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MA4010NI Calculus and Linear Algebra Islington College

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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) = ��2 at 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 ��(��) = ��3 on 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��⃗⃗ and 6��⃗ − 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 

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