Appendix B: The Drill Bank — Graded Practice

Appendix B: The Drill Bank — Graded Practice

Fluency is a muscle. Each chapter below has six drills — Tier A: quick, mechanical, meant to be done fast with answers checked immediately [in brackets] — and one \star challenge — Tier C: a problem with an idea in it, solution sketched. The main chapter exercises are Tier B. Protocol: drills the same day you read the chapter; the \star when you’re feeling strong.

Chapter 1 — Numbers. 1. \frac38 + \frac{5}{12} [\frac{19}{24}]. 2. Write 0.375 as a fraction in lowest terms [\frac38]. 3. 35% of 240 [84]. 4. 2^5 \cdot 2^{-3} [4]. 5. 27^{2/3} [9]. 6. 5 - 2\times 3^2 + 1 [-12]. \star Show that between any two distinct rationals there is another rational. Their midpoint \frac{p+q}{2} is rational and strictly between them — so the rationals have no “nearest neighbors”: they are dense.

Chapter 2 — Algebra. 1. Factor x^2 + 7x + 12 [(x+3)(x+4)]. 2. Solve 4(x-2) = 2x + 10 [x = 9]. 3. Expand (2x-3)^2 [4x^2 - 12x + 9]. 4. Solve |x - 3| < 5 [-2 < x < 8]. 5. Solve 3x - y = 7, x + y = 5 [x=3, y=2]. 6. Factor x^4 - 16 completely [(x-2)(x+2)(x^2+4)]. \star Prove by induction that 3 divides n^3 - n for all n \ge 1. Base: 1 - 1 = 0 ✓. Step: (n+1)^3 - (n+1) = (n^3 - n) + 3(n^2 + n) — the first bracket divisible by hypothesis, the second visibly.

Chapter 3 — Functions. 1. f(x) = 3x - 4: find f(2) and f(-1) [2; -7]. 2. Domain of \frac{1}{\sqrt{4-x}} [x < 4]. 3. With f(x)=x^2, g(x)=2x+1: f(g(1)) and g(f(1)) [9; 3]. 4. Inverse of f(x) = \frac{x+5}{3} [3x - 5]. 5. Slope through (1,4) and (3,10) [3]. 6. Even, odd, or neither: x^3 + x; x^2 + 1; x + 1 [odd; even; neither]. \star A linear f satisfies f(f(x)) = 4x + 9. Find all possibilities. Let f = ax + b: a^2x + ab + b = 4x + 9, so a = \pm 2; a=2 \Rightarrow b = 3; a = -2 \Rightarrow b = -9. Two answers: 2x+3 and -2x - 9.

Chapter 4 — Quadratics. 1. Roots of x^2 - 7x + 10 [2, 5]. 2. Vertex of y = x^2 - 4x + 1 [(2, -3)]. 3. \Delta of 2x^2 + 3x + 5; how many real roots? [-31; none]. 4. Sum and product of roots of 3x^2 - 12x + 7 [4; \frac73]. 5. Solve x^2 + 2x - 1 = 0 [-1 \pm \sqrt2]. 6. Remainder of x^3 + 2x - 5 divided by x - 2 [p(2) = 7]. \star For which k does x^2 + kx + k = 0 have real roots? \Delta = k^2 - 4k = k(k-4) \ge 0: k \le 0 or k \ge 4 — the coefficient’s own quadratic inequality.

Chapter 5 — Exp & Log. 1. \log_3 81 [4]. 2. Solve 2^x = \frac{1}{16} [-4]. 3. \ln(e^3\sqrt e) [3.5]. 4. \log 50 + \log 2 [2]. 5. Solve e^{2x} = 7 [\frac{\ln 7}{2} \approx 0.973]. 6. Half-life 10 days: fraction left after 30 days [\frac18]. \star Which is larger, e^\pi or \pi^e? Compare \frac{\ln x}{x} at e and \pi: the function peaks at x = e (its derivative \frac{1 - \ln x}{x^2} vanishes there), so \frac{\ln e}{e} > \frac{\ln \pi}{\pi}, i.e. \pi > e\ln\pi, i.e. e^\pi > \pi^e. Calculus settling a bar bet.

