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O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for Mathematics [English] Class 12 ISC

CISCE ISC Class 12 Mathematics Solutions Guide

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CISCE ISC Class 12 Mathematics Textbook Solutions

Questions and answers for the ISC Class 12 Mathematics Textbook are on this page. O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics Digest CISCE will help students understand the concepts better.


O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics Chapterwise List | ISC Class 12 Mathematics Digest

The answers to the O.P. Malhotra, S.K. Gupta, Anubhuti Gangal books are the best study material for students. Listed below are the chapter-wise O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Mathematics ISC Class 12 Solutions CISCE.

Chapters

  • RELATIONS AND FUNCTIONS
  •  • 1: Relations
  •  • 2: Functions
  •  • 3: Inverse Trigonometric Functions
  • ALGEBRA
  •  • 4: Matrices
  •  • 5: Determinants
  •  • 6: Adjoint and Inverse of a Matrix
  • CALCULUS
  •  • 7: Continuity and Differentiability of Functions
  •  • 8: Differentiation
  •  • 9: Application of Derivatives
  •  • 10: Maxima and Minima
  •  • 11: Indefinite Integral -1: Standard Forms
  •  • 12: Indefinite Integral - 2: Methods of Integration
  •  • 13: Indefinite Integral – 3: Special Integrals
  •  • 14: Definite Integrals
  •  • 15: Application of Integrals: Area Under the Curve
  •  • 16: Differential Equations
  • VECTOR ALGEBRA
  •  • 17: Vectors
  •  • 18: Scalar or Dot Product
  •  • 19: Vector or Cross Product
  • THREE-DIMENSIONAL GEOMETRY
  •  • 20: Three-Dimensional Geometry - 1: Basic Concepts
  •  • 21: Three-Dimensional Geometry -2: Straight Line in Space
  •  • 22: Three-Dimensional Geometry - 3: The Plane
  • LINEAR PROGRAMMING
  •  • 23: Linear Programming
  • PROBABILITY
  •  • 24: Probability
  •  • 25: Bayes' Theorem and Probability Distribution

Appears in

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O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for Mathematics [English] Class 12 ISC - Shaalaa.com
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Chapters covered in O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for Mathematics [English] Class 12 ISC

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 1: Relations

Concepts covered in Relations are Composition of Functions, Concept of Function, Concept of Relations, Equivalence Class and Relation, Graph of Function, Invertible Functions, Types of Functions, Types of Relations.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 2: Functions

Concepts covered in Functions are Composition of Functions, Concept of Function, Concept of Relations, Equivalence Class and Relation, Graph of Function, Invertible Functions, Types of Functions, Types of Relations.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 3: Inverse Trigonometric Functions

Concepts covered in Inverse Trigonometric Functions are Basics of Inverse Trigonometric Functions, Domain, Range & Principal Value, Graphs of Inverse Trigonometric Functions, Properties of Inverse Trigonometric Functions > Addition–Subtraction Formula for Inverse Sine & Cosine, Properties of Inverse Trigonometric Functions > Addition & Subtraction Formula for Inverse Tangent, Properties of Inverse Trigonometric Functions > Complementary Property, Properties of Inverse Trigonometric Functions > Conversion Property, Properties of Inverse Trigonometric Functions > Double-angle Property, Properties of Inverse Trigonometric Functions > Negative Argument Property, Properties of Inverse Trigonometric Functions > Reciprocal Property, Properties of Inverse Trigonometric Functions > Self-adjusting Property, Properties of Inverse Trigonometric Functions > Triple-angle Property.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 4: Matrices

Concepts covered in Matrices are Concept of Matrices, Equality of Matrices, Invertible Matrices, Operations on Matrices> Addition and Subtraction of Matrices, Operations on Matrices> Matrix Multiplication, Operations on Matrices>Scalar Multiplication, Positive Integral Powers of Matrices, Symmetric and Skew Symmetric Matrices, Transpose of a Matrix, Types of Matrices.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 5: Determinants

Concepts covered in Determinants are Adjoint and Inverse of a Matrix, Application of Determinants to Coordinate Geometry, Concept of Determinant, Inverse of a Square Matrix, Minors and Co-factors, Properties of Determinants, Properties of Invertible Matrices, Relations between a Matrix and its Adjoint, Simplification of Determinants using Properties, Singular and Non-singular Matrix, Solution of System of Linear Equations, Theorems on Determinants.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 6: Adjoint and Inverse of a Matrix

Concepts covered in Adjoint and Inverse of a Matrix are Adjoint and Inverse of a Matrix, Application of Determinants to Coordinate Geometry, Concept of Determinant, Inverse of a Square Matrix, Minors and Co-factors, Properties of Determinants, Properties of Invertible Matrices, Relations between a Matrix and its Adjoint, Simplification of Determinants using Properties, Singular and Non-singular Matrix, Solution of System of Linear Equations, Theorems on Determinants.

