INDIA MATH CLASS XII – EDmama India

A. Introduction to Functions

A.05. Determine the domain and range of a function.
A.06. Graph a linear function and identify its key features.
A.07. Write the equation of a linear function given different information (slope-intercept form, point-slope form, etc.).
A.08. Find the slope of a linear function from its equation or graph.
A.09. Explore the concept of piecewise functions and their real-world applications.
A.01. Identify whether a relation is a function.
A.02. Evaluate functions for given inputs.
A.03. Find values using function graphs.
A.04. Complete a table for a function graph.

B. Transformations of Functions

C. Polynomial Basics

C.01. Find the roots of factored polynomials.
C.02. Evaluate polynomials using synthetic division.
C.03. Write a polynomial from its roots.
C.04. Match polynomials to their corresponding graphs.
C.05. Factor sums and differences of cubes.
C.06. Solve equations involving sums and differences of cubes.
C.07. Factor using a quadratic pattern.
C.08. Solve equations using a quadratic pattern.
C.09. Explore the connection between roots, factors, and the graph of a polynomial.

D. Sequences and Series

D.01. Find terms of a sequence given an explicit formula.
D.02. Find terms of a sequence given a recursive formula.
D.03. Identify a sequence as explicit or recursive.
D.04. Find a recursive formula for a given sequence.
D.05. Find recursive and explicit formulas for a sequence.
D.06. Convert between explicit and recursive formulas.
D.07. Evaluate expressions written in sigma notation.
D.08. Identify arithmetic and geometric series.
D.09. Calculate partial sums of arithmetic series.
D.10. Calculate partial sums of geometric series.
D.11. Find the sum of a finite arithmetic or geometric series.
D.12. Determine whether a geometric series is convergent or divergent.
D.13. Find the value of an infinite geometric series.
D.14. Express a repeating decimal as a fraction using geometric series.
D.15. Apply series to solve real-world problems involving growth and decay.

E. Introduction to Trigonometry

E.01. Convert between radians and degrees.
E.02. Calculate arc length using radians.
E.03. Identify the quadrant of an angle.
E.04. Find coterminal and reference angles.
E.05. Determine trigonometric ratios using right triangles (SOH CAH TOA).
E.06. Determine trigonometric ratios using the unit circle.
E.07. Determine trigonometric ratios using reference angles.
E.08. Solve trigonometric equations.
E.09. Calculate side lengths of right triangles using trigonometric ratios.
E.10. Calculate angle measures in right triangles using inverse trigonometric ratios.
E.11. Solve a right triangle given sufficient information.
E.12. Apply the Law of Sines to solve triangles.
E.13. Apply the Law of Cosines to solve triangles.
E.14. Solve a triangle using appropriate trigonometric laws.
E.15. Calculate the area of a triangle using the sine formula.
E.16. Calculate the area of a triangle using Heron’s formula.
E.17. Model and solve real-world problems involving triangles.

F. Trigonometric Functions

F.01. Identify properties (amplitude, period, phase shift, vertical shift) of sine functions.
F.02. Identify properties (amplitude, period, phase shift, vertical shift) of cosine functions.
F.03. Graph sine functions.
F.04. Graph cosine functions.
F.05. Graph transformations of sine and cosine functions.
F.06. Write equations of sine functions from graphs or properties.
F.07. Write equations of cosine functions from graphs or properties.
F.08. Analyze and compare the graphs of sine and cosine functions.
F.09. Explore real-world applications of trigonometric functions (oscillations, waves, etc.).

G. Matrices

G.01. Understand basic matrix vocabulary (dimensions, elements, etc.).
G.02. Apply matrix operation rules.
G.03. Add and subtract matrices.
G.04. Multiply a matrix by a scalar.
G.05. Form linear combinations of matrices.
G.06. Multiply two matrices.
G.07. Simplify matrix expressions.
G.08. Solve matrix equations.
G.09. Calculate the determinant of a matrix.
G.10. Determine if a matrix is invertible.
G.11. Find the inverse of a 2 x 2 matrix.
G.12. Find the inverse of a 3 x 3 matrix.
G.13. Verify if two matrices are inverses of each other.
G.14. Solve matrix equations using inverses.
G.15. Apply matrices to solve systems of linear equations.

H. Two-Dimensional Vectors

H.01. Find the magnitude of a vector.
H.02. Find the direction angle of a vector.
H.03. Find the component form of a vector.
H.04. Find the component form of a vector from its magnitude and direction angle.
H.05. Find a unit vector in the direction of a given vector.
H.06. Add and subtract vectors.
H.07. Multiply a vector by a scalar.
H.08. Find the magnitude or direction of a vector scalar multiple.
H.09. Find the magnitude and direction of a vector sum.
H.10. Form linear combinations of vectors.
H.11. Graph a resultant vector using the triangle method.
H.12. Graph a resultant vector using the parallelogram method.
H.13. Apply vectors to solve problems involving forces and motion.