Chapter 6 — Trigonometry. 1. \sin\frac{2\pi}{3} [\frac{\sqrt3}{2}]. 2. \cos\theta = \frac35, \theta acute: \sin\theta, \tan\theta [\frac45; \frac43]. 3. Period of \sin 3x [\frac{2\pi}{3}]. 4. Simplify \frac{1 - \sin^2\theta}{\cos\theta} [\cos\theta]. 5. \arctan\sqrt3 [\frac{\pi}{3}]. 6. a = 7, b = 9, C = 60°: find c [\sqrt{67} \approx 8.19]. \star Prove \sin 3\theta = 3\sin\theta - 4\sin^3\theta. Write \sin(2\theta + \theta), expand with the addition formula, substitute the double angles, and replace \cos^2\theta by 1 - \sin^2\theta throughout.

Chapter 7 — Coordinate Geometry. 1. Distance (2,-1) to (5,3) [5]. 2. Midpoint of (-2,4), (6,-2) [(2,1)]. 3. Circle, center (1,-2), radius 3 [(x-1)^2 + (y+2)^2 = 9]. 4. Radius of x^2 + y^2 = 8x [complete the square: 4]. 5. Eccentricity of \frac{x^2}{16} + \frac{y^2}{7} = 1 [c = 3, e = \frac34]. 6. Vertex and focus of y^2 = 8x [(0,0); (2,0)]. \star Find the length of the tangent from (7,1) to the circle x^2 + y^2 = 25. By Pythagoras on the radius–tangent right angle: \sqrt{d^2 - r^2} = \sqrt{49 + 1 - 25} = 5.

Chapter 8 — Sequences & Series. 1. 12th term of 5, 9, 13, \dots [49]. 2. 1 + 2 + \cdots + 100 [5050]. 3. 6th term of 2, 6, 18, \dots [486]. 4. 9 + 3 + 1 + \cdots [\frac{9}{1 - 1/3} = 13.5]. 5. Coefficient of x^2 in (1+x)^6 [15]. 6. Write 0.777\dots as a fraction [\frac79]. \star Derive 1 + 3 + 5 + \cdots + (2n-1) = n^2 from the AP sum formula, and reconcile with Exercise 2.6. S_n = \frac{n}{2}(\text{first} + \text{last}) = \frac{n}{2}(1 + 2n - 1) = n^2 — the induction proof and the Gauss trick certify the same truth from opposite ends.

Chapter 9 — Counting & Probability. 1. P(\text{sum} = 7) with two dice [\frac{6}{36} = \frac16]. 2. \binom{6}{2} [15]. 3. Arrangements of 5 distinct books [120]. 4. P(\text{at least one head in 3 flips}) [1 - \frac18 = \frac78]. 5. E[X] for one die [3.5]. 6. P(A) = 0.5, P(B) = 0.4, independent: P(A \cup B) [0.5 + 0.4 - 0.2 = 0.7]. \star The birthday problem: show P(\text{shared birthday among 23 people}) > \frac12. Complement: P(\text{all distinct}) = \frac{365}{365}\cdot\frac{364}{365}\cdots\frac{343}{365} \approx 0.493, so P(\text{shared}) \approx 0.507 — the pairs matter (\binom{23}{2} = 253 of them), not the people.

Chapter 10 — Vectors. 1. |(3,-4)| [5]. 2. (1,2,3)\cdot(4,-5,6) [12]. 3. Unit vector along (2,-2,1) [\frac13(2,-2,1)]. 4. (1,0,0)\times(0,1,0) [(0,0,1)]. 5. Angle between (1,1) and (1,0) [45°]. 6. Are (2,4) and (3,6) parallel? [yes: (3,6) = \frac32(2,4)]. \star Compute the vector projection of \vec a = (3,4) onto \vec b = (1,0), and say what it means. \left(\frac{\vec a\cdot\vec b}{|\vec b|^2}\right)\vec b = (3,0): the shadow of \vec a on the x-axis — the “how much of \vec a points along \vec b” vector, the primitive behind least squares and attention scores alike.

