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      Class 12 MATHS – JEE

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      • Class 12
      • Class 12 MATHS – JEE
      CoursesClass 12MathsClass 12 MATHS – JEE
      • 1.Relations and Functions
        11
        • Lecture1.1
          Revision – Functions and its Types 38 min
        • Lecture1.2
          Revision – Functions Types 17 min
        • Lecture1.3
          Revision – Sum Related to Relations 04 min
        • Lecture1.4
          Revision – Sums Related to Relations, Domain and Range 22 min
        • Lecture1.5
          Cartesian Product of Sets, Relation, Domain, Range, Inverse of Relation, Types of Relations 34 min
        • Lecture1.6
          Functions, Intervals 39 min
        • Lecture1.7
          Domain 01 min
        • Lecture1.8
          Problem Based on finding Domain and Range 39 min
        • Lecture1.9
          Types of Real function 31 min
        • Lecture1.10
          Odd & Even Function, Composition of Function 32 min
        • Lecture1.11
          Chapter Notes – Relations and Functions
      • 2.Inverse Trigonometric Functions
        16
        • Lecture2.1
          Revision – Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Revision – Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          Revision – Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.4
          Revision – Sums Related to Trigonometrical Identities 18 min
        • Lecture2.5
          Revision – Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.6
          Revision – Trigonometry Equations 44 min
        • Lecture2.7
          Revision – Sum Based on Trigonometry Equations 08 min
        • Lecture2.8
          Introduction to Inverse Trigonometry Function, Range, Domain, Question based on Principal Value 37 min
        • Lecture2.9
          Property -1 of Inverse trigo function 28 min
        • Lecture2.10
          Property -2 to 4 of Inverse trigo function 48 min
        • Lecture2.11
          Questions based on properties of Inverse trigo function 19 min
        • Lecture2.12
          Question based on useful substitution 27 min
        • Lecture2.13
          Numerical problems 19 min
        • Lecture2.14
          Numerical problems 24 min
        • Lecture2.15
          Numerical problems , introduction to Differentiation 44 min
        • Lecture2.16
          Chapter Notes – Inverse Trigonometric Functions
      • 3.Matrices
        9
        • Lecture3.1
          What is matrix 26 min
        • Lecture3.2
          Types of matrix 28 min
        • Lecture3.3
          Operations of matrices 28 min
        • Lecture3.4
          Multiplication of matrices 29 min
        • Lecture3.5
          Properties of a matrices 44 min
        • Lecture3.6
          Numerical problems 19 min
        • Lecture3.7
          Solution of simultaneous linear equation 28 min
        • Lecture3.8
          Solution of simultaneous / Homogenous linear equation 24 min
        • Lecture3.9
          Chapter Notes – Matrices
      • 4.Determinants
        6
        • Lecture4.1
          Introduction of Determinants 23 min
        • Lecture4.2
          Properties of Determinants 29 min
        • Lecture4.3
          Numerical problems 15 min
        • Lecture4.4
          Numerical problems 16 min
        • Lecture4.5
          Applications of Determinants 17 min
        • Lecture4.6
          Chapter Notes – Determinants
      • 5.Continuity
        7
        • Lecture5.1
          Introduction to continuity 28 min
        • Lecture5.2
          Numerical problems 17 min
        • Lecture5.3
          Numerical problems 22 min
        • Lecture5.4
          Basics of continuity 26 min
        • Lecture5.5
          Numerical problems 17 min
        • Lecture5.6
          Numerical problems 11 min
        • Lecture5.7
          Chapter Notes – Continuity and Differentiability
      • 6.Differentiation
        14
        • Lecture6.1
          Introduction to Differentiation 27 min
        • Lecture6.2
          Important formula’s 29 min
