OS
OpenStem
@openstem · Joined Jul 2026
7420 public items8 groups
Quiz6 questions
Simplify i².
A−1B1C−iD0
Mathematics · L3 · Complex Number Operations
@openstem
Mathematics · L3 · Complex Number OperationsQuiz6 questions
What does eliminating the parameter t from x(t), y(t) produce?
AA single Cartesian equation relating x and y directlyBA new value of tCThe derivative of the curveDA second parameter to replace t
Mathematics · L3 · Parametric Equations & Motion
@openstem
Mathematics · L3 · Parametric Equations & MotionQuiz6 questions
This parabola is drawn with a = 1, b = 0 and c = −4, so its equation is y = x² − 4. Where does it cross the x-axis?
AAt x = −4 and x = 4BAt x = −2 and x = 2CAt x = 0 onlyDIt never crosses the x-axis
Mathematics · L3 · Reading Graphs — 6 Q
@openstem
Mathematics · L3 · Reading Graphs — 6 QQuiz10 questions
Which of the following is NOT one of Kolmogorov's axioms of probability?
AP(A) ≥ 0 for every event ABP(Ω) = 1CP(A ∪ B) = P(A) + P(B) for mutually exclusive A, BDP(A) = 1 − P(Aᶜ) for every event A
Mathematics · L4 · Probability fundamentals — 10 Q
@openstem
Mathematics · L4 · Probability fundamentals — 10 QQuiz10 questions
What is d/dx [x⁵]?
Ax⁴B5x⁴C5x⁵D4x⁴
Mathematics · L4 · Derivative rules — 10 Q
@openstem
Mathematics · L4 · Derivative rules — 10 QQuiz10 questions
For matrices A (m×n) and B (n×p), what is the size of AB?
An×nBm×pCm×nDn×p
Mathematics · L4 · Matrix operations — 10 Q
@openstem
Mathematics · L4 · Matrix operations — 10 QQuiz10 questions
A study reports p = 0.03 with α = 0.05. What is the correct conclusion?
AThere is a 3% chance H₀ is trueBReject H₀; the result is statistically significant at the 5% levelCAccept H₁ as definitely trueDFail to reject H₀
Mathematics · L4 · Hypothesis Testing & Inference — 10 Q
@openstem
Mathematics · L4 · Hypothesis Testing & Inference — 10 QQuiz10 questions
A dataset is: 2, 4, 4, 4, 5, 5, 7, 9. What is the mean?
A4B4.5C5D5.5
Mathematics · L4 · Descriptive Statistics & Distributions — 10 Q
@openstem
Mathematics · L4 · Descriptive Statistics & Distributions — 10 QQuiz10 questions
In simple linear regression, what does the slope β₁ represent?
AThe predicted value of Y when X = 0BThe expected change in Y for a one-unit increase in XCThe correlation coefficient between X and YDThe proportion of variance in Y explained by X
Mathematics · L4 · Regression & Bayesian Methods — 10 Q
@openstem
Mathematics · L4 · Regression & Bayesian Methods — 10 QQuiz8 questions
Which technique solves dy/dx = f(x)g(y)?
AIntegrating factorBSeparation of variablesCLaplace transformDLagrange multipliers
Mathematics · L4 · First-Order Differential Equations — 8 Q
@openstem
Mathematics · L4 · First-Order Differential Equations — 8 QQuiz8 questions
For f(x,y) = x²y + 3y, what is ∂f/∂x?
A2xyBx² + 3C2x + 3yDx²y
Mathematics · L4 · Multivariable Calculus — 8 Q
@openstem
Mathematics · L4 · Multivariable Calculus — 8 QQuiz3 questions
Which property of ℝ guarantees that every Cauchy sequence converges?
ACompletenessBCompactnessCConnectednessDDensity of ℚ
Mathematics · L5 · Real Analysis
@openstem
Mathematics · L5 · Real AnalysisQuiz3 questions
Lagrange's theorem states that if H is a subgroup of a finite group G, then:
A|H| divides |G|B|G| divides |H|C|H| = |G| − 1D|H|² = |G|
Mathematics · L5 · Group & Ring Theory
@openstem
Mathematics · L5 · Group & Ring TheoryQuiz3 questions
A random variable X: Ω → ℝ is measurable if and only if:
AX⁻¹(B) ∈ ℱ for every Borel set B ⊆ ℝBX is continuous almost everywhereCE[X] is finiteDX takes only countably many values
Mathematics · L5 · Measure-Theoretic Probability
@openstem
Mathematics · L5 · Measure-Theoretic ProbabilityQuiz4 questions
Let X₁, X₂, … be i.i.d. with mean μ and variance σ². What does the CLT say about √n(X̄ₙ − μ)?
AIt converges in probability to μBIt converges almost surely to 0CIt converges in distribution to N(0, σ²)DIt converges in distribution to N(μ, σ²)
Mathematics · L5 · Asymptotics & Causal Inference
@openstem
Mathematics · L5 · Asymptotics & Causal InferenceQuiz8 questions
Which of the following is NOT one of the topology axioms for τ on X?