Chapter 11 — Complex Numbers. 1. (2+3i) + (4-5i) [6 - 2i]. 2. (1+i)^2 [2i]. 3. |3 - 4i| [5]. 4. i^{2026} [2026 \equiv 2 \pmod 4: -1]. 5. Conjugate of 5 - 2i, and z\bar z [5 + 2i; 29]. 6. Polar form of -2 [2e^{i\pi}]. \star Solve z^2 = i. i = e^{i\pi/2}, so z = e^{i\pi/4} or e^{i5\pi/4}: z = \pm\frac{\sqrt2}{2}(1 + i) — square roots always come in \pm pairs, even in \mathbb{C}.

Chapter 12 — Matrices. 1. \begin{pmatrix}1&2\\3&4\end{pmatrix} + 2\begin{pmatrix}0&1\\1&0\end{pmatrix} [\begin{pmatrix}1&4\\5&4\end{pmatrix}]. 2. \det\begin{pmatrix}5&2\\7&3\end{pmatrix} [1]. 3. What does \begin{pmatrix}2&0\\0&3\end{pmatrix} do to the plane? [stretch x by 2, y by 3]. 4. Inverse of \begin{pmatrix}1&2\\3&4\end{pmatrix} [\begin{pmatrix}-2&1\\3/2&-1/2\end{pmatrix}]. 5. \det of the triangular \begin{pmatrix}1&2&3\\0&4&5\\0&0&6\end{pmatrix} [product of the diagonal: 24]. 6. The shear \begin{pmatrix}1&1\\0&1\end{pmatrix} has \det = 1: what is preserved? [area]. \star Prove (AB)^{-1} = B^{-1}A^{-1}. (AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AA^{-1} = I — socks and shoes again (Exercise 19.5): undo in reverse order.

Chapter 13 — Derivatives. 1. \frac{d}{dx}(x^5 - 3x^2 + 7) [5x^4 - 6x]. 2. \frac{d}{dx}\,x e^{2x} [e^{2x}(1 + 2x)]. 3. \frac{d}{dx}\ln(3x^2+1) [\frac{6x}{3x^2+1}]. 4. Slope of y = \sqrt x at x = 4 [\frac14]. 5. \lim_{x\to0}\frac{\sin 5x}{x} [5]. 6. Critical points of x^3 - 12x [x = \pm 2: max at -2, min at 2]. \star A 13 m ladder: the foot slides away at 2 m/s. How fast is the top descending when the foot is 5 m out? x^2 + y^2 = 169; differentiate: 2xx' + 2yy' = 0. At x = 5, y = 12: y' = -\frac{5\cdot 2}{12} = -\frac56 m/s — related rates, i.e. the chain rule with time as the hidden variable.

Chapter 14 — Integrals. 1. \int(6x^2 - 4x + 1)dx [2x^3 - 2x^2 + x + C]. 2. \int_0^{\pi/2}\cos x\,dx [1]. 3. \int e^{3x}dx [\frac{e^{3x}}{3} + C]. 4. \int\frac{2x}{x^2+1}dx [\ln(x^2+1) + C]. 5. \int_1^4 \frac{dx}{\sqrt x} [[2\sqrt x]_1^4 = 2]. 6. Average value of x^2 on [0,3] [\frac13\int_0^3 x^2 dx = 3]. \star Show \int_0^\infty x e^{-x}dx = 1. Parts with u = x: [-xe^{-x}]_0^\infty + \int_0^\infty e^{-x}dx = 0 + 1. This integral is the mean of the exponential distribution (Chapter 20) and the value \Gamma(2) — a first handshake with the Gamma function.

Chapter 15 — Real Analysis. 1. \lim \frac{3n^2+1}{n^2+n} [3]. 2. Does \sum \frac{1}{n^{1.5}} converge? [yes: p > 1]. 3. Does \sum\frac{n}{n+1} converge? [no: terms \to 1 \ne 0]. 4. First three nonzero Taylor terms of e^{-x} [1 - x + \frac{x^2}{2}]. 5. Supremum of \{1 - \frac1n : n \in \mathbb{N}\} [1, not attained]. 6. Is |x| continuous at 0? Differentiable there? [yes; no — a corner]. \star Show \sum_{n=1}^{\infty}\frac{1}{n(n+1)} = 1. Partial fractions: \frac{1}{n(n+1)} = \frac1n - \frac1{n+1}; the partial sums telescope to 1 - \frac{1}{N+1} \to 1 — an infinite sum computed exactly, by cancellation rather than formula.