        • Lecture6.3
          Numerical problems 29 min
        • Lecture6.4
          Numerical problems 31 min
        • Lecture6.5
          Differentiation by using trigonometric substitution 21 min
        • Lecture6.6
          Differentiation of implicit function 21 min
        • Lecture6.7
          Differentiation of logarthmetic function 31 min
        • Lecture6.8
          Differentiation of log function 25 min
        • Lecture6.9
          Infinite series & parametric function 26 min
        • Lecture6.10
          Infinite series & parametric function 27 min
        • Lecture6.11
          Higher order derivatives 27 min
        • Lecture6.12
          Differentiation of function of a function 16 min
        • Lecture6.13
          Numerical problems 27 min
        • Lecture6.14
          Numerical problems 04 min
      • 7.Mean Value Theorem
        4
        • Lecture7.1
          Lagrange theorem 24 min
        • Lecture7.2
          Rolle’s theorem 20 min
        • Lecture7.3
          Lagrange theorem 24 min
        • Lecture7.4
          Rolle’s theorem 20 min
      • 8.Applications of Derivatives
        6
        • Lecture8.1
          Rate of change of Quantities 30 min
        • Lecture8.2
          Rate of change of Quantities 18 min
        • Lecture8.3
          Rate of change of Quantities 18 min
        • Lecture8.4
          Approximation 10 min
        • Lecture8.5
          Approximation 05 min
        • Lecture8.6
          Chapter Notes – Applications of Derivatives
      • 9.Increasing and Decreasing Function
        5
        • Lecture9.1
          Introduction 36 min
        • Lecture9.2
          Numerical Problem 26 min
        • Lecture9.3
          Numerical Problem 22 min
        • Lecture9.4
          Numerical Problem 22 min
        • Lecture9.5
          Numerical Problem 21 min
      • 10.Tangents and Normal
        3
        • Lecture10.1
          Introduction 34 min
        • Lecture10.2
          Numerical Problems 32 min
        • Lecture10.3
          Angle of intersection of two curves 27 min
      • 11.Maxima and Minima
        10
        • Lecture11.1
          Introduction 28 min
        • Lecture11.2
          Local maxima & Local Minima 27 min
        • Lecture11.3
          Numerical Problems 37 min
        • Lecture11.4
          Maximum & minimum value in closed interval 19 min
        • Lecture11.5
          Application of Maxima & Minima 10 min
        • Lecture11.6
          Application of Maxima & Minima 14 min
        • Lecture11.7
          Numerical Problems 17 min
        • Lecture11.8
          Numerical Problems 18 min
        • Lecture11.9
          Numerical Problems 15 min
        • Lecture11.10
          Numerical Problems 14 min
      • 12.Integrations
        19
        • Lecture12.1
          Introduction to Indefinite Integration 37 min
        • Lecture12.2
          Integration by substitution 25 min
        • Lecture12.3
          Numerical problems on Substitution 39 min
        • Lecture12.4
          Numerical problems on Substitution 04 min
        • Lecture12.5
          Integration various types of particular function (Identities) 31 min
        • Lecture12.6
          Integration by parts-1 18 min
        • Lecture12.7
          Integration by parts-2 10 min
        • Lecture12.8
          Integration by parts-2 16 min
        • Lecture12.9
          Integration by parts-2 08 min
        • Lecture12.10
          ILATE Rule 12 min
        • Lecture12.11
          Integration of some special function 07 min
        • Lecture12.12
          Integration of some special function 06 min
        • Lecture12.13
          Integration by substitution using trigonometric 14 min
        • Lecture12.14
          Evaluation of some specific Integration 12 min
        • Lecture12.15
          Evaluation of some specific Integration 29 min
        • Lecture12.16
          Integration by partial fraction 27 min
        • Lecture12.17
          Integration of some special function 11 min
        • Lecture12.18
          Numerical Problems based on partial fraction 20 min
