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COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics Syllabus

The Mathematics - Calculus, Coordinate Geometry and Statistics syllabus for COMEDK UGET in full: 5 chapters, 16 topics and 41 sub-topics, every one listed below, plus 50 Mathematics - Calculus, Coordinate Geometry and Statistics flashcards with answers.

5Chapters
16Topics
41Sub-topics
50Flashcards

Mathematics - Calculus, Coordinate Geometry and Statistics syllabus: all 5 chapters

Expand any chapter to see its topics and sub-topics.

  1. Limits, Continuity and Differentiation

     3 topics
    • Limits and Continuity
      • Evaluation of limits
      • Continuity and differentiability
    • Differentiation
      • Derivatives of standard functions
      • Chain rule, product and quotient rules
      • Implicit and logarithmic differentiation
    • Applications of Derivatives
      • Rate of change and tangents and normals
      • Increasing and decreasing functions
      • Maxima and minima

  2. Integral Calculus

     3 topics
    • Indefinite Integration
      • Standard integrals and substitution
      • Integration by parts
      • Integration by partial fractions
    • Definite Integration
      • Properties of definite integrals
      • Definite integral as a limit of sum
    • Applications of Integrals
      • Area under simple curves
      • Area between two curves

  3. Differential Equations and Vectors

     3 topics
    • Differential Equations
      • Order and degree
      • Variable separable equations
      • Linear differential equations
    • Vector Algebra
      • Addition and scalar multiplication
      • Dot and cross products
      • Scalar and vector triple products
    • Applications of Vectors
      • Area of triangle and parallelogram
      • Coplanarity and geometric applications

  4. Coordinate Geometry

     4 topics
    • Straight Lines
      • Slope and forms of a line
      • Distance and angle between lines
      • Family of lines
    • Circles
      • Equation of a circle
      • Tangents and normals
    • Conic Sections
      • Parabola
      • Ellipse
      • Hyperbola
    • Three-Dimensional Geometry
      • Direction cosines and ratios
      • Equation of a line in space
      • Equation of a plane and distance

  5. Probability and Statistics

     3 topics
    • Probability
      • Addition and multiplication theorems
      • Conditional probability and Bayes' theorem
      • Random variables and binomial distribution
    • Statistics
      • Measures of central tendency
      • Mean deviation, variance and standard deviation
    • Linear Programming
      • Formulation of LPP
      • Graphical solution of LPP

COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics flashcards — 50 questions with answers

  1. State the definition of the limit of a function f(x) as x approaches a (epsilon-delta).

    lim(x→a) f(x) = L means: for every ε>0 there exists δ>0 such that 0<|x−a|<δ implies |f(x)−L|<ε.

  2. What three conditions must hold 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).

  3. State the value of lim(x→0) (sin x)/x and lim(x→0) (tan x)/x.

    Both limits equal 1 (with x in radians).

  4. What is lim(x→0) (1 − cos x)/x²?

    1/2.

  5. Give the standard limits lim(x→0) (a^x − 1)/x and lim(x→0) (e^x − 1)/x.

    (a^x − 1)/x → ln a, and (e^x − 1)/x → 1.

  6. State the first principle (definition) of the derivative of f at x.

    f'(x) = lim(h→0) [f(x+h) − f(x)] / h.

  7. State the product rule and quotient rule for differentiation.

    Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v².

  8. 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).

  9. Give the derivatives of sin x, cos x, tan x, and sec x.

    d/dx sin x = cos x; cos x → −sin x; tan x → sec²x; sec x → sec x tan x.

  10. Give the derivatives of e^x, a^x, ln x, and log_a x.

    e^x → e^x; a^x → a^x ln a; ln x → 1/x; log_a x → 1/(x ln a).

  11. Give the derivatives of arcsin x, arctan x, and arcsec x.

    arcsin x → 1/√(1−x²); arctan x → 1/(1+x²); arcsec x → 1/(|x|√(x²−1)).

