = 𝟑 + 𝟒𝒊 and 𝒛𝟐 = 𝟐𝒆𝟎.𝟖𝒊 . Find
(a) Find 𝒛𝟑 = 𝒛𝟏 ∙ 𝒛𝟐 , with 𝒛𝟑 expressed in modulus argument form (3 marks)
(b) Divide the complex conjugate of 𝒛𝟏 by 𝒛 = 2 + 5𝑖, expressing your answer
in the form 𝑎 + 𝑖𝑏. (3 marks)
Question 1 Continued Overleaf
1
THE UNIVERSITY OF WARWICK
First Year Examinations: Summer 2018
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) Determine the position of any extremum points of the function
𝑦 = 12𝑥 + 3𝑥 2
and ascertain if they are a maximum or a minimum. (4 marks)
y sin cos3x .
dy
(ii) Find the derivative for the function (4 marks)
dx
(iii) A quantity 𝑧 = 𝑓(𝑥, 𝑦), has 𝑓(𝑥, 𝑦) defined as
𝑓(𝑥, 𝑦) = 𝑥 2 + 3𝑥𝑦 2 − 𝑦 4 + 4 .
Find the total differential, 𝑑𝑧. (4 marks)
dy
(iv) Use implicit differentiation to find where x and y are related through
dx
x 2e y 3x y 2 2 0 . (4 marks)
(v) Three vectors are given as:
𝒂 = 3𝒊 + 4𝒋 + 2𝒌, 𝒃 = 2𝒊 + 5𝒋 + 3𝒌 and 𝒄 = 4𝒊 + 10𝒋 − 5𝒌 .
(a) Calculate the cross product 𝒂 × 𝒃 (3 marks)
(b) Calculate the triple product (𝒂 × 𝒃) ∙ 𝒄 (3 marks)
3 −4 𝑥 −6 4
(vi) Given: ( ) ( ) = 𝐴𝑇 with 𝐴 = (2 1) ( ).
2 −1 𝑦 3 −9
Find 𝑥 and 𝑦. (6 marks)
(vii) Two complex numbers are given by 𝒛𝟏 = 𝟑 + 𝟒𝒊 and 𝒛𝟐 = 𝟐𝒆𝟎.𝟖𝒊 . Find
(a) Find 𝒛𝟑 = 𝒛𝟏 ∙ 𝒛𝟐 , with 𝒛𝟑 expressed in modulus argument form (3 marks)
(b) Divide the complex conjugate of 𝒛𝟏 by 𝒛 = 2 + 5𝑖, expressing your answer
in the form 𝑎 + 𝑖𝑏. (3 marks)
Question 1 Continued Overleaf
1