        • Lecture12.19
          Chapter Notes – Integrals
      • 13.Definite Integrals
        11
        • Lecture13.1
          Introduction 24 min
        • Lecture13.2
          Properties of Definite Integration 19 min
        • Lecture13.3
          Numerical problem based on properties 22 min
        • Lecture13.4
          Area under the curve 16 min
        • Lecture13.5
          Area under the curve (Ellipse) 20 min
        • Lecture13.6
          Area under the curve (Parabola) 10 min
        • Lecture13.7
          Area under the curve (Parabola & Circle) 40 min
        • Lecture13.8
          Area bounded by lines 10 min
        • Lecture13.9
          Numerical problems 25 min
        • Lecture13.10
          Area under the curve (Circle ) 02 min
        • Lecture13.11
          Chapter Notes – Application of Integrals
      • 14.Differential Equations
        6
        • Lecture14.1
          Introduction to chapter 38 min
        • Lecture14.2
          Solution of D.E. – Variable separation methods 14 min
        • Lecture14.3
          Solution of D.E. – Variable separation methods 27 min
        • Lecture14.4
          Solution of D.E. – Second order 21 min
        • Lecture14.5
          Homogeneous D.E. 31 min
        • Lecture14.6
          Chapter Notes – Differential Equations
      • 15.Vectors
        12
        • Lecture15.1
          Introduction , Basic concepts , types of vector 34 min
        • Lecture15.2
          Position vector, distance between two points, section formula 44 min
        • Lecture15.3
          Numerical problem 02 min
        • Lecture15.4
          collinearity of points and coplanarity of vector 34 min
        • Lecture15.5
          Direction cosine 18 min
        • Lecture15.6
          Projection , Dot product, Cauchy- Schwarz inequality 24 min
        • Lecture15.7
          Numerical problem (dot product) 20 min
        • Lecture15.8
          Vector (Cross) product , Lagrange’s Identity 15 min
        • Lecture15.9
          Numerical problem (cross product) 38 min
        • Lecture15.10
          Numerical problem (cross product) 06 min
        • Lecture15.11
          Numerical problem (cross product) 22 min
        • Lecture15.12
          Chapter Notes – Vectors
      • 16.Three Dimensional Geometry
        7
        • Lecture16.1
          Introduction to 3D, axis in 3D, plane in 3D, Distance between two points 32 min
        • Lecture16.2
          Numerical problems , section formula , centroid of a triangle 35 min
        • Lecture16.3
          projection , angle between two lines 40 min
        • Lecture16.4
          Numerical Problem based on Direction ratio & cosine 02 min
        • Lecture16.5
          locus of any point 15 min
        • Lecture16.6
          Numerical Problem based on locus 16 min
        • Lecture16.7
          Chapter Notes – Three Dimensional Geometry
      • 17.Direction Cosine
        2
        • Lecture17.1
          Introduction 34 min
        • Lecture17.2
          Angle Between two vectors 25 min
      • 18.Plane
        3
        • Lecture18.1
          Introduction to plane , general equation of a plane , normal form 31 min
        • Lecture18.2
          Angle between two planes 30 min
        • Lecture18.3
          Distance of a point from a plane 29 min
      • 19.Straight Lines
        22
        • Lecture19.1
          Revision – Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture19.2
          Revision – Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture19.3
          Revision – Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture19.4
          Revision – Sums Related Finding the Equation of Line 27 min
        • Lecture19.5
          Revision – Sums based on Previous Concepts of Straight line 32 min
        • Lecture19.6
          Revision – Parametric Form of a Straight Line 16 min
        • Lecture19.7
          Revision – Sums Related to Parametric Form of a Straight Line 17 min
        • Lecture19.8
          Revision – Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture19.9
