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KEAM Mathematics: Calculus, Geometry and Statistics Syllabus
The Mathematics: Calculus, Geometry and Statistics syllabus for KEAM in full: 4 chapters, 12 topics and 31 sub-topics, every one listed below, plus 50 Mathematics: Calculus, Geometry and Statistics flashcards with answers.
Mathematics: Calculus, Geometry and Statistics syllabus: all 4 chapters
Expand any chapter to see its topics and sub-topics.
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Coordinate Geometry
3 topics- Straight Lines
- Slope and forms of equation of a line
- Distance and angle between lines
- Conic Sections
- Circle and its equations
- Parabola and ellipse
- Hyperbola
- Three Dimensional Geometry
- Direction cosines and ratios
- Equation of a line in space
- Equation of a plane and shortest distance
- Straight Lines
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Differential Calculus
3 topics- Limits and Continuity
- Evaluation of limits and standard limits
- Continuity and differentiability
- Differentiation
- Derivatives of standard functions
- Chain, product and quotient rules
- Logarithmic and implicit differentiation
- Applications of Derivatives
- Rate of change and tangents/normals
- Maxima, minima and monotonicity
- Approximations and Rolle's/mean value theorems
- Limits and Continuity
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Integral Calculus and Differential Equations
3 topics- Indefinite Integration
- Standard integrals and substitution
- Integration by parts and partial fractions
- Definite Integration and Areas
- Properties of definite integrals
- Area under and between curves
- Differential Equations
- Order, degree and formation
- Variable separable and homogeneous equations
- Linear differential equations
- Indefinite Integration
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Vectors, Statistics and Probability
3 topics- Vector Algebra
- Addition and scalar multiplication of vectors
- Scalar (dot) and vector (cross) products
- Scalar triple product
- Statistics
- Measures of central tendency
- Mean deviation, variance and standard deviation
- Probability
- Addition and multiplication theorems
- Conditional probability and Bayes' theorem
- Random variables and binomial distribution
- Vector Algebra
KEAM Mathematics: Calculus, Geometry and Statistics flashcards — 50 questions with answers
What is the slope-intercept form of the equation of a straight line, and what does each symbol represent?
y = mx + c, where m is the slope (tan of the angle the line makes with the positive x-axis) and c is the y-intercept (the y-coordinate where the line crosses the y-axis).
State the formula for the distance of a point (x1, y1) from the line ax + by + c = 0.
Distance = |a·x1 + b·y1 + c| / sqrt(a^2 + b^2).
What is the angle between two lines with slopes m1 and m2, and the conditions for them to be parallel or perpendicular?
tan θ = |(m1 − m2) / (1 + m1·m2)|. Parallel when m1 = m2; perpendicular when m1·m2 = −1.
Give the equation of a line in normal form and explain p and ω.
x·cos ω + y·sin ω = p, where p is the perpendicular distance from the origin to the line and ω is the angle that the perpendicular makes with the positive x-axis.
What is the standard equation of a circle with centre (h, k) and radius r, and the general equation of a circle?
Standard: (x − h)^2 + (y − k)^2 = r^2. General: x^2 + y^2 + 2gx + 2fy + c = 0, with centre (−g, −f) and radius sqrt(g^2 + f^2 − c).
For the parabola y^2 = 4ax, give the focus, directrix, vertex, axis, and length of the latus rectum.
Focus: (a, 0); directrix: x = −a; vertex: (0, 0); axis: the x-axis; length of latus rectum: 4a.
For the ellipse x^2/a^2 + y^2/b^2 = 1 (a > b), state the eccentricity, foci, and length of the latus rectum.
Eccentricity e = sqrt(1 − b^2/a^2); foci: (±ae, 0); length of latus rectum: 2b^2/a.
For the hyperbola x^2/a^2 − y^2/b^2 = 1, state the eccentricity, foci, and equations of the asymptotes.
Eccentricity e = sqrt(1 + b^2/a^2); foci: (±ae, 0); asymptotes: y = ±(b/a)x.
Define the eccentricity of a conic and give its value for a circle, parabola, ellipse, and hyperbola.
Eccentricity e is the ratio of the distance from a point on the conic to the focus to its distance from the directrix. Circle: e = 0; parabola: e = 1; ellipse: 0 < e < 1; hyperbola: e > 1.
