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Mat.402 • Class 12science Stream16 Lessons

Credit Hours: 5

Mathematics

गणित (कक्षा १२)

Algebra (Permutation/Combination, Binomial Theorem, Complex Numbers, Sequences, Linear Systems), Trigonometry (Properties of Triangles), Analytic Geometry (Conic Sections), Vectors (Dot & Cross Products), Statistics & Probability, Advanced Calculus (Integration Techniques, Differential Equations), and Computational Methods (Gauss Elimination, Simplex LPP) or Mechanics.

Solved Q&A (3)Assignments (1)Important Questions (1)

Prescribed Chapters

Showing 16 Lessons

Unit 1

Permutations and Combinations

1 Lesson

1Lesson 1Syllabus Mapped 📋

Permutations and Combinations

Basic counting principles, permutations of distinct and repeated objects, circular permutations, combinations, and conditional selections.

Fundamental Principles of Counting (Addition and Multiplication)Permutations of n Distinct Objects Taken r at a Time: P(n, r)Permutations with Repetition: n! / (p! q! r!)Circular Permutations: (n - 1)! and (n - 1)! / 2

~60 mins study5 Marks in Grid

Unit 2

Binomial Theorem & Exponential/Logarithmic Series

1 Lesson

2Lesson 2Syllabus Mapped 📋

Binomial Theorem & Exponential/Logarithmic Series

Binomial theorem for positive integral index, general term, middle term, properties of binomial coefficients, binomial theorem for any rational index, expansions of e^x, a^x, and ln(1 + x).

Binomial Theorem Statement and Proof by Mathematical InductionGeneral Term: T_{r+1} = C(n, r) a^{n-r} b^rMiddle Term(s) Determination in Binomial ExpansionsSum of Binomial Coefficients: C₀ + C₁ + ... + Cₙ = 2ⁿ

~65 mins study6 Marks in Grid

Unit 3

Complex Numbers & De Moivre's Theorem

1 Lesson

3Lesson 3Notes Ready ✓1 Solved Q&A

Complex Numbers & De Moivre's Theorem

Polar and exponential forms, De Moivre theorem for rational exponents, roots of complex numbers, cube roots of unity (1, ω, ω²) and their algebraic properties.

Polar (Trigonometric) Form: z = r(cos θ + i sin θ)Euler's Formula: e^{iθ} = cos θ + i sin θDe Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)Roots of Complex Numbers: z^{1/n}

~60 mins study6 Marks in Grid

Unit 4

Sequence and Series & Mathematical Induction

1 Lesson

4Lesson 4Syllabus Mapped 📋

Sequence and Series & Mathematical Induction

Sum of natural numbers Σn, squares Σn², and cubes Σn³, method of differences, arithmetico-geometric series, principle of mathematical induction.

Sum of First n Natural Numbers: Σn = n(n+1)/2Sum of Squares: Σn² = n(n+1)(2n+1)/6Sum of Cubes: Σn³ = [n(n+1)/2]²Arithmetico-Geometric Series (A.G.P.) and Sum to Infinity

~55 mins study5 Marks in Grid

Unit 5

System of Linear Equations

1 Lesson

5Lesson 5Syllabus Mapped 📋

System of Linear Equations

Cramer's rule for 3 variables, row-equivalent matrix method (Gaussian elimination to echelon form), and inverse matrix method (A X = B).

Cramer's Rule: Determinant Ratios x = D_x/D, y = D_y/D, z = D_z/DConsistent, Inconsistent, and Dependent SystemsRow-Equivalent Operations and Row Echelon FormGaussian Elimination for 3x3 Linear Systems

~60 mins study6 Marks in Grid

Unit 6

Properties of a Triangle & Solution of Triangles

1 Lesson

6Lesson 6Syllabus Mapped 📋

Properties of a Triangle & Solution of Triangles

Sine rule, cosine rule, tangent rule (Napier analogies), projection formulas, half-angle formulas for sin(A/2), cos(A/2), tan(A/2), area of triangle, and solution of triangles.

The Sine Law: a / sin A = b / sin B = c / sin C = 2RThe Cosine Law: cos A = (b² + c² - a²) / 2bcProjection Laws: a = b cos C + c cos BNapier's Analogies (Tangent Laws)

~60 mins study5 Marks in Grid

Unit 7

Conic Sections (Parabola, Ellipse, Hyperbola)

1 Lesson

7Lesson 7Syllabus Mapped 📋

Conic Sections (Parabola, Ellipse, Hyperbola)

Condition of tangency of line y = mx + c to a circle, standard equations, focus, directrix, eccentricity, vertex, latus rectum, tangent, and normal of Parabola y² = 4ax, Ellipse x²/a² + y²/b² = 1, and Hyperbola x²/a² - y²/b² = 1.

Circle: Tangency Condition c² = a²(1 + m²) and Tangent EquationParabola: Standard Equation y² = 4ax, Focus (a, 0), Directrix x = -aParabola Tangency Condition: c = a / m and Tangent y = mx + a/mEllipse: Standard Form x²/a² + y²/b² = 1, Eccentricity e = √(1 - b²/a²)

~70 mins study7 Marks in Grid

Unit 8

Product of Vectors (Scalar & Vector Products)

1 Lesson

8Lesson 8Syllabus Mapped 📋

Product of Vectors (Scalar & Vector Products)

Scalar dot product and vector cross product, angle between vectors, geometric interpretations, area of triangle and parallelogram, vector proofs of sine and cosine laws.

