JEE Main Algebra: Complex Numbers and Quadratic Equations

• JEE Main is an entrance examination in India for admission to various undergraduate engineering and architecture courses.
• Mathematics is a crucial section of the JEE Main exam.

Algebra

• Complex Numbers:

  • Imaginary number i = √-1
  • Complex number representation: z = a + ib, where a and b are real numbers.
  • Argand plane: graphical representation of complex numbers.
  • Modulus: |z| = √(a² + b²)
  • Argument: θ = tan⁻¹(b/a)
  • Polar form: z = r(cos θ + i sin θ)
  • Euler's form: z = re^(iθ)
  • De Moivre's Theorem: (cos θ + i sin θ)^n = cos nθ + i sin nθ
  • nth roots of unity and their properties

• Quadratic Equations

  • Standard form: ax² + bx + c = 0, where a ≠ 0.
  • Discriminant: D = b² - 4ac
  • Nature of roots:
    • D > 0: real and distinct
    • D = 0: real and equal
    • D < 0: complex conjugate
  • Sum of roots: α + β = -b/a
  • Product of roots: αβ = c/a
  • Formation of quadratic equation: x² - (α + β)x + αβ = 0
  • Maximum/minimum value of a quadratic expression.

• Sequences and Series

  • Arithmetic Progression (AP): a, a + d, a + 2d,...
    • nth term: aₙ = a + (n - 1)d
    • Sum of n terms: Sₙ = n/2 [2a + (n - 1)d] = n/2 [a + l], where l is the last term.
  • Geometric Progression (GP): a, ar, ar², ar³,...
    • nth term: aₙ = ar^(n-1)
    • Sum of n terms: Sₙ = a(1 - rⁿ) / (1 - r), r ≠ 1
    • Sum to infinity: S∞ = a / (1 - r), |r| < 1
  • Harmonic Progression (HP): Reciprocals of terms are in AP.
  • Arithmetic Mean (AM), Geometric Mean (GM), Harmonic Mean (HM) and their relationships.
  • AM ≥ GM ≥ HM

• Permutations and Combinations

  • Fundamental principle of counting.
  • Permutation: Arrangement of objects in a specific order. ⁿPᵣ = n! / (n - r)!
  • Combination: Selection of objects without regard to order. ⁿCᵣ = n! / [r! (n - r)!]
  • Properties of combinations: ⁿCᵣ = ⁿCₙ₋ᵣ, ⁿC₀ = ⁿCₙ = 1, ⁿCₓ = ⁿCᵧ ⇒ x = y or x + y = n
  • Circular permutations.

• Binomial Theorem

  • (x + y)ⁿ = ∑(k=0 to n) ⁿCₖ x^(n-k) yᵏ
  • General term: Tₖ₊₁ = ⁿCₖ x^(n-k) yᵏ
  • Middle term(s) in binomial expansion.
  • Properties of binomial coefficients.

• Matrices and Determinants

  • Matrix: Rectangular array of numbers.
  • Types of matrices: Row, column, square, diagonal, scalar, identity, zero.
  • Matrix operations: Addition, subtraction, multiplication.
  • Transpose of a matrix: Aᵀ
  • Determinant of a matrix: |A|
  • Properties of determinants.
  • Adjoint of a matrix: adj(A)
  • Inverse of a matrix: A⁻¹ = adj(A) / |A|
  • Solving system of linear equations using matrices: Cramer's rule, matrix inversion method.

• Probability

  • Sample space and events.
  • Probability of an event: P(E) = n(E) / n(S)
  • Conditional probability: P(A|B) = P(A ∩ B) / P(B)
  • Independent events: P(A ∩ B) = P(A)P(B)
  • Bayes' theorem.
  • Random variables and probability distributions.
  • Expected value and variance.
  • Bernoulli trials and binomial distribution.

Calculus

• Functions

  • Definition and types of functions: one-to-one, onto, into, many-to-one.
  • Domain and range of a function.
  • Composite functions.
  • Inverse of a function.

• Limits, Continuity, and Differentiability

  • Limits of functions.
  • Standard limits.
  • Continuity and differentiability of functions.
  • Relationship between continuity and differentiability.

• Differentiation

  • Derivatives of algebraic, trigonometric, exponential, and logarithmic functions.
  • Chain rule.
  • Product rule and quotient rule.
  • Second order derivatives.
  • L'Hôpital's rule.

• Application of Derivatives

  • Rate of change of quantities.
  • Increasing and decreasing functions.
  • Tangents and normals.
  • Maxima and minima.
  • Mean value theorems: Rolle's theorem, Lagrange's mean value theorem.

• Indefinite Integration

  • Integration as reverse process of differentiation.
  • Standard integrals.
  • Integration by substitution, by parts, and by partial fractions.

• Definite Integration

  • Properties of definite integrals.
  • Fundamental Theorem of Calculus.
  • Applications of definite integrals: area under curves, area between two curves.

• Differential Equations

  • Order and degree of a differential equation.
  • Formation of differential equations.
  • Solution of differential equations: variable separable, homogeneous, linear.

Trigonometry

• Trigonometric Ratios and Identities

  • Trigonometric ratios: sine, cosine, tangent, cotangent, secant, cosecant.
  • Trigonometric identities.
  • Trigonometric functions of compound angles.
  • Trigonometric functions of multiple and sub-multiple angles.
  • Conditional Trigonometric Identities

• Inverse Trigonometric Functions

  • Definition and properties of inverse trigonometric functions.
  • Principal values of inverse trigonometric functions.

• Trigonometric Equations

  • General solutions of trigonometric equations.

Coordinate Geometry

• Straight Lines

  • Different forms of equations of a line.
  • Slope of a line.
  • Angle between two lines.
  • Distance of a point from a line.
  • Family of lines.
  • Concurrency of lines.

• Circles

  • Equation of a circle in different forms.
  • Tangents and normals to a circle.
  • Condition of tangency.
  • Family of circles.

• Conic Sections

  • Parabola: Standard equation, focus, directrix, latus rectum.
  • Ellipse: Standard equation, foci, directrices, latus rectum, eccentricity.
  • Hyperbola: Standard equation, foci, directrices, latus rectum, eccentricity, asymptotes.
  • Rectangular hyperbola.

Vector Algebra

• Vectors and Scalars

  • Magnitude and direction of a vector.
  • Types of vectors: unit vector, zero vector, equal vectors.
  • Position vector of a point.
  • Components of a vector.
  • Addition and subtraction of vectors.
  • Scalar multiplication of vectors.

• Dot Product and Cross Product

  • Dot product (scalar product) of two vectors.
  • Cross product (vector product) of two vectors.
  • Applications of dot and cross products: area of triangle, volume of parallelepiped.

3D Geometry

• Direction Cosines and Direction Ratios

  • Relation between direction cosines.
  • Direction ratios of a line.

• Equations of a Line and a Plane

  • Equation of a line in space.
  • Equation of a plane in different forms.
  • Angle between a line and a plane.
  • Distance of a point from a plane.

Statistics

• Measures of Dispersion
  • Mean, median, mode.
  • Standard deviation and variance.
  • Coefficient of variation.