  12. What is the geometric meaning of the derivative f'(a)?

    It is the slope of the tangent line to y=f(x) at x=a (instantaneous rate of change).

  13. State the equations of the tangent and normal to y = f(x) at point (x₁, y₁).

    Tangent: y − y₁ = f'(x₁)(x − x₁). Normal: y − y₁ = −1/f'(x₁) · (x − x₁).

  14. What is the second-derivative test for a local extremum at a critical point c?

    If f'(c)=0: f''(c)>0 ⇒ local minimum; f''(c)<0 ⇒ local maximum; f''(c)=0 ⇒ test inconclusive.

  15. Define a point of inflection in terms of the second derivative.

    A point where the concavity changes; typically f''(x)=0 (or undefined) and f'' changes sign there.

  16. When is a function increasing or decreasing on an interval?

    Increasing where f'(x) > 0; decreasing where f'(x) < 0 on that interval.

  17. State Rolle's Theorem.

    If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

  18. State the Mean Value Theorem (Lagrange).

    If f is continuous on [a,b] and differentiable on (a,b), then ∃ c in (a,b) with f'(c) = [f(b)−f(a)]/(b−a).

  19. Give ∫xⁿ dx (n ≠ −1) and ∫(1/x) dx.

    ∫xⁿ dx = x^(n+1)/(n+1) + C; ∫(1/x) dx = ln|x| + C.

  20. Give ∫sin x dx, ∫cos x dx, and ∫sec²x dx.

    ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C; ∫sec²x dx = tan x + C.

Show the other 30 Mathematics - Calculus, Coordinate Geometry and Statistics flashcards

  1. Give ∫e^x dx, ∫a^x dx, and ∫tan x dx.

    ∫e^x dx = e^x + C; ∫a^x dx = a^x/ln a + C; ∫tan x dx = ln|sec x| + C.

  2. State the formula for integration by parts.

    ∫u dv = uv − ∫v du; choose u by the ILATE/LIATE priority order.

  3. Give ∫1/(x²+a²) dx and ∫1/(x²−a²) dx.

    ∫1/(x²+a²) dx = (1/a) arctan(x/a) + C; ∫1/(x²−a²) dx = (1/2a) ln|(x−a)/(x+a)| + C.

  4. Give ∫1/√(a²−x²) dx and ∫1/√(x²+a²) dx.

    ∫1/√(a²−x²) dx = arcsin(x/a) + C; ∫1/√(x²+a²) dx = ln|x + √(x²+a²)| + C.

  5. State the Fundamental Theorem of Calculus (evaluation form).

    If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

  6. State the property ∫ₐᵇ f(x) dx in terms of ∫₀ᵃ for f(a−x).

    ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx (king property).

  7. State the rule for ∫₋ₐᵃ f(x) dx for even and odd functions.

    If f is even: 2∫₀ᵃ f(x) dx. If f is odd: 0.

  8. Give the formula for the area under y = f(x) ≥ 0 between x = a and x = b.

    Area = ∫ₐᵇ f(x) dx.

  9. Give the formula for the area between two curves y = f(x) and y = g(x) with f ≥ g on [a,b].

    Area = ∫ₐᵇ [f(x) − g(x)] dx.

  10. What is the order and degree of a differential equation?

    Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives (rational, radical-free).

  11. Describe the method for solving a variable-separable differential equation.

    Write dy/dx = g(x)h(y), separate as dy/h(y) = g(x) dx, then integrate both sides.

  12. State the standard form and integrating factor of a linear first-order ODE.

    dy/dx + P(x)y = Q(x); integrating factor I.F. = e^∫P dx; solution: y·(I.F.) = ∫Q·(I.F.) dx + C.

  13. What substitution solves a homogeneous differential equation dy/dx = F(y/x)?

    Let y = vx, so dy/dx = v + x(dv/dx), converting it into a variable-separable equation in v and x.