I. Complex Numbers

I.01. Add and subtract complex numbers.
I.02. Multiply and divide complex numbers.
I.03. Simplify powers of i.
I.04. Find complex conjugates.
I.05. Calculate the absolute value (modulus) of a complex number.
I.06. Graph complex numbers on the Argand plane.
I.07. Perform addition and subtraction graphically on the Argand plane.
I.08. Understand the geometric interpretation of complex conjugates.
I.09. Determine the midpoint between two complex numbers on the Argand plane.
I.10. Calculate the distance between two complex numbers on the Argand plane.

J. Conic Sections

J.01. Identify and define the four conic sections: parabola, circle, ellipse, hyperbola.
J.02. Find the properties of parabolas (vertex, focus, directrix).
J.03. Write equations of parabolas in vertex form.
J.04. Graph parabolas.
J.05. Find the properties of circles (center, radius).
J.06. Write equations of circles in standard form.
J.07. Graph circles.
J.08. Find the properties of ellipses (center, vertices, foci, major axis, minor axis).
J.09. Calculate the eccentricity of an ellipse.
J.10. Write equations of ellipses in standard form.
J.11. Find the properties of hyperbolas (center, vertices, foci, transverse axis, conjugate axis, asymptotes).
J.12. Calculate the eccentricity of a hyperbola.
J.13. Write equations of hyperbolas in standard form.
J.14. Convert equations of conic sections from general to standard form.
J.15. Identify conic sections from their general equations.
J.16. Apply conic sections to model real-world phenomena (satellite orbits, lenses, etc.).

K. Introduction to Calculus: Limits and Continuity

K.09. Evaluate limits graphically and numerically.
K.01. Identify graphs of continuous functions.
K.02. Determine continuity using graphs.
K.03. Determine one-sided continuity using graphs.
K.04. Find and analyze points of discontinuity using graphs.
K.05. Determine continuity on an interval using graphs.
K.06. Determine the continuity of a piecewise function at a point.
K.07. Make a piecewise function continuous.
K.08. Apply the Intermediate Value Theorem.

L. Introduction to Calculus: Derivatives

M. Derivative Rules

N. Calculating Derivatives

N.01. Find derivatives of polynomials.
N.02. Find derivatives of rational functions.
N.03. Find derivatives of trigonometric functions.
N.04. Find derivatives of exponential functions.
N.05. Find derivatives of logarithmic functions.
N.06. Find derivatives of inverse trigonometric functions.
N.07. Find derivatives of radical functions.
N.08. Apply the product rule to find derivatives.
N.09. Apply the quotient rule to find derivatives.
N.10. Apply the chain rule to find derivatives.
N.11. Simplify expressions after applying derivative rules.

O. Advanced Derivative Techniques

P. Higher-Order Derivatives

Q. Probability

Q.01. Understand the basic concepts of probability.
Q.02. Calculate probabilities of events.
Q.03. Calculate combinations and permutations.
Q.04. Find probabilities using combinations and permutations.
Q.05. Find probabilities using two-way frequency tables.
Q.06. Identify independent events.
Q.07. Find conditional probabilities.
Q.08. Determine independence using conditional probability.
Q.09. Find conditional probabilities using two-way frequency tables.
Q.10. Find probabilities using the addition rule.
Q.11. Solve real-world problems involving probability.

R. Probability Distributions

R.01. Identify discrete and continuous random variables.
R.02. Write a discrete probability distribution.
R.03. Graph a discrete probability distribution.
R.04. Calculate the expected value of a random variable.
R.05. Calculate the variance of a random variable.
R.06. Calculate the standard deviation of a random variable.
R.07. Write the probability distribution for a game of chance.
R.08. Calculate expected values for a game of chance.
R.09. Determine the better bet in a gambling scenario.
R.10. Find probabilities using the binomial distribution.
R.11. Calculate the mean, variance, and standard deviation of binomial distributions.
R.12. Find probabilities using the normal distribution.
R.13. Find z-values corresponding to given probabilities.
R.14. Find values of normal variables.
R.15. Understand the concept of distributions of sample means.
R.16. Apply the Central Limit Theorem.
R.17. Use normal distributions to approximate binomial distributions.
R.18. Apply probability distributions to model and analyze real-world data.

S. Statistics

T. Trigonometric Identities

U. Polar Form of Complex Numbers

V. Three-Dimensional Vectors

V.01. Find the magnitude of a three-dimensional vector.
V.02. Find the component form of a three-dimensional vector.
V.03. Find a three-dimensional unit vector.
V.04. Add and subtract three-dimensional vectors.
V.05. Multiply a three-dimensional vector by a scalar.
V.06. Form linear combinations of three-dimensional vectors.
V.07. Calculate dot products and cross products of three-dimensional vectors.
V.08. Apply three-dimensional vectors to solve problems in geometry and physics.
V.09. Visualize three-dimensional vectors in space.