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O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 7: Continuity and Differentiability of Functions

Concepts covered in Continuity and Differentiability of Functions are Algebra of Continuous Functions, Basic Rules of Differentiation, Concept of Differentiability, Continuous and Discontinuous Functions, Derivative of Inverse Trigonometric Function, Derivative of Trigonometric Function, Derivatives of Composite Functions, Derivatives of Functions in Parametric Forms, Differentiation "ab initio" of Inverse Circular Functions, Differentiation by Transformation/Substitution, Differentiation of Implicit Functions, Differentiation with Respect to Another Function, Exponential and Logarithmic Functions, Higher Order Derivatives, Logarithmic Differentiation, Properties of Continuous Function, Relationship Between Continuity and Differentiability, Types of Discontinuity.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 8: Differentiation

Concepts covered in Differentiation are Algebra of Continuous Functions, Basic Rules of Differentiation, Concept of Differentiability, Continuous and Discontinuous Functions, Derivative of Inverse Trigonometric Function, Derivative of Trigonometric Function, Derivatives of Composite Functions, Derivatives of Functions in Parametric Forms, Differentiation "ab initio" of Inverse Circular Functions, Differentiation by Transformation/Substitution, Differentiation of Implicit Functions, Differentiation with Respect to Another Function, Exponential and Logarithmic Functions, Higher Order Derivatives, Logarithmic Differentiation, Properties of Continuous Function, Relationship Between Continuity and Differentiability, Types of Discontinuity.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 9: Application of Derivatives

Concepts covered in Application of Derivatives are Application/Practical Problems on Maxima and Minima, Approximations, Increasing and Decreasing Functions, Maxima and Minima, Rate Measure, Sign of the Derivative, Tangents and Normals.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 10: Maxima and Minima

Concepts covered in Maxima and Minima are Application/Practical Problems on Maxima and Minima, Approximations, Increasing and Decreasing Functions, Maxima and Minima, Rate Measure, Sign of the Derivative, Tangents and Normals.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 11: Indefinite Integral -1: Standard Forms

Concepts covered in Indefinite Integral -1: Standard Forms are Different Types of Integrals, Evaluation of Definite Integrals, Evaluation of Quadratic Equation, Fundamental Theorem of Calculus, Integrals ∫√tanx dx and ∫√cot x dx, Integration, Integration by Substitution(Change of Variable), Integration of Irrational Functions > Type 9: sin²x, cos²x Denominator Forms, Integration of Irrational Function> Type 1: Standard Radical Forms, Integration of Irrational Function> Type 2: 1 / √(Quadratic), Integration of Irrational Function> Type 3: Linear / √(Quadratic), Integration of Irrational Function> Type 4: √(a² ± x²) and ∫√(x² − a²) dx Forms, Integration of Irrational Function> Type 5: √(Quadratic), Integration of Irrational Function> Type 6: Linear × √(Quadratic), Integration of Irrational Function> Type 7: 1 / Linear Trigonometric Forms, Integration of Irrational Function> Type 8: Linear Trig / Linear Trig, Integration of the form where K is Constant, Integration Using Standard Algebraic Forms, Methods of Integration> Integration by Parts, Methods of Integration> Integration by Substitution, Methods of Integration> Integration Using Partial Fraction, Properties of Definite Integral, Properties of Indefinite Integral, Some Special Integrals, Standard Forms of Integrals.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 12: Indefinite Integral - 2: Methods of Integration

