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KEAM Mathematics: Calculus, Geometry and Statistics Flashcards
50 question-and-answer cards covering Mathematics: Calculus, Geometry and Statistics as it is examined in KEAM. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics: Calculus, Geometry and Statistics deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
State the first derivative test for local maxima and minima.
At a critical point where f'(x) = 0: if f'(x) changes from positive to negative, it is a local maximum; if from negative to positive, a local minimum; if no sign change, it is neither (a point of inflection).
State the second derivative test for maxima and minima.
At a point where f'(c) = 0: if f''(c) < 0 it is a local maximum; if f''(c) > 0 it is a local minimum; if f''(c) = 0 the test is inconclusive.
Define increasing and decreasing functions in terms of the derivative.
f is increasing on an interval if f'(x) > 0 throughout it, and decreasing if f'(x) < 0 throughout it.
What is the relationship dy ≈ f'(x)·dx used for in applications of derivatives?
It gives the approximate change in y (differential dy) for a small change dx in x, used to approximate values and estimate errors.
State the power rule for indefinite integration and the integral of 1/x.
∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ −1); ∫ (1/x) dx = ln|x| + C.
State the integrals of sin x, cos x, and sec^2 x.
∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ sec^2 x dx = tan x + C.
State the formula for integration by parts.
∫ u dv = u·v − ∫ v du. The first function u is chosen by the ILATE rule (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).
State the integral ∫ dx/(x^2 + a^2) and ∫ dx/sqrt(a^2 − x^2).
∫ dx/(x^2 + a^2) = (1/a) tan⁻¹(x/a) + C; ∫ dx/sqrt(a^2 − x^2) = sin⁻¹(x/a) + C.
State the integrals ∫ e^x dx and ∫ a^x dx.
∫ e^x dx = e^x + C; ∫ a^x dx = a^x / ln a + C.
State the Fundamental Theorem of Calculus (second part) for definite integrals.
If F is an antiderivative of f, then ∫ from a to b of f(x) dx = F(b) − F(a).
State the property of definite integrals ∫ from a to b f(x) dx in terms of ∫ from a to b f(a + b − x) dx.
∫ from a to b f(x) dx = ∫ from a to b f(a + b − x) dx.
State the definite-integral property for ∫ from 0 to a f(x) dx when f is split using f(a − x).
∫ from 0 to a f(x) dx = ∫ from 0 to a f(a − x) dx; in particular ∫ from 0 to a [f(x)] dx = ∫ from 0 to a [f(a−x)] dx is used to simplify symmetric integrands.
State the rule for ∫ from −a to a f(x) dx depending on whether f is even or odd.
If f is even: ∫ from −a to a f(x) dx = 2 ∫ from 0 to a f(x) dx. If f is odd: ∫ from −a to a f(x) dx = 0.
How do you find the area bounded by the curve y = f(x), the x-axis, and lines x = a and x = b?
Area = ∫ from a to b |f(x)| dx (take the magnitude where the curve lies below the x-axis).
Define the order and degree of a differential equation.
Order is the highest derivative present in the equation. Degree is the power of the highest-order derivative after the equation is made polynomial (free of radicals/fractions in the derivatives).
How do you solve a variable-separable differential equation?
Write it as f(y) dy = g(x) dx, then integrate both sides: ∫ f(y) dy = ∫ g(x) dx + C.
State the standard form of a linear first-order differential equation and its integrating factor.
Standard form: dy/dx + P(x)·y = Q(x). Integrating factor IF = e^(∫ P dx); solution: y·(IF) = ∫ Q·(IF) dx + C.
State the scalar (dot) product of two vectors and a formula for the angle between them.
a · b = |a||b| cos θ = a1·b1 + a2·b2 + a3·b3. Hence cos θ = (a · b) / (|a||b|).
State the vector (cross) product magnitude and its geometric meaning.
|a × b| = |a||b| sin θ, equal to the area of the parallelogram with sides a and b; a × b is perpendicular to both a and b (direction by right-hand rule).
What is the scalar triple product [a b c], and what does its value represent geometrically?
[a b c] = a · (b × c), equal to the volume of the parallelepiped with edges a, b, c. If [a b c] = 0, the three vectors are coplanar.
State the formulas for mean, variance, and standard deviation of ungrouped data.
Mean x̄ = (Σ xi)/n; Variance σ^2 = (Σ (xi − x̄)^2)/n; Standard deviation σ = sqrt(variance).
State the formula for the variance using the shortcut (raw moments) method.
σ^2 = (Σ xi^2)/n − ( (Σ xi)/n )^2 = mean of squares minus square of the mean.
State the addition theorem of probability for two events A and B, and for mutually exclusive events.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
State the conditional probability formula and the multiplication theorem of probability.
Conditional: P(A|B) = P(A ∩ B)/P(B), P(B) ≠ 0. Multiplication: P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B). For independent events, P(A ∩ B) = P(A)·P(B).
What this deck covers
The Mathematics: Calculus, Geometry and Statistics deck follows the KEAM Mathematics: Calculus, Geometry and Statistics syllabus — 4 chapters and 12 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 12.5 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 110 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematics: Calculus, Geometry and Statistics flashcards FAQ
How many Mathematics: Calculus, Geometry and Statistics flashcards are in this KEAM deck?
50 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these KEAM flashcards free?
Yes. The preview here is free to read with no signup, and the full 50-card deck is free inside the Examius app.
What do the Mathematics: Calculus, Geometry and Statistics cards cover?
They follow the KEAM Mathematics: Calculus, Geometry and Statistics syllabus — 4 chapters and 12 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.