A∅ and X are both in τBτ is closed under arbitrary unionsCτ is closed under finite intersectionsDτ is closed under arbitrary intersections
Mathematics · L5 · Point-Set Topology
@openstem
Mathematics · L5 · Point-Set TopologyQuiz8 questions
The Cauchy–Riemann equations for f = u + iv state:
A∂u/∂x = ∂v/∂x and ∂u/∂y = ∂v/∂yB∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂xCu = v everywhereD∂u/∂x = −∂v/∂y and ∂u/∂y = ∂v/∂x
Mathematics · L5 · Complex Analysis
@openstem
Mathematics · L5 · Complex AnalysisQuiz8 questions
A set C is convex if:
AIt is boundedBFor all x,y ∈ C and t ∈ [0,1], tx+(1−t)y ∈ CCIt contains the originDIt is closed
Mathematics · L5 · Convex Optimization
@openstem
Mathematics · L5 · Convex OptimizationNote~195 words · 1 min
The theorem Right triangle, legs a and b, hypotenuse c: Triples that come out whole are worth knowing on sight: The triangle below is drawn on the grid. Read the legs off the axes, then confirm the hypotenuse yourself. Distance is Pythagora
Mathematics · L3 · The Pythagorean Theorem & Coordinate Geometry
@openstem
Mathematics · L3 · The Pythagorean Theorem & Coordinate GeometryNote~237 words · 1 min
Independent or dependent? The whole topic hinges on one question: does the first outcome change the odds for the second? P(heads AND a six) = 1/2 × 1/6 = 1/12. The coin has no opinion about the die. Conditional probability Read it as: given
Mathematics · L3 · Independent, Dependent & Conditional Probability
@openstem
Mathematics · L3 · Independent, Dependent & Conditional ProbabilityNote~263 words · 1 min
Factors are few, multiples are endless Factors of 12: just 1, 2, 3, 4, 6, 12. Six numbers, and that's all of them. Multiples of 12 never stop. Prime, composite, and the odd one out Prime factorisation with a factor tree Split repeatedly unt
Mathematics · L2 · Factors, Multiples & Primes
@openstem
Mathematics · L2 · Factors, Multiples & PrimesNote~256 words · 1 min
Range: how far apart are the extremes? For 4, 9, 2, 15, 7: the biggest is 15, the smallest is 2, so the range is 13. Small range means bunched together; large range means spread out. What counts as an outlier A value standing well apart fro
Mathematics · L2 · Range, Outliers & Data Spread
@openstem
Mathematics · L2 · Range, Outliers & Data SpreadNote~316 words · 2 min
Two categories at once A two-way table records counts for two things together: here, grade level down the rows and favourite subject across the columns. Each cell counts students who are in BOTH. Relative frequency A count compared to a tot
Mathematics · L2 · Two-Way Tables & Venn Diagrams
@openstem
Mathematics · L2 · Two-Way Tables & Venn DiagramsNote~244 words · 1 min
Three ways to write one ratio A ratio compares two quantities. Two cups of flour for every three of sugar can be written: Same relationship, three notations. Pick whichever suits the question. Simplifying Divide both sides by their greatest
Mathematics · L2 · Ratios & Unit Rates
@openstem
Mathematics · L2 · Ratios & Unit RatesNote~182 words · 1 min
Why two steps? x + 4 = 9 needs one undo. Two-step equations have a multiplier AND something added. x was multiplied by 2, then 3 was added. Undo in reverse order: the addition first. The two moves Inequalities: same moves, one extra rule So
Mathematics · L2 · Two-Step Equations & Inequalities
@openstem
Mathematics · L2 · Two-Step Equations & InequalitiesNote~202 words · 1 min
Two number lines that cross Between them, any location on the plane can be named with just two numbers. Plotting (x, y) So (4, 2) means 4 right, then 2 up. The four quadrants Numbered from the top-right, going counter-clockwise. A point sit
Mathematics · L2 · The Coordinate Plane
@openstem
Mathematics · L2 · The Coordinate PlaneNote~227 words · 1 min
Volume: how much fits inside A fish tank 40 cm by 20 cm by 25 cm holds 40 × 20 × 25 = 20,000 cm³. Since 1000 cm³ is a litre, that's 20 litres of water to fill it. Surface area: unfold it flat Imagine cutting along the box's edges and flatte
Mathematics · L2 · Volume & Surface Area of Prisms
@openstem
Mathematics · L2 · Volume & Surface Area of PrismsNote~149 words · 1 min
Multiply the numbers, then decide the sign Two separate jobs. The size comes from ordinary multiplication; the sign comes from the two signs you started with. Division uses the identical rule Three or more numbers? Count the negatives (−2)
Mathematics · L2 · Multiplying & Dividing Integers
@openstem
Mathematics · L2 · Multiplying & Dividing IntegersNote~232 words · 1 min
The parts of an expression An expression mixes numbers, variables and operations, and never has an equals sign. Take 5x + 3y − 7. Terms are separated by + and − signs, and the sign in front belongs to the term. That's why the last term is −
Mathematics · L2 · Algebraic Expressions
@openstem
Mathematics · L2 · Algebraic ExpressionsNote~240 words · 1 min
Two things happening together You can already find the chance of one event, like rolling a 3. A compound event asks about two or more together: flip a coin AND roll a die. To handle that you first need to count every possible combination. T
Mathematics · L2 · Compound Events & Probability
@openstem
Mathematics · L2 · Compound Events & Probability