Concepts covered in Indefinite Integral - 2: Methods of Integration are Different Types of Integrals, Evaluation of Definite Integrals, Evaluation of Quadratic Equation, Fundamental Theorem of Calculus, Integrals ∫√tanx dx and ∫√cot x dx, Integration, Integration by Substitution(Change of Variable), Integration of Irrational Functions > Type 9: sin²x, cos²x Denominator Forms, Integration of Irrational Function> Type 1: Standard Radical Forms, Integration of Irrational Function> Type 2: 1 / √(Quadratic), Integration of Irrational Function> Type 3: Linear / √(Quadratic), Integration of Irrational Function> Type 4: √(a² ± x²) and ∫√(x² − a²) dx Forms, Integration of Irrational Function> Type 5: √(Quadratic), Integration of Irrational Function> Type 6: Linear × √(Quadratic), Integration of Irrational Function> Type 7: 1 / Linear Trigonometric Forms, Integration of Irrational Function> Type 8: Linear Trig / Linear Trig, Integration of the form where K is Constant, Integration Using Standard Algebraic Forms, Methods of Integration> Integration by Parts, Methods of Integration> Integration by Substitution, Methods of Integration> Integration Using Partial Fraction, Properties of Definite Integral, Properties of Indefinite Integral, Some Special Integrals, Standard Forms of Integrals.

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O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 13: Indefinite Integral – 3: Special Integrals

Concepts covered in Indefinite Integral – 3: Special Integrals are Different Types of Integrals, Evaluation of Definite Integrals, Evaluation of Quadratic Equation, Fundamental Theorem of Calculus, Integrals ∫√tanx dx and ∫√cot x dx, Integration, Integration by Substitution(Change of Variable), Integration of Irrational Functions > Type 9: sin²x, cos²x Denominator Forms, Integration of Irrational Function> Type 1: Standard Radical Forms, Integration of Irrational Function> Type 2: 1 / √(Quadratic), Integration of Irrational Function> Type 3: Linear / √(Quadratic), Integration of Irrational Function> Type 4: √(a² ± x²) and ∫√(x² − a²) dx Forms, Integration of Irrational Function> Type 5: √(Quadratic), Integration of Irrational Function> Type 6: Linear × √(Quadratic), Integration of Irrational Function> Type 7: 1 / Linear Trigonometric Forms, Integration of Irrational Function> Type 8: Linear Trig / Linear Trig, Integration of the form where K is Constant, Integration Using Standard Algebraic Forms, Methods of Integration> Integration by Parts, Methods of Integration> Integration by Substitution, Methods of Integration> Integration Using Partial Fraction, Properties of Definite Integral, Properties of Indefinite Integral, Some Special Integrals, Standard Forms of Integrals.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 14: Definite Integrals

Concepts covered in Definite Integrals are Different Types of Integrals, Evaluation of Definite Integrals, Evaluation of Quadratic Equation, Fundamental Theorem of Calculus, Integrals ∫√tanx dx and ∫√cot x dx, Integration, Integration by Substitution(Change of Variable), Integration of Irrational Functions > Type 9: sin²x, cos²x Denominator Forms, Integration of Irrational Function> Type 1: Standard Radical Forms, Integration of Irrational Function> Type 2: 1 / √(Quadratic), Integration of Irrational Function> Type 3: Linear / √(Quadratic), Integration of Irrational Function> Type 4: √(a² ± x²) and ∫√(x² − a²) dx Forms, Integration of Irrational Function> Type 5: √(Quadratic), Integration of Irrational Function> Type 6: Linear × √(Quadratic), Integration of Irrational Function> Type 7: 1 / Linear Trigonometric Forms, Integration of Irrational Function> Type 8: Linear Trig / Linear Trig, Integration of the form where K is Constant, Integration Using Standard Algebraic Forms, Methods of Integration> Integration by Parts, Methods of Integration> Integration by Substitution, Methods of Integration> Integration Using Partial Fraction, Properties of Definite Integral, Properties of Indefinite Integral, Some Special Integrals, Standard Forms of Integrals.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 15: Application of Integrals: Area Under the Curve

Concepts covered in Application of Integrals: Area Under the Curve are Area Bounded by Two Curves, Area Under Simple Curves, Geometrical Interpretation of a Definite Integral., Sketching of Curves.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 16: Differential Equations

Concepts covered in Differential Equations are Applications of Differential Equation, Basic Concepts of Differential Equations, Formation of Differential Equations, Forms of Solving Differential Equation>Reducible to variables separable, Forms of Solving Differential Equations> Homogeneous Differential Equations, Forms of Solving Differential Equations>Linear Differential Equations, Forms of Solving Differential Equations> Variable Separable, Order and Degree of a Differential Equation, Solution of a Differential Equation.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 17: Vectors

