MA4010NI Calculus and Linear Algebra Islington College

  1^stSit Problem Sheets Question Paper Year Long 2023/2024 

Module code: MA4010NI Module title: Calculus and Linear Algebra Module leader: Mr. Indra Dhakal (Islington College)

Coursework Type: Individual  Coursework Weight:This coursework accounts for 25%of your total module  grades. Submission Date: Week 4, 8, 12, 20, 24 for each 5 problem sheets. When Coursework is  Week 3 given out: Submission  Submit the following in the MST portal of your classroom Instructions: before the due date: ● A report (document) in .pdf format in the  Google Classroom or through any medium which the  module leader specifies. Warning:London Metropolitan University and Islington College  takes Plagiarism seriously. Offenders will be dealt with  sternly.

© London Metropolitan University

Page 1 of 6 

Problem Sheet 1 (5% of the module mark)[20 marks]Deadline: Week 4 

Question 1 

a) Define composite functions. 

b) It is given that ��(��) = √�� + 1 , ��(��) =¹_��.  

Write the formulas for �� ∘ �� and �� ∘ ��. 

c) Find the domain of each. 

d) Let ��(��) =^�� 

_��−2. Find a function �� = ��(��) so that (�� ∘ ��)(��) = ��. 

Question 2 

a) Consider the function  

��(��) = 

i) Find the domain of ��. 

^√3 + ��¹³− 2 _{�� − 1}. 

ii) Use a graphing utility to graph the function. iii) Find lim _{��→−27+}��(��). 

iv) Find lim_��→1��(��).

Page 2 of 6 

Problem Sheet 2 (5% of the module mark)[20 marks]Deadline: Week 8 

Question 1 

a) If ��(��) =^�� 

_��−1, find the derivative of �� from the first principle. 

b) i) Determine the slope of the graph of 3(��²+ ��²)²= 100���� at the point (3, 1). ii) Find ��′′ if ��²+ ��²= 25. 

Question 2 

a) Find the tangent and normal line to the graph of ��²(��²+ ��²) = ��²at the point  (^√2₂,^√2₂). 

b) Determine the points of inflection and discuss the concavity of the graph  ��(��) = ��⁴− 4��³.

Page 3 of 6 

Problem Sheet 3 (5% of the module mark) [20 marks] 

Deadline: Week 12 

Question 1 

a) Evaluate: 

_∫1 

√�� + 1 + √_{�� − 1}���� 

b)Using partial fraction method integrate ∫^��2+��−1 

��³+��²_−6������. 

Question 2 

a) Show that 

√�� 

∫ _{�� ln �� ���� =}¹4 

1 

b) Find the area of the surface formed by revolving the graph of ��(��) = ��³on the  interval [0, 1] about �� −axis.

Page 4 of 6 

Problem Sheet 4 (5% of the module mark)(20 marks)Deadline: Week 20 

Question 1 

a) Express the complex number ^{5−√3��} 

1−√_3��in �� + ���� form. Also, find the modulus of  

the number. 

b) Find the six sixth roots if �� = −8 and graph these roots in the complex plane. 

Question 2 

_{a) If the position vector of the points��, ��and��are−2��⃗+ ��⃗ − ��}^⃗⃗_{, −4��⃗ + 2��⃗ + 2��}⃗⃗ _{and6��⃗ − 3��⃗ − 13��}^⃗⃗respectively.  

_{i) Find����}^{⃗⃗⃗⃗⃗⃗}_{and����}^{⃗⃗⃗⃗⃗⃗}. 

_{ii) Are����}^{⃗⃗⃗⃗⃗⃗}_{and����}^{⃗⃗⃗⃗⃗⃗}perpendicular? 

_{iii) If����}^{⃗⃗⃗⃗⃗⃗}_{= ������}^{⃗⃗⃗⃗⃗⃗}, then find the value of ��. 

2 

b) Check whether the vectors ��⃗⃗ = ( 

2 

1 

independent.

) , ��⃗ = ( 

−4 6 

5 

) , ��⃗⃗ = ( 

−2 8 

6 

) are linearly  

Page 5 of 6 

Problem Sheet 5 (5% of the module mark)[20 marks] 

Deadline: Week 24 

Question 1 

Consider the system of equations  

x + 2y – 3z = 4 

3x – y + 5z = 2 

4x + y + (��²-14) z = ��+2 

a) Reduce the system to echelon form.  

b) Find the values of the constant �� that will give no solutions.  

c) Find the values of the constant �� that will give an infinite number of solutions.  d) Find the values of the constant �� that will give a unique solution.  

Question 2 

Determine a basis and dimension of the solution space of the homogeneous system ��₁− 4��₂− 3��₃− 7��₄= 0 

2��₁− ��₂+ ��₃+ 7��₄= 0 

��₁+ 2��₂+ 3��₃+ 11��₄= 0 

Express the general solution of each system as a span or using arbitrary constants.  End of paper

Page 6 of 6