          Revision – Different condition for Angle b/w two lines 04 min
        • Lecture19.10
          Revision – Sums Based on Angle b/w Two Lines 36 min
        • Lecture19.11
          Revision – Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture19.12
          Revision – Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture19.13
          Revision – Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture19.14
          Revision – Finding the Distance of a point from the line 35 min
        • Lecture19.15
          Revision – Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture19.16
          Introduction to straight line , symmetric form , Angle between the lines 27 min
        • Lecture19.17
          Numerical Problem 18 min
        • Lecture19.18
          Angle between two lines 32 min
        • Lecture19.19
          Unsymmetric form of Line 26 min
        • Lecture19.20
          Numerical problem , perpendicular distance of a point from a line 22 min
        • Lecture19.21
          Numerical Problem 21 min
        • Lecture19.22
          Numerical problem , Condition for a line lie on a plane 26 min
      • 20.Straight Lines (Vector)
        4
        • Lecture20.1
          Vector and Cartesian equation of a straight line 27 min
        • Lecture20.2
          Angle between two straight line 25 min
        • Lecture20.3
          Numerical problems 37 min
        • Lecture20.4
          Shortest Distance between two lines 22 min
      • 21.Linear Programming
        5
        • Lecture21.1
          Introduction to L.P. 30 min
        • Lecture21.2
          Numerical Problems 43 min
        • Lecture21.3
          Numerical Problems 23 min
        • Lecture21.4
          Numerical Problems 17 min
        • Lecture21.5
          Chapter Notes – Linear Programming
      • 22.Probability
        23
        • Lecture22.1
          Introduction to probability 41 min
        • Lecture22.2
          Types of events 42 min
        • Lecture22.3
          Numerical problems 30 min
        • Lecture22.4
          Conditional probability 12 min
        • Lecture22.5
          Numerical problems 09 min
        • Lecture22.6
          Numerical problems (conditional Probability) 04 min
        • Lecture22.7
          Numerical problems (conditional Probability) 06 min
        • Lecture22.8
          Numerical problems (conditional Probability) 05 min
        • Lecture22.9
          Numerical problems (conditional Probability) 06 min
        • Lecture22.10
          Bayes’ Theorem 17 min
        • Lecture22.11
          Numerical problem ( conditional Probability) 04 min
        • Lecture22.12
          Numerical problem ( Baye’s Theorem) 19 min
        • Lecture22.13
          Numerical problem ( Baye’s Theorem) 18 min
        • Lecture22.14
          Numerical problem ( Baye’s Theorem) 10 min
        • Lecture22.15
          Mean and Variance of a random variable 09 min
        • Lecture22.16
          Mean and Variance of a random variable 09 min
        • Lecture22.17
          Mean and Variance of a random variable 08 min
        • Lecture22.18
          Mean and Variance of a discrete random variable 07 min
        • Lecture22.19
          Numerical problem 18 min
        • Lecture22.20
          Bernoulli’s Trials & Binomial Distribution 11 min
        • Lecture22.21
          Numerical problem 12 min
        • Lecture22.22
          Mean and Variance of Binomial Distribution 05 min
        • Lecture22.23
          Chapter Notes – Probability
      • 23.Limits
        4
        • Lecture23.1
          Introduction to limits 35 min
        • Lecture23.2
          Numerical problems 27 min
        • Lecture23.3
          Rationalization 33 min
        • Lecture23.4
          Limits in trigonometry 29 min
      • 24.Partial Fractions
        4
        • Lecture24.1
          Introduction to partial fraction 27 min
        • Lecture24.2
          Partial Fractions 02 29 min
        • Lecture24.3
          Partial Fractions 03 17 min
        • Lecture24.4
          Improper partial fraction 20 min

