Practice, by level

Practice2 questions

1. Solve: 3x − 5 = 16

   x = 7x = 11/3x = 63

2. To solve 4(x + 2) = 20, the gentlest first move is…

   divide both sides by 4subtract 2 from both sidesmultiply out and panic

Practice2 questions

1. A meal has 4 starters, 5 mains, 3 desserts. How many three-course meals?

   60125040

2. How many ways can 4 distinct books be arranged on a shelf?

   24 — that's 4!16 — that's 4²4

Practice1 question

1. How many ways to award gold, silver, bronze among 6 athletes?

   120 — that's 6 × 5 × 420720

Practice2 questions

1. A pizza lets you choose 3 toppings from 10. How many topping combinations?

   120 — that's C(10, 3)720 — that's P(10, 3)30

2. Which situation needs combinations (not permutations)?

   choosing a 5-person committee from 20awarding 1st, 2nd, 3rd placesetting a 4-digit ordered PIN

Practice2 questions

1. How many distinct arrangements of the letters in 'LEVEL'?

   30 — that's 5!/(2!·2!)120 — that's 5!60

2. Flipping a coin 3 times, P(exactly 1 head) is…

   3/81/31/2

Practice2 questions

1. Legs 6 and 8 — the hypotenuse is…

   101448

2. How far is (0,0) from (5,12)?

   131760

Practice1 question

1. An online poll on a news site finds 80% oppose a policy. The biggest concern is…

   self-selection bias — who chooses to vote isn't representativethe sample is too small80% is too round to be real

Practice1 question

1. Mean = $71k but median = $58k. The distribution is…

   right-skewed — high earners pull the mean upleft-skewedsymmetric

Practice2 questions

1. Counting independent rare events (calls per minute) is best modeled by…

   the Poisson distributionthe Normal distributionthe Uniform distribution

2. The expected value of one fair six-sided die roll is…

   3.5 — a balance point, not an achievable roll3 — the rounded middle6 — the maximum

Practice1 question

1. A rare disease, a very accurate test, a positive result. Your true risk is…

   often far below the test's accuracy — the base rate dominatesequal to the test's accuracyalways above 90%

Practice2 questions

1. To halve your poll's margin of error, you need…

   4× the sample2× the samplea bigger population

2. About what fraction of bell-curve values fall within ±2σ?

   95%68%99.7%

Practice2 questions

1. Demanding stronger evidence before declaring an effect real will…

   reduce false alarms (Type I) but increase misses (Type II)reduce both error types at oncehave no effect on errors

2. Training a classifier by minimizing cross-entropy is equivalent to…

   maximum likelihood estimationcomputing a confidence intervala Type II error

Practice2 questions

1. Cities with more police have more crime — a positive correlation. The best reading is…

   a confounder (population size) drives bothpolice cause crimecrime is random noise

2. Regression's least-squares line minimizes…

   the sum of squared residuals (vertical errors)the correlation coefficientthe number of data points

Practice2 questions

1. Training accuracy 98%, test accuracy 72%. The diagnosis is…

   overfitting — high variance, memorizing training noiseunderfitting — the model is too simplethe test set is broken

2. Why hold out a test set never used in training?

   to fairly estimate population (real-world) performanceto have more data to train on laterto make training faster

Practice2 questions

1. In y = 3 sin(2x), the amplitude and period are…

   amplitude 3, period πamplitude 2, period 3amplitude 3, period 2π

2. Which everyday thing is NOT naturally modeled by sin/cos?

   a one-time population crashAC wall voltagedaylight hours over a year

Practice1 question

1. Two identical waves, half a cycle out of phase, are added. The result is…

   silence — total cancellationdouble volumea doubled frequency

Practice2 questions

1. A curve is momentarily flat at a point. Its derivative there is…

   01undefined

2. f′(2) = −3 means that near x = 2, f is…

   decreasing, 3 output units per input unitequal to −3increasing slowly

Practice3 questions

1. Match the derivative:

   $$\frac{d}{dx}\,x^5 = \;?$$
   $5x^4$$x^4$$5x^5$

2. Which function is its own derivative?

   $e^x$$x^e$$\ln x$

3. Match the derivative:

   $$\frac{d}{dx}\left(\frac{1}{x}\right) = \;?$$
   $-\frac{1}{x^2}$$\frac{1}{x^2}$$\ln x$