W. Systems of Inequalities and Linear Programming

X. Nonlinear Inequalities

Y. Advanced Polynomial Concepts

Y.01. Divide polynomials using long division.
Y.02. Divide polynomials using synthetic division.
Y.03. Apply the Rational Root Theorem to find possible rational roots of a polynomial.
Y.04. Apply the Complex Conjugate Theorem to find complex roots of a polynomial.
Y.05. Apply Descartes’ Rule of Signs to determine the number of positive and negative real roots of a polynomial.
Y.06. Understand and apply the Fundamental Theorem of Algebra.
Y.07. Apply the Binomial Theorem to expand binomial expressions.
Y.08. Use Pascal’s Triangle to determine binomial coefficients.

Course Content

A. Introduction to Functions

A.05. Determine the domain and range of a function.

A.06. Graph a linear function and identify its key features.

A.07. Write the equation of a linear function given different information (slope-intercept form, point-slope form, etc.).

A.08. Find the slope of a linear function from its equation or graph.

A.09. Explore the concept of piecewise functions and their real-world applications.

A.01. Identify whether a relation is a function.

A.02. Evaluate functions for given inputs.

A.03. Find values using function graphs.

A.04. Complete a table for a function graph.

B. Transformations of Functions

B.01. Identify and describe translations of functions.

B.02. Identify and describe reflections of functions.

B.03. Identify and describe dilations of functions.

B.04. Apply function transformation rules to sketch transformed graphs.

B.05. Synthesize multiple transformations of functions.

B.06. Generalize transformation rules for various function families.

C. Polynomial Basics

C.01. Find the roots of factored polynomials.

C.02. Evaluate polynomials using synthetic division.

C.03. Write a polynomial from its roots.

C.04. Match polynomials to their corresponding graphs.

C.05. Factor sums and differences of cubes.

C.06. Solve equations involving sums and differences of cubes.

C.07. Factor using a quadratic pattern.

C.08. Solve equations using a quadratic pattern.

C.09. Explore the connection between roots, factors, and the graph of a polynomial.

D. Sequences and Series

D.01. Find terms of a sequence given an explicit formula.

D.02. Find terms of a sequence given a recursive formula.

D.03. Identify a sequence as explicit or recursive.

D.04. Find a recursive formula for a given sequence.

D.05. Find recursive and explicit formulas for a sequence.

D.06. Convert between explicit and recursive formulas.

D.07. Evaluate expressions written in sigma notation.

D.08. Identify arithmetic and geometric series.

D.09. Calculate partial sums of arithmetic series.

D.10. Calculate partial sums of geometric series.

D.11. Find the sum of a finite arithmetic or geometric series.

D.12. Determine whether a geometric series is convergent or divergent.

D.13. Find the value of an infinite geometric series.

D.14. Express a repeating decimal as a fraction using geometric series.

D.15. Apply series to solve real-world problems involving growth and decay.

E. Introduction to Trigonometry

E.01. Convert between radians and degrees.

E.02. Calculate arc length using radians.

E.03. Identify the quadrant of an angle.

E.04. Find coterminal and reference angles.

E.05. Determine trigonometric ratios using right triangles (SOH CAH TOA).

E.06. Determine trigonometric ratios using the unit circle.

E.07. Determine trigonometric ratios using reference angles.

E.08. Solve trigonometric equations.

E.09. Calculate side lengths of right triangles using trigonometric ratios.

E.10. Calculate angle measures in right triangles using inverse trigonometric ratios.

E.11. Solve a right triangle given sufficient information.

E.12. Apply the Law of Sines to solve triangles.

E.13. Apply the Law of Cosines to solve triangles.

E.14. Solve a triangle using appropriate trigonometric laws.

E.15. Calculate the area of a triangle using the sine formula.

E.16. Calculate the area of a triangle using Heron’s formula.

E.17. Model and solve real-world problems involving triangles.

F. Trigonometric Functions

F.01. Identify properties (amplitude, period, phase shift, vertical shift) of sine functions.

F.02. Identify properties (amplitude, period, phase shift, vertical shift) of cosine functions.

F.03. Graph sine functions.

F.04. Graph cosine functions.

F.05. Graph transformations of sine and cosine functions.

F.06. Write equations of sine functions from graphs or properties.

F.07. Write equations of cosine functions from graphs or properties.

F.08. Analyze and compare the graphs of sine and cosine functions.

F.09. Explore real-world applications of trigonometric functions (oscillations, waves, etc.).

G. Matrices

G.01. Understand basic matrix vocabulary (dimensions, elements, etc.).

G.02. Apply matrix operation rules.

G.03. Add and subtract matrices.

G.04. Multiply a matrix by a scalar.

G.05. Form linear combinations of matrices.

G.06. Multiply two matrices.

G.07. Simplify matrix expressions.