  14. Give the magnitude of vector a = a₁i + a₂j + a₃k and the unit vector along a.

    |a| = √(a₁²+a₂²+a₃²); unit vector = a/|a|.

  15. Define the dot product a·b and give its geometric formula.

    a·b = a₁b₁+a₂b₂+a₃b₃ = |a||b|cos θ, where θ is the angle between them.

  16. Define the cross product a×b: its magnitude and direction.

    |a×b| = |a||b|sin θ; direction perpendicular to both a and b (right-hand rule). It equals the determinant |i j k; a₁ a₂ a₃; b₁ b₂ b₃|.

  17. How do you test whether two vectors are perpendicular or parallel?

    Perpendicular ⇔ a·b = 0. Parallel ⇔ a×b = 0 (components proportional).

  18. Give the formula for the projection (scalar) of a onto b.

    Scalar projection = (a·b)/|b|; vector projection = [(a·b)/|b|²] b.

  19. What does the scalar triple product [a b c] represent geometrically, and when is it zero?

    [a b c] = a·(b×c) = volume of the parallelepiped; it is 0 when the three vectors are coplanar.

  20. Give the area of a triangle with two sides as vectors a and b from one vertex.

    Area = (1/2)|a × b|.

  21. Give the slope-intercept and point-slope forms of a straight line.

    Slope-intercept: y = mx + c. Point-slope: y − y₁ = m(x − x₁).

  22. Give the distance from point (x₁, y₁) to line ax + by + c = 0.

    Distance = |ax₁ + by₁ + c| / √(a²+b²).

  23. What is the condition for two lines with slopes m₁ and m₂ to be perpendicular, and the angle formula between them?

    Perpendicular ⇔ m₁m₂ = −1; angle θ: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.

  24. Give the standard equation of a circle with centre (h, k) and radius r, and the general form.

    (x−h)² + (y−k)² = r². General: x²+y²+2gx+2fy+c=0 with centre (−g,−f) and r = √(g²+f²−c).

  25. State the standard equation of a parabola opening rightward and its focus and directrix.

    y² = 4ax; focus (a, 0); directrix x = −a; vertex at origin.

  26. Give the equation of an ellipse (a>b) with its eccentricity formula.

    x²/a² + y²/b² = 1; e = √(1 − b²/a²), with 0 < e < 1; foci at (±ae, 0).

  27. Give the equation of a hyperbola and its eccentricity formula.

    x²/a² − y²/b² = 1; e = √(1 + b²/a²), with e > 1; foci at (±ae, 0).

  28. Give the direction cosines relation and the distance between points (x₁,y₁,z₁) and (x₂,y₂,z₂) in 3D.

    l²+m²+n² = 1 (direction cosines); distance = √[(x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)²].

  29. State Bayes' Theorem.

    P(Eᵢ|A) = [P(Eᵢ)·P(A|Eᵢ)] / Σⱼ P(Eⱼ)·P(A|Eⱼ), for a partition {Eⱼ}.

  30. State the conditions and objective of a Linear Programming Problem (LPP), and where the optimum occurs.

    Maximize/minimize a linear objective function subject to linear inequality constraints with non-negative variables; the optimum (if it exists) occurs at a corner (vertex) of the feasible region.

COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics FAQ

What is in the COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics syllabus?

COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics is split into 5 chapters: Limits, Continuity and Differentiation, Integral Calculus, Differential Equations and Vectors, Coordinate Geometry and Probability and Statistics. Together they contain 16 topics and 41 sub-topics.

How many chapters are in COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics?

5 chapters, containing 16 topics and 41 sub-topics. The largest chapter is Coordinate Geometry, with 4 topics.

Are there flashcards for COMEDK UGET Mathematics - Calculus, Coordinate Geometry and Statistics?

Yes — a 50-card Mathematics - Calculus, Coordinate 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.