Chapter 16 — Linear Algebra. 1. Are (1,2) and (2,4) independent? [no]. 2. Dimension of span\{(1,0,0),(0,1,0),(1,1,0)\} [2]. 3. Eigenvalues of \begin{pmatrix}3&0\\0&7\end{pmatrix} [3, 7 — diagonal matrices confess immediately]. 4. Eigen-structure of \begin{pmatrix}2&1\\0&2\end{pmatrix} [\lambda = 2 repeated, but only one eigenvector direction (1,0): not diagonalizable]. 5. Rank of \begin{pmatrix}1&2\\2&4\end{pmatrix} [1]. 6. For \begin{pmatrix}1&2\\2&1\end{pmatrix} (\lambda = 3, -1): verify trace =\sum\lambda and \det = \prod\lambda [2 = 3-1 ✓; -3 = 3\cdot(-1) ✓]. \star If A is invertible with eigenvalue \lambda, show A^{-1} has eigenvalue \frac{1}{\lambda}. From A\vec v = \lambda\vec v, apply A^{-1} to both sides: \vec v = \lambda A^{-1}\vec v, so A^{-1}\vec v = \frac1\lambda \vec v (and \lambda \ne 0, else A would kill \vec v and not be invertible).

Chapter 17 — Multivariable. 1. f = x^2y^3: both partials [2xy^3; 3x^2y^2]. 2. \nabla(x^2+y^2) at (1,1) [(2,2)]. 3. Direction of steepest ascent of f = xy at (2,3) [along \nabla f = (3,2)]. 4. \int_0^1\!\int_0^1 xy\,dy\,dx [\frac14]. 5. Critical point of x^2 + y^2 - 2x + 4y [(1,-2), a minimum]. 6. Classify (0,0) for f = x^2 - y^2 [D = -4 < 0: saddle]. \star Find the volume under z = 4 - x^2 - y^2 above the disc r \le 2. Polar: \int_0^{2\pi}\!\int_0^2 (4 - r^2)\,r\,dr\,d\theta = 2\pi\left[2r^2 - \frac{r^4}{4}\right]_0^2 = 2\pi(8 - 4) = 8\pi — the paraboloid “cap” of the gradient figure, measured.

Chapter 18 — Differential Equations. 1. Solve y' = 3y, y(0) = 2 [2e^{3x}]. 2. Solve y' = x^2, y(0) = 1 [\frac{x^3}{3} + 1]. 3. Characteristic roots of y'' - 5y' + 6y = 0 [2 and 3]. 4. General solution of y'' + 9y = 0 [c_1\cos 3x + c_2\sin 3x]. 5. Is y = e^{2x} a solution of y'' - 4y = 0? [yes: 4e^{2x} - 4e^{2x} = 0]. 6. Doubling time for y' = 0.05y [\frac{\ln 2}{0.05} \approx 13.9]. \star For the logistic equation y' = y(1-y), find the equilibria and their stability. Equilibria where y' = 0: y = 0 and y = 1. Near 0, y' \approx y > 0: solutions flee — unstable. Near 1, write y = 1+\varepsilon: y' \approx -\varepsilon — pulled back: stable. Every S-shaped growth curve in biology is this picture.

Chapter 19 — Abstract Algebra. 1. 17 \bmod 5 [2]. 2. Inverse of 3 in (\mathbb{Z}_7, \times) [5, since 15 \equiv 1]. 3. Order of 2 in (\mathbb{Z}_7\setminus\{0\}, \times) [2, 4, 1: order 3]. 4. \gcd(84, 30) by Euclid [84 = 2\cdot30+24; 30 = 24+6; 24 = 4\cdot6: \gcd = 6]. 5. Is (\mathbb{Z}_4\setminus\{0\}, \times) a group? [no: 2\times2 = 0 escapes the set — not even closed]. 6. Last digit of 7^{100} [powers of 7 end 7,9,3,1 with period 4; 100 \equiv 0: last digit 1]. \star Compute 2^{100} \bmod 101. 101 is prime, so Fermat’s little theorem (Lagrange in costume) gives 2^{100} \equiv 1 \pmod{101} — no computation needed. That shortcut, industrialized, is modular exponentiation in RSA.