Scalar (Dot) Product: a · b = |a| |b| cos θVector (Cross) Product: a × b = |a| |b| sin θ n̂Properties of Dot and Cross Products in Component FormArea of Parallelogram: |a × b| and Triangle: 1/2 |a × b|

~55 mins study5 Marks in Grid

Unit 9

Correlation and Regression

1 Lesson

9Lesson 9Syllabus Mapped 📋

Correlation and Regression

Karl Pearson correlation coefficient r, Spearman rank correlation coefficient R, lines of regression of y on x and x on y by least squares method, regression coefficients b_yx and b_xy.

Karl Pearson's Coefficient of Correlation: r = Cov(x, y) / (σ_x σ_y)Properties of Correlation Coefficient: -1 ≤ r ≤ +1Spearman's Rank Correlation: R = 1 - 6Σd² / [n(n² - 1)]Lines of Regression: (y - ȳ) = b_yx (x - x̄) and (x - x̄) = b_xy (y - ȳ)

~60 mins study6 Marks in Grid

Unit 10

Probability & Conditional Probability

1 Lesson

10Lesson 10Syllabus Mapped 📋

Probability & Conditional Probability

Conditional probability P(A|B) = P(A ∩ B) / P(B), multiplication theorem, independent events, total probability theorem, and combinatorial probability problems.

Axiomatic Definition of ProbabilityConditional Probability: P(A | B) = P(A ∩ B) / P(B)Multiplication Theorem of ProbabilityIndependent Events: P(A ∩ B) = P(A) · P(B)

~55 mins study5 Marks in Grid

Unit 11

Derivatives, Hyperbolic Functions & Applications

1 Lesson

11Lesson 11Syllabus Mapped 📋

Derivatives, Hyperbolic Functions & Applications

Hyperbolic and inverse hyperbolic functions, L'Hospital rule for indeterminate forms 0/0 and ∞/∞, differentials, equations of tangents and normals to curves, derivative as rate measure, maxima and minima.

Hyperbolic Functions: sinh x, cosh x, tanh x and DerivativesInverse Hyperbolic Functions and DerivativesL'Hospital's Rule for Indeterminate Forms 0/0 and ∞/∞Differentials and Linear Approximations

~70 mins study7 Marks in Grid

Unit 12

Anti-Derivatives & Standard Integrals

1 Lesson

12Lesson 12Syllabus Mapped 📋1 Important Qs ⭐

Anti-Derivatives & Standard Integrals

Standard algebraic and trigonometric integrals, integration of rational functions by partial fractions, trigonometric integrals, integrals reducible to standard forms.

Standard Integrals: ∫ dx / (x² ± a²), ∫ dx / √(a² - x²), ∫ dx / √(x² ± a²)Quadratic Integrals: ∫ dx / (a x² + b x + c)Linear over Quadratic Integrals: ∫ (px + q) dx / (a x² + b x + c)Integration by Partial Fractions (Non-repeated and repeated linear/quadratic factors)

~75 mins study8 Marks in Grid

Unit 13

Differential Equations

1 Lesson

13Lesson 13Syllabus Mapped 📋1 Solved Q&A

Differential Equations

Order and degree, first-order first-degree equations: variable separable, homogeneous equations y = vx, and first-order linear differential equations dy/dx + Py = Q using integrating factor e^(∫P dx).

Order and Degree of Differential EquationsVariable Separable Method: f(x) dx + g(y) dy = 0Homogeneous Differential Equations: Substitution y = v xFirst-Order Linear Differential Equations: dy/dx + P(x) y = Q(x)

~65 mins study6 Marks in Grid

Unit 14

Computational Methods (Gauss Elimination & Simplex Method)

1 Lesson

14Lesson 14Syllabus Mapped 📋1 Solved Q&A

Computational Methods (Gauss Elimination & Simplex Method)

Solving systems of linear equations by Gauss Elimination with partial pivoting, Gauss-Seidel iterative method, and Linear Programming Problem (LPP) by Simplex Method with two decision variables.

Gauss Elimination with Augmented Matrix [A | B]Partial Pivoting to Avoid Zero/Small PivotsBack-Substitution to Find UnknownsGauss-Seidel Iterative Method and Convergence Condition (Diagonally Dominant)

~70 mins study7 Marks in Grid

Unit 15

Mechanics: Statics (Forces, Lami's Theorem)

1 Lesson

15Lesson 15Syllabus Mapped 📋

Mechanics: Statics (Forces, Lami's Theorem)

Composition and resolution of forces, triangle and polygon laws of forces, Lami's theorem P/sin α = Q/sin β = R/sin γ, moments and couples, conditions of equilibrium of coplanar forces.

Parallelogram Law of Forces and Resultant: R = √(P² + Q² + 2PQ cos θ)Triangle Law and Polygon Law of ForcesLami's Theorem for Three Coplanar Concurrent Forces in EquilibriumMoment of a Force and Varignon Theorem

~55 mins study5 Marks in Grid

Unit 16

Mechanics: Dynamics (Newton's Laws, Projectiles)

1 Lesson

16Lesson 16Syllabus Mapped 📋

Mechanics: Dynamics (Newton's Laws, Projectiles)

Newton's laws of motion, conservation of linear momentum, work, energy and power, projectile motion on horizontal and inclined planes, range, height, time of flight.

Newton's Laws of Motion in Component FormConservation of Linear Momentum and ImpulseWork-Energy Theorem for Variable ForcesProjectile Motion on Horizontal Plane: Trajectory, Height, Time, Range

~60 mins study6 Marks in Grid