1

Makerere
University

Department of Electrical and Computer Engineering

B.Sc. in Electrical Engineering, B.Sc. in Computer & Communications

Engineering and B.Sc. Biomedical Engineering

EMT1101 – ENGINEERING MATHEMATICS I

Coursework Set 1

Instructions:

a) Work in groups of 2–3 members.

b) Each group must include members from different programmes (BELE, BCCE, BBI).

c) Clearly list each member’s name, registration number, and programme on the cover page.

d) Solutions may be handwritten or typed.

e) Submission deadline: 30th October 2025 at 8:00 AM (strictly).

1. For power system engineers, it is essential to ensure maximum power transfer from the

source to the load. Figure 1 shows a circuit in which a non-ideal voltage source is

connected to a variable load resistor with resistance 𝑅𝐿. The source voltage is 𝑉 and its

internal resistance is 𝑅𝑆. Calculate the value of 𝑅𝐿 which results in the maximum power

being transferred from the voltage source to the load resistor.
6 Marks

Figure 1

2. Find the dimensions of the right-circular cylinder of largest volume that can be inscribed

in a sphere of radius R.
4 Marks

3. Sketch graphs of the functions

2

(i) 𝑦 =

𝑥2−𝑥−6

𝑥+1
5 Marks

(ii)𝑦 =

𝑥−1

𝑥2−4
5 Marks

4. Given the system of linear equations

4𝑥 − 5𝑦 + 7𝑧 = −14

9𝑥 + 2𝑦 − 3𝑧 = 47

𝑥 − 𝑦 − 5𝑧 = 11

Solve the equation using

(i) Crammer’s rule
5 Marks

(ii)Gauss elimination method
5 Marks

5. (a) Sketch graphs of the following functions

(i) 𝑓(𝑥) =

𝑥
|𝑥|
2 Marks

(ii) 𝑓(𝑥) = √4 − 𝑥2
2 Marks

(iii)𝑓(𝑥) = {𝑥2,
𝑥 1

2,
𝑥 ≤ 1
2 Marks

(b) An open box is to be made from an 8𝑐𝑚 × 15𝑐𝑚 piece of sheet metal by cutting out

squares with sides of length 𝑥 from each of the four corners and bending up the sides.

Express the volume 𝑉 of the box as a function 𝑥, and state the domain and range of the

function.
4 Marks

6. Evaluate the following integrals

(i) ∫
sin2 3𝑥 cos 3𝑥 𝑑𝑥

𝜋 2
⁄

0
4 Marks

(ii) ∫

cos 2𝑥

√7−3 sin 2𝑥 𝑑𝑥

𝜋 4
⁄

0
4 Marks

(iii)∫

𝑥2

√4−3𝑥 𝑑𝑥

1
0
4 Marks

(iv) ∫ √tan 𝑥 sec2 𝑥 𝑑𝑥
4 Marks

(v) ∫ 𝑒𝑎𝑥 cos 𝑏𝑥 𝑑𝑥
4 Marks

Question

1

Makerere 
University

Department of Electrical and Computer Engineering

B.Sc. in Electrical Engineering, B.Sc. in Computer & Communications

Engineering and B.Sc. Biomedical Engineering

EMT1101 – ENGINEERING MATHEMATICS I

Coursework Set 1

Instructions:

a) Work in groups of 2–3 members.

b) Each group must include members from different programmes (BELE, BCCE, BBI).

c) Clearly list each member’s name, registration number, and programme on the cover page.

d) Solutions may be handwritten or typed.

e) Submission deadline: 30th October 2025 at 8:00 AM (strictly).

1. For power system engineers, it is essential to ensure maximum power transfer from the

source to the load. Figure 1 shows a circuit in which a non-ideal voltage source is

connected to a variable load resistor with resistance 𝑅𝐿. The source voltage is 𝑉 and its

internal resistance is 𝑅𝑆. Calculate the value of 𝑅𝐿 which results in the maximum power

