Exam format examples | Kwangwoon University NiiL

Problem solving

Example 1

Provided references

Let the water volume be $V(t)$ [L] and the inflow and outflow per minute be $q_{\mathrm{in}},q_{\mathrm{out}}$ [L/min]. Then $\frac{dV}{dt}=q_{\mathrm{in}}-q_{\mathrm{out}}$ .

Problem

A tank contains 4 L of water, with 3 L flowing in and 1 L flowing out per minute. These rates remain constant for 2 minutes, and the tank does not overflow. Set the starting time to t=0.

Find an expression for $V(t)$ and calculate the water volume after 2 minutes. Show how you apply the initial value, and include units.

Example answer

Since $\frac{dV}{dt}=3-1=2\,\mathrm{L/min}$ , $V(t)=2t+C$ . At $V(0)=4$ , $C=4$ , so $V(t)=4+2t$ [L]. Therefore, $V(2)=4+2\times2=8\,\mathrm L$ .

Example 2

Provided references

$\mathrm{Accuracy}=\frac{\text{number of correctly classified records}}{\text{total number of records}}$ .

Problem

Of 20 equipment measurement records, 4 are actual faults and 16 are normal. The decisions of two models are shown below. Each number is the count of corresponding records.

Actual status	Model decision	A	B
Fault	Fault	0	3
Fault	Normal	4	1
Normal	Fault	0	5
Normal	Normal	16	11

Calculate the accuracy of both models and show your calculations.
Missing a fault by classifying it as normal costs 10 per case; raising a false alarm by classifying a normal case as faulty costs 1 per case. Calculate each model's total error cost, and state which model has the lower cost and why you chose it.

Example answer

A: $\frac{0+16}{20}=0.8$ , B: $\frac{3+11}{20}=0.7$ .
A's cost is $4\times10+0\times1=40$ , and B's is $1\times10+5\times1=15$ . Choose B based on cost. B has more false alarms but reduces costly missed faults.

Example 3

Provided references

The average rate of change between two times is $\frac{\text{change in measured value}}{\text{time interval}}$ .

Problem

Voltage measurements over time are shown below. Represent values between measurement times by connecting each pair of points with a straight line.

Time [s]	Voltage [V]
0	0
1	2
2	2
3	2

Draw a graph connecting the four measurement points, with time [s] on the horizontal axis and voltage [V] on the vertical axis. Label the axes and units, and mark the measurement points.
Calculate the average rate of change in voltage from 0 seconds to 1 second, and include units.

Example answer

It rises linearly from 0 V at 0 seconds to 2 V at 1 second, then remains constant at 2 V until 3 seconds.
$\frac{2-0}{1-0}=2\,\mathrm{V/s}$ .

MATLAB

Example 1

Provided references

mean(v): If v is a row vector, returns the arithmetic mean of all its elements as a scalar.

Problem

The MATLAB code below selects the first and third elements of x and computes their mean.

x = [2 4 6];
picked = x([1 3]);
a = mean(picked);

Write the values and array sizes of picked and a. Express each size as rows × columns.

Example answer

picked=[2 6], with size 1×2. $a=\frac{2+6}{2}=4$ , with size 1×1.

Example 2

Provided references

numel(v): Returns the number of elements in array v.

Problem

The MATLAB code below adds only the positive values in x.

x = [1 -1 2];
total = 0;
for k = 1:numel(x)
    if x(k) > 0
        total = total + ____;
    end
end

Fill in the blank to add only positive values.
Tabulate total by k immediately after each iteration.

Example answer

The blank is x(k).
At k=2, x(2)=-1, so the if body is skipped and total remains 1.
k x(k) total
1 1 1
2 -1 1
3 2 3

k	x(k)	total
1	1	1
2	-1	1
3	2	3

Example 3

Provided references

a:b: Creates a row vector from a to b in increments of 1.

plot(x,y): Uses values at matching positions in x and y as coordinates, then connects the points in order with straight lines.

Problem

In the MATLAB code below, t represents time [s] and y represents voltage [V].

t = 0:2;
y = 2*t;
plot(t,y);

Write the values of t and y, and sketch the graph drawn by plot. Label the axes and units, and mark the three points.

Example answer

t=[0 1 2], y=[0 2 4]. Connecting the three points gives a straight line through the origin.

Python

Example 1

Provided references

Multiplying or adding a scalar to a NumPy array applies the operation to each element.

v[mask]: Selects elements of v in order at positions where the Boolean array mask of the same length is True.

Problem

The Python code below selects elements of y corresponding to values in array x that are at least 2. np refers to NumPy.

x = np.array([1, 2, 3])
y = 2*x + 1
mask = x >= 2
selected = y[mask]

Write the values of y, mask, and selected in execution order. Also explain why the position where x=2 is selected.

Example answer

y=[3,5,7], mask=[False,True,True], selected=[5,7]. >= includes the boundary value 2, so the second position is also selected.

Example 2

Provided references

d.get(key,default): Returns the stored value if key is in dictionary d; otherwise, returns default. It does not modify the dictionary itself.

Problem

The Python code below counts how often each character appears in a list.

counts = {}
for name in ["A", "B", "A"]:
    counts[name] = counts.get(name, 0) + ____

Fill in the blank so that the count increases by 1 each time a character is read. Write the value of counts after the loop ends.

Example answer

The blank is 1. After the first A, the dictionary is {"A":1}; after B, {"A":1,"B":1}; and after the final A, {"A":2,"B":1}. The final value is {"A":2,"B":1}; the key order may differ.

Example 3

Provided references

sum(v) returns the sum of the elements, and len(v) returns the number of elements. For an empty list [], both are 0.

Problem

The Python function below returns the mean of the input list.

def average(values):
    return sum(values) / len(values)

result = average([2, 4])

Write the value of result and show the calculation.
Explain why average([]) cannot return a mean, including the calculation it attempts.

Example answer

result=3.0. The input list's sum, 6, is divided by its 2 elements, giving $\frac{2+4}{2}=3$ .
For an empty list, computing $\frac{0}{0}$ causes division by 0. It does not return a mean of 0.

Example 4

Provided references

x.mean(dim=k): Reduces the selected axis by averaging and removes it from the result. With dim=0, it averages elements in each column, leaving one value per column; with dim=1, it averages elements in each row, leaving one value per row.

Problem

These are temperatures [°C] measured by two sensors at two times. Each row of x represents a time point, and each column represents a sensor. torch refers to PyTorch.

x = torch.tensor([[1., 3.],
                  [5., 7.]])
m = x.mean(dim=0)

Write the value and shape of m. Explain which measurements from which sensor are averaged in each element of m.

Example answer

m=[3.,5.], with shape (2,). The first element averages sensor 1's values at two times, $\frac{1+5}{2}=3$ ; the second averages sensor 2's, $\frac{3+7}{2}=5$ . Reducing the time axis leaves two means, one per sensor.