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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 – Probability

        Event: A subset of the sample space associated with a random experiment is called an event or a case.
        e.g. In tossing a coin, getting either head or tail is an event.

        Equally Likely Events: The given events are said to be equally likely if none of them is expected to occur in preference to the other.
        e.g. In throwing an unbiased die, all the six faces are equally likely to come.

        Mutually Exclusive Events: A set of events is said to be mutually exclusive, if the happening of one excludes the happening of the other, i.e. if A and B are mutually exclusive, then (A ∩ B) = Φ
        e.g. In throwing a die, all the 6 faces numbered 1 to 6 are mutually exclusive, since if any one of these faces comes, then the possibility of others in the same trial is ruled out.

        Exhaustive Events: A set of events is said to be exhaustive if the performance of the experiment always results in the occurrence of at least one of them.
        If E1, E2, …, En are exhaustive events, then E1 ∪ E2 ∪……∪ En = S.
        e.g. In throwing of two dice, the exhaustive number of cases is 62 = 36. Since any of the numbers 1 to 6 on the first die can be associated with any of the 6 numbers on the other die.

        Complement of an Event: Let A be an event in a sample space S, then the complement of A is the set of all sample points of the space other than the sample point in A and it is denoted by A’or \bar { A }.
        i.e. A’ = {n : n ∈ S, n ∉ A]

        Note:
        (i) An operation which results in some well-defined outcomes is called an experiment.
        (ii) An experiment in which the outcomes may not be the same even if the experiment is performed in an identical condition is called a random experiment.

        Probability of an Event
        If a trial result is n exhaustive, mutually exclusive and equally likely cases and m of them are favourable to the happening of an event A, then the probability of happening of A is given by
        Probability Class 12 Notes Maths Chapter 13 1

        Note:
        (i) 0 ≤ P(A) ≤ 1
        (ii) Probability of an impossible event is zero.
        (iii) Probability of certain event (possible event) is 1.
        (iv) P(A ∪ A’) = P(S)
        (v) P(A ∩ A’) = P(Φ)
        (vi) P(A’)’ = P(A)
        (vii) P(A ∪ B) = P(A) + P(B) – P(A ∩ S)

         

        Conditional Probability: Let E and F be two events associated with the same sample space of a random experiment. Then, probability of occurrence of event E, when the event F has already occurred, is called a conditional probability of event E over F and is denoted by P(E/F).
        Probability Class 12 Notes Maths Chapter 13 2
        Similarly, conditional probability of event F over E is given as
        Probability Class 12 Notes Maths Chapter 13 3

        Properties of Conditional Probability: If E and E are two events of sample space S and G is an event of S which has already occurred such that P(G) ≠ 0, then
        (i) P[(E ∪ F)/G] = P(F/G) + P(F/G) – P[(F ∩ F)/G], P(G) ≠ 0
        (ii) P[(E ∪ F)/G] = P(F/G) + P(F/G), if E and F are disjoint events.
        (iii) P(F’/G) = 1 – P(F/G)
        (iv) P(S/E) = P(E/E) = 1

        Multiplication Theorem: If E and F are two events associated with a sample space S, then the probability of simultaneous occurrence of the events E and F is
        P(E ∩ F) = P(E) . P(F/E), where P(F) ≠ 0
        or
        P(E ∩ F) = P(F) . P(F/F), where P(F) ≠ 0
        This result is known as multiplication rule of probability.

        Multiplication Theorem for More than Two Events: If F, F and G are three events of sample space, then
        Probability Class 12 Notes Maths Chapter 13 4

        Independent Events: Two events E and F are said to be independent, if probability of occurrence or non-occurrence of one of the events is not affected by that of the other. For any two independent events E and F, we have the relation
        (i) P(E ∩ F) = P(F) . P(F)
        (ii) P(F/F) = P(F), P(F) ≠ 0
        (iii) P(F/F) = P(F), P(F) ≠ 0
        Also, their complements are independent events,
        i.e. P(\bar { E } ∩ \bar { F }) = P(\bar { E }) . P(\bar { F })
        Note: If E and F are dependent events, then P(E ∩ F) ≠ P(F) . P(F).