being transferred from the voltage source to the load resistor.
6 Marks

Figure 1

2. Find the dimensions of the right-circular cylinder of largest volume that can be inscribed

in a sphere of radius R.
4 Marks

3. Sketch graphs of the functions

2

(i) 𝑦 =

𝑥2−𝑥−6

𝑥+1
5 Marks

(ii)𝑦 =

𝑥−1

𝑥2−4
5 Marks

4. Given the system of linear equations

4𝑥 − 5𝑦 + 7𝑧 = −14

9𝑥 + 2𝑦 − 3𝑧 = 47

𝑥 − 𝑦 − 5𝑧 = 11

Solve the equation using

(i) Crammer’s rule 
5 Marks

(ii)Gauss elimination method
5 Marks

5. (a) Sketch graphs of the following functions

(i) 𝑓(𝑥) =

𝑥
|𝑥|
2 Marks

(ii) 𝑓(𝑥) = √4 − 𝑥2
2 Marks

(iii)𝑓(𝑥) = {𝑥2,
𝑥 > 1

2,
𝑥 ≤ 1
2 Marks

(b) An open box is to be made from an 8𝑐𝑚 × 15𝑐𝑚 piece of sheet metal by cutting out

squares with sides of length 𝑥 from each of the four corners and bending up the sides.

Express the volume 𝑉 of the box as a function 𝑥, and state the domain and range of the

function.
4 Marks

6. Evaluate the following integrals

(i) ∫
sin2 3𝑥 cos 3𝑥 𝑑𝑥

𝜋 2
⁄

0
4 Marks

(ii) ∫

cos 2𝑥

√7−3 sin 2𝑥 𝑑𝑥

𝜋 4
⁄

0
4 Marks

(iii)∫

𝑥2

√4−3𝑥 𝑑𝑥

1
0
4 Marks

(iv) ∫ √tan 𝑥 sec2 𝑥 𝑑𝑥
4 Marks

(v) ∫ 𝑒𝑎𝑥 cos 𝑏𝑥 𝑑𝑥
4 Marks

StudyXY · Accepted Answer

: To calculate the value of $R\_L$ which results in the maximum power being transferred from the voltage source to the load resistor, we need to find the value of $R\_L$ that makes the voltage across the load resistor equal to half of the source voltage. : Solving for $R\_L$ gives: : Therefore, the value of $R\_L$ which results in the maximum power being transferred from the voltage source to the load resistor is: : To find the dimensions of the right-circular cylinder of largest volume that can be inscribed in a sphere of radius $R$, we need to maximize the volume of the cylinder subject to the constraint that the cylinder fits entirely within the sphere. : To maximize the volume of the cylinder, we need to find the values of $r$ and $h$ that satisfy the constraint and maximize the volume. : Taking the partial derivatives of the Lagrangian function with respect to $r$, $h$, and $\lambda$, and setting them equal to zero gives: : Solving these equations simultaneously gives: : Therefore, the dimensions of the right-circular cylinder of largest volume that can be inscribed in a sphere of radius $R$ are: : The volume of the cylinder is: : Therefore, the volume of the cylinder is: : For part (iii), we need to sketch the graph of the function: : The graph of the function consists of two parts: : The graph of the function is shown below: : The volume of the box is given by: : The domain of the function is: : The range of the function is: : For part (vi), we need to evaluate the following integrals: : Using the double angle identity for cosine, we can rewrite the integral as: : Let $u = \sin 3x$, then $du = 3 \cos 3x dx$, and the integral becomes: : For part (vi), we need to evaluate the following integrals: : Let $u = \sin 2x$, then $du = 2 \cos 2x dx$, and the integral becomes: : For part (vi), we need to evaluate the following integrals: : Let $u = \sqrt{4 - 3x}$, then $du = -\frac{3}{2} dx$, and the integral becomes: : For part (vi), we need to evaluate the following integrals: : Let $u = \sqrt{	an x}$, then $du = \frac{\sec^2 x}{2 \sqrt{	an x}} dx$, and the integral becomes: : For part (vi), we need to evaluate the following integrals: : Using integration by parts, we can rewrite the integral as: : Using integration by parts again, we can rewrite the second integral as: : Therefore, the integral is: : The solutions to the given problems are presented in steps above.

Q
QuestionAnatomy and Physiology

Answer

Full Solution Locked

Step 1
: To calculate the value of $R\_L$ which results in the maximum power being transferred from the voltage source to the load resistor, we need to find the value of $R\_L$ that makes the voltage across the load resistor equal to half of the source voltage.

Step 2
: Solving for $R\_L$ gives:

Final Answer

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