KEAM Mathematics: Calculus, Geometry and Statistics Syllabus

Mathematics: Calculus, Geometry and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics: Calculus, Geometry and Statistics in KEAM, not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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

Mathematics: Calculus, Geometry and Statistics flashcards for KEAM

19 of 50 cards from the Mathematics: Calculus, Geometry and Statistics deck — real questions with worked answers.

1. 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).

2. 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).

3. 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.

4. 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.

5. 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).

6. 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.

7. 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.

8. 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.

9. 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.

10. 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].

11. 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.

12. 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).

13. 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.

14. 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.

15. 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.

16. 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).

17. 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.

18. 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).

19. 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).

See more Mathematics: Calculus, Geometry and Statistics flashcards →

Planning Mathematics: Calculus, Geometry and Statistics for KEAM

Mathematics: Calculus, Geometry and Statistics (KEAM) FAQ