Practice3 questions

1. Match the antiderivative:

   $$\int x^3\,dx = \;?$$
   $\frac{x^4}{4} + C$$3x^2 + C$$x^4 + C$

2. Which integral produces ln|x| + C?

   $\int \frac{1}{x}\,dx$$\int \ln x\,dx$$\int \frac{1}{x^2}\,dx$

3. Two antiderivatives of the same function always differ by…

   a constanta factor of xnothing — they're identical

Practice2 questions

1. The area under a speed–time curve gives…

   distance traveledaccelerationaverage speed

2. What is ∫₀² 3 dx? (Sketch it.)

   639

Practice2 questions

1. An inductive proof needs…

   a base case and an n → n+1 stepchecking many examplesa formula for every n

2. How many halvings get 1,000,000 down to 1?

   about 20about 1,000about 500,000

Practice2 questions

1. A transformation has determinant 0. Solving Ax = b will…

   fail or give infinitely many answersgive exactly one answergive exactly two answers

2. The dot product of (1, 2) and (4, −2) is…

   0 — they're perpendicular8(4, −4)

Practice1 question

1. A system's update matrix has eigenvalues 0.9 and 0.4. Repeated updates…

   decay toward zero, the 0.9 direction fading slowestgrow without boundoscillate forever

Practice1 question

1. Which regularizer tends to drive weights to exactly zero (feature selection)?

   L1 (‖w‖₁) — its sharp corners land on the axesL2 (‖w‖₂) — it shrinks but rarely zeroesneither affects sparsity

Practice1 question

1. A 10,000 × 10,000 ratings matrix is well-approximated by rank 20. This means…

   about 20 latent patterns explain most of the datathe matrix has only 20 nonzero entries20 users rated everything

Practice2 questions

1. The singular values of a data matrix decay quickly. This tells you the data is…

   Need a hint?
   near low-rank — compressible, with a few dominant patternspure random noiseimpossible to approximate

2. PCA's principal components are, essentially…

   the top singular directions of the centered datathe rows with the largest valuesrandomly chosen axes

Practice1 question

1. Squared-error loss, compared to absolute-error loss, …

   punishes large mistakes much more heavilyignores large mistakesis only for classification

Practice2 questions

1. To decrease the loss as fast as possible, step in the direction of…

   the negative gradient, −∇Lthe positive gradient, +∇Lany random direction

2. For L = w₁² + 3w₂², the partial ∂L/∂w₂ is…

   Need a hint?
   6w₂2w₁w₂²

Practice2 questions

1. For y = (3x + 1)², dy/dx at x = 1 is…

   2486

2. Backpropagation is best described as…

   the chain rule applied backward, reusing shared factorsa new differentiation rule for networksrandom weight updates

Practice2 questions

1. For L(w) = w² (gradient 2w), one step from w = 3 with η = 0.1 gives…

   Need a hint?
   2.43.60.6

2. Training loss suddenly explodes to huge values. The most likely culprit is…

   a learning rate that's too largea learning rate that's too smalltoo much training data

Practice2 questions

1. On a convex loss surface, gradient descent…

   is guaranteed to reach the global minimumoften gets stuck in local minimacannot be used

2. Lagrange multipliers are used to…

   optimize subject to constraintscompute gradients fasterguarantee convexity

Practice2 questions

1. Which situation obeys y′ = ky?

   interest compounding on a balancea car at steady speeda ball in free fall

2. A culture doubles every 3 hours, starting at 1,000. After 9 hours…

   8,0003,00027,000

Practice1 question

1. A characteristic equation gives r = −3 ± 4i. The system…

   oscillates at 4 rad/s inside a decaying e^{−3t} envelopegrows at rate 3decays with no oscillation

Practice1 question

1. Two instruments play the same note but sound different because…

   their harmonic amplitude recipes differtheir fundamentals differ slightlyhuman ears are unreliable

Practice1 question

1. 1,000 samples at 1,000 Hz are FFT'd. Bin spacing is…

   1 Hz1,000 Hzdepends on the FFT implementation

Practice1 question

1. A sine completes exactly 7 cycles in your DFT frame. Leakage is…

   absent — one clean binseveredependent on amplitude

Practice1 question

1. x[n] with square brackets signals…

   a discrete signal — a numbered list of samplesan absolute valuea continuous signal

Practice1 question

1. An LTI system's response to a single unit impulse is…

   its kernel h — and that determines everythingalways a smoothed stepunpredictable without more tests

Practice1 question

1. Multiplying a phasor by i does what to its wave?

   shifts its phase a quarter cycledoubles its frequencymakes it imaginary, i.e. unphysical

Practice2 questions

1. A moving-average filter mainly…

   passes low frequencies, suppresses high onespasses high frequenciesshifts pitch downward

2. Feed a 100 Hz sine into an LTI filter. The output contains…

   only 100 Hz, rescaled and phase-shifted100 Hz plus new harmonicswhatever frequencies the filter prefers