        Three events E, F and G are said to be mutually independent, if
        (i) P(E ∩ F) = P(E) . P(F)
        (ii) P(F ∩ G) = P(F) . P(G)
        (iii) P(E ∩ G) = P(E) . P(G)
        (iv)P(E ∩ F ∩ G) = P(E) . P(F) . P(G)
        If atleast one of the above is not true for three given events, then we say that the events are not independent.
        Note: Independent and mutually exclusive events do not have the same meaning.

        Baye’s Theorem and Probability Distributions
        Partition of Sample Space: A set of events E1, E2,…,En is said to represent a partition of the sample space S, if it satisfies the following conditions:
        (i) Ei ∩ Ej = Φ; i ≠ j; i, j = 1, 2, …….. n
        (ii) E1 ∪ E2 ∪ …… ∪ En = S
        (iii) P(Ei) > 0, ∀ i = 1, 2,…, n

         

        Theorem of Total Probability: Let events E1, E2, …, En form a partition of the sample space S of an experiment.If A is any event associated with sample space S, then
        Probability Class 12 Notes Maths Chapter 13 5

        Baye’s Theorem: If E1, E2,…,En are n non-empty events which constitute a partition of sample space S, i.e. E1, E2,…, En are pairwise disjoint E1 ∪ E2 ∪ ……. ∪ En = S and P(Ei) > 0, for all i = 1, 2, ….. n Also, let A be any non-zero event, the probability
        Probability Class 12 Notes Maths Chapter 13 6

        Random Variable: A random variable is a real-valued function, whose domain is the sample space of a random experiment. Generally, it is denoted by capital letter X.
        Note: More than one random variables can be defined in the same sample space.

        Probability Distributions: The system in which the values of a random variable are given along with their corresponding probabilities is called probability distribution.
        Let X be a random variable which can take n values x1, x2,…, xn.
        Let p1, p2,…, pn be the respective probabilities.
        Then, a probability distribution table is given as follows:
        Probability Class 12 Notes Maths Chapter 13 7
        such that P1 + p2 + P3 +… + pn = 1
        Note: If xi is one of the possible values of a random variable X, then statement X = xi is true only at some point(s) of the sample space. Hence ,the probability that X takes value x, is always non-zero, i.e. P(X = xi) ≠ 0

        Mean and Variance of a Probability Distribution: Mean of a probability distribution is
        Probability Class 12 Notes Maths Chapter 13 8

        Bernoulli Trial: Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions:
        (i) There should be a finite number of trials.
        (ii) The trials should be independent.
        (iii) Each trial has exactly two outcomes, success or failure.
        (iv) The probability of success remains the same in each trial.

        Binomial Distribution: The probability distribution of numbers of successes in an experiment consisting of n Bernoulli trials obtained by the binomial expansion (p + q)n, is called binomial distribution.
        Let X be a random variable which can take n values x1, x2,…, xn. Then, by binomial distribution, we have P(X = r) = nCr prqn-r
        where,
        n = Total number of trials in an experiment
        p = Probability of success in one trial
        q = Probability of failure in one trial
        r = Number of success trial in an experiment
        Also, p + q = 1
        Binomial distribution of the number of successes X can be represented as
        Probability Class 12 Notes Maths Chapter 13 9

        Mean and Variance of Binomial Distribution
        (i) Mean(μ) = Σ xipi = np
        (ii) Variance(σ2) = Σ xi2 pi – μ2 = npq
        (iii) Standard deviation (σ) = √Variance = √npq
        Note: Mean > Variance

        Prev Mean and Variance of Binomial Distribution
        Next Introduction to limits

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