What is the distance formula between two points (x1, y1, z1) and (x2, y2, z2) in three dimensions?
Distance = sqrt[(x2 − x1)^2 + (y2 − y1)^2 + (z2 − z1)^2].
Define direction cosines of a line and state the identity they satisfy.
Direction cosines l, m, n are the cosines of the angles the line makes with the positive x, y, z axes respectively. They satisfy l^2 + m^2 + n^2 = 1.
How are direction cosines obtained from direction ratios a, b, c?
l = a/sqrt(a^2+b^2+c^2), m = b/sqrt(a^2+b^2+c^2), n = c/sqrt(a^2+b^2+c^2) (with the appropriate sign).
Give the vector and Cartesian equation of a line through point with position vector a, parallel to vector b.
Vector form: r = a + λb. Cartesian form: (x − x1)/a1 = (y − y1)/a2 = (z − z1)/a3, where (a1, a2, a3) are direction ratios.
State the vector and Cartesian equation of a plane in normal form with unit normal n̂ and distance d from origin.
Vector form: r · n̂ = d. Cartesian form: lx + my + nz = d, where l, m, n are direction cosines of the normal and d is the perpendicular distance from the origin.
What is the angle between a line with direction vector b and a plane with normal vector n?
sin θ = |b · n| / (|b| |n|), where θ is the angle between the line and the plane.
State the formula for the distance of a point (x1, y1, z1) from the plane Ax + By + Cz + D = 0.
Distance = |A·x1 + B·y1 + C·z1 + D| / sqrt(A^2 + B^2 + C^2).
State the limit value of (sin x)/x as x approaches 0, and of (1 − cos x)/x^2 as x approaches 0.
lim(x→0) (sin x)/x = 1; lim(x→0) (1 − cos x)/x^2 = 1/2.
State the limits lim(x→0) (e^x − 1)/x and lim(x→0) (a^x − 1)/x.
lim(x→0) (e^x − 1)/x = 1; lim(x→0) (a^x − 1)/x = ln a (loge a).
What are the three conditions for a function f to be continuous at x = a?
(1) f(a) is defined; (2) lim(x→a) f(x) exists; (3) lim(x→a) f(x) = f(a).
State the limit lim(x→0) (1 + x)^(1/x) and lim(x→∞) (1 + 1/x)^x.
Both equal e.
Show the other 30 Mathematics: Calculus, Geometry and Statistics flashcards
State the derivatives of sin x, cos x, tan x, and sec x.
d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec^2 x; d/dx(sec x) = sec x · tan x.
State the product rule and quotient rule for differentiation.
Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv') / v^2.
State the chain rule for differentiating a composite function y = f(g(x)).
dy/dx = f'(g(x)) · g'(x), i.e. dy/dx = (dy/du)·(du/dx) where u = g(x).
State the derivatives of ln x, e^x, a^x, and x^n.
d/dx(ln x) = 1/x; d/dx(e^x) = e^x; d/dx(a^x) = a^x · ln a; d/dx(x^n) = n·x^(n−1).
State the derivatives of sin⁻¹ x and tan⁻¹ x.
d/dx(sin⁻¹ x) = 1/sqrt(1 − x^2); d/dx(tan⁻¹ x) = 1/(1 + x^2).
What is the geometric meaning of the first derivative f'(x), and how do you find the slope of a tangent at a point?
f'(x) gives the slope of the tangent to the curve y = f(x) at the point x. The slope of the tangent at x = a is f'(a).
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).
KEAM Mathematics: Calculus, Geometry and Statistics FAQ
What is in the KEAM Mathematics: Calculus, Geometry and Statistics syllabus?
KEAM Mathematics: Calculus, Geometry and Statistics is split into 4 chapters: Coordinate Geometry, Differential Calculus, Integral Calculus and Differential Equations and Vectors, Statistics and Probability. Together they contain 12 topics and 31 sub-topics.
How many chapters are in KEAM Mathematics: Calculus, Geometry and Statistics?
4 chapters, containing 12 topics and 31 sub-topics. The largest chapter is Coordinate Geometry, with 3 topics.
Are there flashcards for KEAM Mathematics: Calculus, Geometry and Statistics?
Yes — a 50-card Mathematics: Calculus, Geometry and Statistics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.