        Chapter Notes – Determinants

        Every square matrix A is associated with a number, called its determinant and it is denoted by det (A) or |A| .

        Only square matrices have determinants. The matrices which are not square do not have determinants

        (i) First Order Determinant

        If A = [a], then det (A) = |A| = a

        (ii) Second Order Determinant

        Second Order Determinant

        |A| = a11a22 – a21a12

        (iii) Third Order Determinant

        Third Order Determinant

        Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule

        Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule

        then determinant can be formed by enlarging the matrix by adjoining the first two columns on the right and draw lines as show below parallel and perpendicular to the diagonal.

        Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule

        The value of the determinant, thus will be the sum of the product of element. in line parallel to the diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus,

        Δ = a11a22a33 + a12a23a31 + a13a21a32 – a13a22a31 – a11a23a32 – a12a21a33.
        Note This method doesn’t work for determinants of order greater than 3.

        Properties of Determinants

        (i) The value of the determinant remains unchanged, if rows are changed into columns and columns are changed into rows e.g., |A’| = |A|
        (ii) If A = [aij]n x n , n > 1 and B be the matrix obtained from A by interchanging two of its rows or columns, then
        det (B) = – det (A)
        (iii) If two rows (or columns) of a square matrix A are proportional, then |A| = O.
        (iv) |B| = k |A| ,where B is the matrix obtained from A, by multiplying one row (or column) of A by k.
        (v) |kA| = kn|A|, where A is a matrix of order n x n.
        (vi) If each element of a row (or column) of a determinant is the sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants, e.g.,

        Properties of Determinants

        (vii) If the same multiple of the elements of any row (or column) of a determinant are added to the corresponding elements of any other row (or column), then the value of the new determinant remains unchanged, e.g.,

        Properties of Determinants

        (viii) If each element of a row (or column) of a determinant is zero, then its value is zero.
        (ix) If any two rows (columns) of a determinant are identical, then its value is zero.
        (x) If each element of row (column) of a determinant is expressed as a sum of two or more terms, then the determinant can be expressed as the sum of two or more determinants.

        Important Results on Determinants

        (i) |AB| = |A||B| , where A and B are square matrices of the same order.
        (ii) |An| = |A|n
        (iii) If A, B and C are square matrices of the same order such that ith column (or row) of A is the sum of i th columns (or rows) of B and C and all other columns (or rows) of A, Band C are identical, then |A| =|B| + |C|
        (iv) |In| = 1,where In is identity matrix of order n
        (v) |On| = 0, where On is a zero matrix of order n
        (vi) If Δ(x) be a 3rd order determinant having polynomials as its elements.
        (a) If Δ(a) has 2 rows (or columns) proportional, then (x – a) is a factor of Δ(x).
        (b) If Δ(a) has 3 rows (or columns) proportional, then (x – a)2 is a factor of Δ(x). ,
        (vii) A square matrix A is non-singular, if |A| ≠ 0 and singular, if |A| =0.
        (viii) Determinant of a skew-symmetric matrix of odd order is zero and of even order is a nonzero perfect square.
        (ix) In general, |B + C| ≠ |B| + |C|
        (x) Determinant of a diagonal matrix = Product of its diagonal elements
        (xi) Determinant of a triangular matrix = Product of its diagonal elements
        (xii) A square matrix of order n, is non-singular, if its rank r = n i.e., if |A| ≠ 0, then rank (A) = n

        Important Results on Determinants

        Important Results on Determinants

        Important Results on Determinants

        (xiv) If A is a non-singular matrix, then |A-1| = 1 / |A| = |A|-1
        (xv) Determinant of a orthogonal matrix = 1 or – 1.
        (xvi) Determinant of a hermitian matrix is purely real .
        (xvii) If A and B are non-zero matrices and AB = 0, then it implies |A| = 0 and |B| = 0.

        Minors and Cofactors

        Minors and Cofactors

        then the minor Mij of the element aij is the determinant obtained by deleting the i row and jth column.

        Minors and Cofactors

        The cofactor of the element aij is Cij = (- 1)i + j Mij

        Adjoint of a Matrix :-

        Adjoint of a matrix is the transpose of the matrix of cofactors of the give matrix, i.e.,

        Adjoint of a Matrix

        Properties of Minors and Cofactors

        (i) The sum of the products of elements of .any row (or column) of a determinant with the cofactors of the corresponding elements of any other row (or column) is zero, i.e., if

        Properties of Minors and Cofactors

        then a11C31 + a12C32 + a13C33 = 0 ans so on.
        (ii) The sum of the product of elements of any row (or column) of a determinant with the cofactors of the corresponding elements of the same row (or column) is Δ

        Properties of Minors and Cofactors

        Properties of Minors and Cofactors

        Differentiation of Determinant

        Differentiation of Determinant

        Integration of Determinant

        Integration of Determinant

        If the elements of more than one column or rows are functions of x, then the integration can be
        done only after evaluation/expansion of the determinant.