Concepts covered in Vectors are Algebra of Vectors > Addition & Subtraction of Two Vectors, Algebra of Vectors > Scalar Multiplication, Basic Concepts of Vector Algebra, Collinearity and Coplanarity of Vectors, Components of Vector in Algebra, Direction Ratios, Direction Cosine & Direction Angles in Vector, Parallel Vectors, Product of Two Vectors > Scalar (Dot) Product, Product of Two Vectors > Vector (Cross) Product, Section Formula in Vector Algebra, Types of Vectors in Algebra, Vectors in Coordinate Geometry.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 18: Scalar or Dot Product

Concepts covered in Scalar or Dot Product are Algebra of Vectors > Addition & Subtraction of Two Vectors, Algebra of Vectors > Scalar Multiplication, Basic Concepts of Vector Algebra, Collinearity and Coplanarity of Vectors, Components of Vector in Algebra, Direction Ratios, Direction Cosine & Direction Angles in Vector, Parallel Vectors, Product of Two Vectors > Scalar (Dot) Product, Product of Two Vectors > Vector (Cross) Product, Section Formula in Vector Algebra, Types of Vectors in Algebra, Vectors in Coordinate Geometry.

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O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 19: Vector or Cross Product

Concepts covered in Vector or Cross Product are Algebra of Vectors > Addition & Subtraction of Two Vectors, Algebra of Vectors > Scalar Multiplication, Basic Concepts of Vector Algebra, Collinearity and Coplanarity of Vectors, Components of Vector in Algebra, Direction Ratios, Direction Cosine & Direction Angles in Vector, Parallel Vectors, Product of Two Vectors > Scalar (Dot) Product, Product of Two Vectors > Vector (Cross) Product, Section Formula in Vector Algebra, Types of Vectors in Algebra, Vectors in Coordinate Geometry.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 20: Three-Dimensional Geometry - 1: Basic Concepts

Concepts covered in Three-Dimensional Geometry - 1: Basic Concepts are Angle Between Two Lines, Angle Between Two Planes, Coplanar and Intersection of Two Lines, Different Forms of Equation of a Plane, Direction Cosines and Direction Ratios of a Line, Distance of a Point from a Plane, Foundation of Coordinate Geometry, Plane with Normal Form, Shortest Distance Between Two Lines, Straight Line In Space.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 21: Three-Dimensional Geometry -2: Straight Line in Space

Concepts covered in Three-Dimensional Geometry -2: Straight Line in Space are Angle Between Two Lines, Angle Between Two Planes, Coplanar and Intersection of Two Lines, Different Forms of Equation of a Plane, Direction Cosines and Direction Ratios of a Line, Distance of a Point from a Plane, Foundation of Coordinate Geometry, Plane with Normal Form, Shortest Distance Between Two Lines, Straight Line In Space.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 22: Three-Dimensional Geometry - 3: The Plane

Concepts covered in Three-Dimensional Geometry - 3: The Plane are Angle Between Two Lines, Angle Between Two Planes, Coplanar and Intersection of Two Lines, Different Forms of Equation of a Plane, Direction Cosines and Direction Ratios of a Line, Distance of a Point from a Plane, Foundation of Coordinate Geometry, Plane with Normal Form, Shortest Distance Between Two Lines, Straight Line In Space.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 23: Linear Programming

Concepts covered in Linear Programming are Concepts of Linear Programming, Graphical Solution of Linear Inequations, Linear Programming Problem and Its Mathematical Formulation, Methods to Solve LPP (Graphical / Corner Point Method), System of Inequations.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 24: Probability

Concepts covered in Probability are Bayes’ Theorem, Concept of Probability, Conditional Probability, Finding Probability of Dependent Events, Laws of Probability, Mean (Exception) of Probability Distribution, Multiplication Theorem of Probability, Random Variables, The Law of Total Probability.

O.P. Malhotra, S.K. Gupta, Anubhuti Gangal Solutions for ISC Class 12 Mathematics (12th) 25: Bayes' Theorem and Probability Distribution

Concepts covered in Bayes' Theorem and Probability Distribution are Bayes’ Theorem, Concept of Probability, Conditional Probability, Finding Probability of Dependent Events, Laws of Probability, Mean (Exception) of Probability Distribution, Multiplication Theorem of Probability, Random Variables, The Law of Total Probability.

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