Mathematical Methods and Applications

Answer ALL questions
Full marks may be obtained for
complete and correct solutions that show method and accuracy
Marks for parts of questions are shown in brackets, e.g. [..%] This assessment aims to test your ability to correctly and accurately apply techniques of
algebra, geometry, functions, differentiation, and integration.
This assessment is for individual students and contributes a maximum of 25% to the final
Module Grade. The submission must be your own work (it is your responsibility to
familiarise yourself with University policies on plagiarism) and should provide sufficient working
to demonstrate that you understand concepts, can correctly analyse problems, and can
correctly implement methods.
YOUR parameters ?, ?, ?, and ? to be used in the assessment questions are obtained from
the last four digits of your 8-digit UB number, ? ? ? ? ? ? ? ?, according to
? = 1 + ?, ? = 1 + ?, ? = 1 + ?, ? = 1 + ?
It is your responsibility to ensure that you use the correct parameter values in the problem
descriptions below. Illustration, if the last 4 digits of your UB number are 9306 then YOUR
parameter values would be ? = 10, ? = 4, ? = 1, ? = 7.
As you answer each question, FIRST replace coefficients ?, ?, ?, and ? by YOUR values.
Question 1: Essential Algebra
(a) Simplify the expression
?√5 + ?√2
?√6 − √5
[5] (b) Find the exact value of ? that satisfies the equation
3!”#
27!$# =
81?!
9!”& [5] ENM4004-B / December 2020 Assessment 01 Page 2 of 3
Question 2: Coordinate Geometry
(a) Complete the square and write down the centre and radius of the circle
?& + ?& − 2?? + 2?? − ? = 0 [5] (b) Determine the equation of an ellipse that is centred on the origin with
semi-major axis of length ? + ? and foci located at (0,±?). [5] (c) Determine all intersections of the circle identified in part (a) with the
straight line ? + 2? = ?. [5] Question 3: Functions
(a) For the function ?(?) below, state its domain and range, and identify roots,
intercepts, asymptotes, and turning points.
?(?) =
2??& − 1
?& − 1
Use the information to sketch the function
[10] [5] (b) Determine the inverse of the function ?(?) in part (a). [5] (c) Determine all values of ? in the interval [−?, ?] that satisfy the equation
? cos& ? − ??sin 2? = 0 [6] Question 4: Differentiation
Evaluate the limiting expressions:
(a) (i) lim
!→?
<
? − ?
?$ − ?$? [5] (ii) lim
!→(
C
sinh??
cosh??
E [5] (b) Locate all stationary points of the function ?(?) = ?) − ??& + ? and
determine their nature (maximum, minimum, or inflexion) [7] (c) For the function ?(?) = ?!!?, use logarithmic differentiation to find an
expression for *+
*!. [8] ENM4004-B / December 2020 Assessment 01 Page 3 of 3
Question 5: Integration
(a) Evaluate the integral
FC?
?&
+ sin??E ??
[4] (b) Make use of the substitution ? = √1 + ?? to determine the exact value of
the definite integral
F
cos √1 + ??
√1 + ??
-!
)?$#?
/
??
[8] (c) Use integration by parts to evaluate the definite integral
F?&?$?!
&
/
??
[12]