        Solution of Linear equations by Determinant/Cramer’s Rule

        Case 1.

        The solution of the system of simultaneous linear equations
        a1x + b1y = C1 …(i)
        a2x + b2y = C2 …(ii)
        is given by x = D1 / D, Y = D2 / D

        Solution of Linear equations by Determinant/Cramer’s Rule

        (i) If D ≠ 0, then the given system of equations is consistent and has a unique solution given by x = D1 / D, y = D2 / D
        (ii) If D = 0 and Dl = D2 = 0, then the system is consistent and has infinitely many solutions.
        (iii) If D = 0 and one of Dl and D2 is non-zero, then the system is inconsistent.

        Case II.

        Let the system of equations be
        a1x + b1y + C1z = d1
        a2x + b2y + C2z = d2
        a3x + b3y + C3z = d3
        Then, the solution of the system of equation is x = D1 / D, Y = D2 / D, Z = D3 / D, it is called Cramer’s rule.

        Solution of Linear equations by Determinant/Cramer’s Rule

        (i) If D ≠ 0, then the system of equations is consistent with unique solution.
        (ii) If D = 0 and atleast one of the determinant D1, D2, D3 is non-zero, then the given system is inconsistent, i.e., having no solution.
        (iii) If D = 0 and D1 = D2 = D3 = 0, then the system is consistent, with infinitely many solutions.
        (iv) If D ≠ 0 and D1 = D2 = D3 = 0, then system has only trivial solution, (x = y = z = 0).

        Cayley-Hamilton Theorem

        Every matrix satisfies its characteristic equation, i.e., if A be a square matrix, then |A – xl| = 0 is the characteristics equation of A. The values of x are called eigenvalues of A.
        i.e., if x3 – 4x2 – 5x – 7 = 0 is characteristic equation for A, then A3 – 4A2 + 5A – 7I = 0

        Properties of Characteristic Equation

        (i) The sum of the eigenvalues of A is equal to its trace.
        (ii) The product of the eigenvalues of A is equal to its determinant.
        (iii) The eigenvalues of an orthogonal matrix are of unit modulus.
        (iv) The feigen values of a unitary matrix are of unit modulus.
        (v) A and A’ have same eigenvalues.
        (vi) The eigenvalues of a skew-hermitian matrix are either purely imaginary or zero.
        (vii) If x is an eigenvalue of A, then x is the eigenvalue of A* .
        (viii) The eigenvalues of a triangular matrix are its diagonal elements.
        (ix) If x is the eigenvalue of A and |A| ≠ 0, then (1 / x) is the eigenvalue of A-1.
        (x) If x is the eigenvalue of A and |A| ≠ 0, then |A| / x is the eigenvalue of adj (A).
        (xi) If x1, x2,x3, … ,xn are eigenvalues of A, then the eigenvalues of A2 are x2 2, x2 2,…, xn 2.

        Cyclic Determinants

        Cyclic Determinants

        Cyclic Determinants

        Applications of Determinant in Geometry

        Let three points in a plane be A(x1, y1), B(x2, y2) and C(x3, y3), then

        Applications of Determinant in Geometry

        = 1 / 2 [x1 (y2 – y3) + x2 (y3 – y1) + x3 (y1 – y2)]

        Applications of Determinant in Geometry

        Maximum and Minimum Value of Determinants

        Maximum and Minimum Value of Determinants

        where ais ∈ [α1, α2,…, αn]
        Then, |A|max when diagonal elements are
        { min (α1, α2,…, αn)}
        and non-diagonal elements are
        { max (α1, α2,…, αn)}
        Also, |A|min = – |A|max

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