Chapter 20 — Statistics. 1. Mean and median of 2,4,4,5,10 [5; 4 — the outlier drags the mean, not the median]. 2. Population \sigma of \{1,3,5\} [\sqrt{8/3} \approx 1.63]. 3. P(Z > 1) [\approx 0.159]. 4. IQ \sim N(100, 15): P(X > 130) [P(Z>2) \approx 0.023]. 5. Standard error of the mean: n = 25, \sigma = 10 [2]. 6. E[X] = 5, \mathrm{Var}(X) = 4: find E[2X+3] and \mathrm{Var}(2X+3) [13; 16 — shifts don’t spread]. \star Find the variance of the sum of two fair dice. Independent variances add: 2 \times \frac{35}{12} = \frac{35}{6} \approx 5.83. (Means add always; variances add only under independence — the fine print the CLT depends on.)

Chapter 21 — Vectors, Matrices, and Data. 1. Cosine similarity of (1,0) and (0,1) [0]. 2. Parameters in a layer \mathbb{R}^{100} \to \mathbb{R}^{50} [5000 + 50 = 5050]. 3. Fit y = wx (no intercept) to (1,2), (2,3) [w = \frac{\sum xy}{\sum x^2} = \frac{8}{5} = 1.6]. 4. Covariance \begin{pmatrix}4&0\\0&1\end{pmatrix}: first PC and its variance share [the x-axis; \frac{4}{5} = 80\%]. 5. Which pair is more similar: (1,1) vs (2,2), or (1,1) vs (3,0)? [first: \cos = 1 vs \approx 0.707]. 6. If X is n \times d, what shape is X^TX? [d \times d — the feature-by-feature Gram matrix]. \star Show the mean minimizes \sum_i (x_i - c)^2 over c. Differentiate: -2\sum(x_i - c) = 0 \Rightarrow c = \bar x; second derivative 2n > 0 confirms the minimum. “The mean is the least-squares summary of data” — the one-number version of regression.

Chapter 22 — The Calculus of Learning. 1. \sigma(0) [0.5]. 2. ReLU(-3) and ReLU(2) [0; 2]. 3. One GD step: L = w^2, w = 4, \eta = 0.1 [w \leftarrow 4 - 0.1\cdot 8 = 3.2]. 4. \frac{\partial}{\partial w}(y - wx)^2 [-2x(y - wx) — the workhorse gradient of regression]. 5. A 20-layer chain of factors 0.8 [0.8^{20} \approx 0.012: vanishing]. 6. Is L(w) = |w| convex? [yes — though not differentiable at 0, where subgradients take over]. \star For L(w) = w^2, find exactly which learning rates converge. The update is w \leftarrow w - 2\eta w = (1 - 2\eta)w: a geometric sequence with ratio 1 - 2\eta, converging iff |1 - 2\eta| < 1, i.e. 0 < \eta < 1 — Chapter 8’s convergence condition reborn as a hyperparameter rule.

Chapter 23 — Probability, Information, Learning. 1. softmax of (0, 0) [(0.5, 0.5)]. 2. Entropy of a fair 4-sided die [\log_2 4 = 2 bits]. 3. Cross-entropy loss when q_{\text{true}} = 0.5 [-\ln 0.5 \approx 0.693]. 4. D_{KL}(p\,\|\,p) [0 — no surprise in believing the truth]. 5. Higher entropy: (0.9, 0.1) or (0.6, 0.4)? [the second — closer to uniform]. 6. L2 penalty \lambda|\vec w|^2 for \lambda = 0.1, \vec w = (3,4) [0.1 \times 25 = 2.5]. \star Show by calculus that the entropy of a coin is maximized at p = \frac12. H(p) = -p\log_2 p - (1-p)\log_2(1-p); \frac{dH}{dp} = \log_2\frac{1-p}{p} = 0 exactly when p = \frac12, and H'' < 0 throughout — matching the figure in Chapter 23, and the intuition: a fair coin is maximal ignorance.