Mid-Course Assessment (Weeks 1–4)

A cumulative assessment covering Weeks 1–4: algebra and notation, functions and activations, NumPy shapes and broadcasting, vectors and norms, matrices, and linear systems. It mixes direct calculation, shape reasoning, and short interpretation questions. Aim for 80% or better; each explanation points to the chapter to revisit if you miss it.

1.
Evaluate $2^3 + 6 \div 2 \times 3$ using correct operator precedence.
2.
Using log rules, simplify $\ln(e^5)$ .
3.
Expand the weighted sum $\displaystyle\sum_{i=1}^{3} w_i x_i$ for $\mathbf{w} = [2, 0, -1]$ and $\mathbf{x} = [3, 5, 4]$ .
4.
Interpretation: A model outputs the vector $[0.2, 0.5, 0.3]$ of class scores (indexed from $0$ ). Using $\arg\max$ , which class does it predict, and why?
5.
What are the range of $\tanh$ and the range of the sigmoid $\sigma$ ?
6.
Shape reasoning: In NumPy, X has shape (100, 4) and w has shape (4,). What is the shape of X @ w?
7.
In NumPy, adding an array of shape (5, 3) to one of shape (3,) gives what result?
8.
Compute the Euclidean (L2) distance between $\mathbf{p} = [1, 2]$ and $\mathbf{q} = [4, 6]$ .
9.
The dot product of $\mathbf{a}$ and $\mathbf{b}$ is negative. What does the sign tell you about the angle $\theta$ between them?
10.
Compute the matrix–vector product $A\mathbf{x}$ for $A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$ .
11.
Shape reasoning: $A$ is $(3, 4)$ , $B$ is $(4, 2)$ , and $C$ is $(2, 5)$ . What is the shape of the chained product $ABC$ ?
12.
Interpretation: A square system $A\mathbf{x} = \mathbf{b}$ comes from two lines that are identical (one equation is a multiple of the other). How many solutions, and why?