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      Class 11 MATHS – JEE

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      • Class 11
      • Class 11 MATHS – JEE
      CoursesClass 11MathsClass 11 MATHS – JEE
      • 1.Sets, Relation and Functions
        12
        • Lecture1.1
          Introduction to sets, Description of sets 32 min
        • Lecture1.2
          Types of Sets, Subsets 39 min
        • Lecture1.3
          Intervals, Venn Diagrams, Operations on Sets 37 min
        • Lecture1.4
          Laws of Algebra of Sets 26 min
        • Lecture1.5
          Introduction to sets and its types, operations of sets, Venn Diagrams 28 min
        • Lecture1.6
          Functions and its Types 38 min
        • Lecture1.7
          Functions Types 17 min
        • Lecture1.8
          Cartesian Product of Sets, Relation, Domain and Range 40 min
        • Lecture1.9
          Sum Related to Relations 04 min
        • Lecture1.10
          Sums Related to Relations, Domain and Range 22 min
        • Lecture1.11
          Chapter Notes – Sets, Relation and Functions
        • Lecture1.12
          NCERT Solutions – Sets, Relation and Functions
      • 2.Trigonometric Functions
        28
        • Lecture2.1
          Introduction, Some Identities and Some Sums 16 min
        • Lecture2.2
          Some Sums Related to Trigonometry Identities, trigonometry Functions Table and Its Quadrants 35 min
        • Lecture2.3
          NCERT Sums Ex.3.3 (Q.1-5)Based on Trigometry table and Their Quadrants, Trigonometry Identities of Sum and Diff. of two Angles 21 min
        • Lecture2.4
          NCERT Sums Ex-3.2 Based on Trigonometry Function of Lower & Higher Angles 22 min
        • Lecture2.5
          NCERT Sums Ex-3.3 (Q.6 – 10) Based on Radian Angles 11 min
        • Lecture2.6
          NCERT Sums Ex-3.3 (Q.11-13)Based on Trigonometry Identities 16 min
        • Lecture2.7
          NCERT Sums Ex-3.3 (Q. 14)Based on Trigonometry Identities 14 min
        • Lecture2.8
          NCERT Sums Ex-3.3 (Q.16) Based on Trigonometry Identities 05 min
        • Lecture2.9
          NCERT Sums Ex-3.3 (Q.17 -21) Based on Trigonometry Identities 12 min
        • Lecture2.10
          NCERT Sums Ex-3.4 (Q. 1 – 9), Trigonometry Equation 25 min
        • Lecture2.11
          Sums Based on Trigonometry Equations 24 min
        • Lecture2.12
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.13
          Sums Based on Trigonometry Equations 11 min
        • Lecture2.14
          Sums Based on Trigonometry Equations 17 min
        • Lecture2.15
          Equations Having two Variable Angle which satisfy both equations 10 min
        • Lecture2.16
          Trigonometrical Identities-Some important relations and Its related Sums 16 min
        • Lecture2.17
          Sums Related to Trigonometrical Identities 18 min
        • Lecture2.18
          Properties of Triangles and Solution of Triangles-Sine formula, Napier Analogy and Sums 17 min
        • Lecture2.19
          Relation Between Degree and Radian, Quadrant and NCERT Sum Ex.3.1, 3.2 41 min
        • Lecture2.20
          Trigonometric Functions Table 09 min
        • Lecture2.21
          Some Trigonometric Identities and its related Sums 42 min
        • Lecture2.22
          Sums Related to Trigonometrical Identities 19 min
        • Lecture2.23
          Sums Related to Trigonometrical Identities 41 min
        • Lecture2.24
          Sums Related to Trigonometrical Identities 23 min
        • Lecture2.25
          Trigonometry Equations 44 min
        • Lecture2.26
          Sum Based on Trigonometry Equations 07 min
        • Lecture2.27
          Sums Based on Trigonometry function of Lower Angle 03 min
        • Lecture2.28
          Chapter Notes – Trigonometric Functions
      • 3.Mathematical Induction
        5
        • Lecture3.1
          Introduction to PMI 25 min
        • Lecture3.2
          NCERT Solution of EX- 4.1 14 min
        • Lecture3.3
          NCERT Solution of EX- 4.1 22 min
        • Lecture3.4
          NCERT Solution of EX- 4.1 13 min
        • Lecture3.5
          Chapter Notes – Mathematical Induction
      • 4.Complex Numbers and Quadratic Equation
        15
        • Lecture4.1
          Introduction, Nature of Roots, Numbers, Introduction of i 27 min
        • Lecture4.2
          Sum Related to Relations, Real and Imaginary part of C-N, Conjugate of a C-N 33 min
        • Lecture4.3
          Absolute value or Modulus of a C-N and Related Sums 29 min
        • Lecture4.4
          Sums Related To Multiplicative Inverse 29 min
        • Lecture4.5
          Polar Form of a C-N 32 min
        • Lecture4.6
          Sums Related To Polar Form 32 min
        • Lecture4.7
          Square Roots of C-N and its Related Sums 28 min
        • Lecture4.8
          De Moivris Theorem and its related Sums 31 min
        • Lecture4.9
          Introduction, Nature of Roots, Numbers, Introduction of i and its Sums, Real and Imaginary Part of C-N 35 min
        • Lecture4.10
          Sums Related to Real and Imaginary Part of C-N and Operations on C-N 13 min
        • Lecture4.11
          Sums Related To Multiplicative Inverse 07 min
        • Lecture4.12
          Sums Related To Multiplicative Inverse and Modulus and Argument of a C-N 33 min
        • Lecture4.13
          Polar form of a C-N, Nature of Roots 38 min
        • Lecture4.14
          Sums Based on Roots of Quadratic Equations, Sums of Polar form 10 min
        • Lecture4.15
          Chapter Notes – Complex Numbers and Quadratic Equation
      • 5.Linear Inequalities
        4
        • Lecture5.1
          Introduction, Solve some Linear Inequalities and its Graph 42 min
        • Lecture5.2
          Solve some Linear Inequalities and its Graph 12 min
        • Lecture5.3
          Solve some Linear Inequalities and its Graph and Introduction-Permutations and Combinations 35 min
        • Lecture5.4
          Chapter Notes – Linear Inequalities
      • 6.Permutations and Combinations
        5
        • Lecture6.1
          NCERT Sums Ex-7.3, Equation 41 min
        • Lecture6.2
          NCERT Sums Ex-7.3, Equation 02 min
        • Lecture6.3
          Combination and NCERT Sums Ex-7.4 40 min
        • Lecture6.4
          NCERT Sums Ex-7.1 & 7.2 22 min
        • Lecture6.5
          Chapter Notes – Permutations and Combinations
      • 7.Binomial Theorem
        19
        • Lecture7.1
          Introduction to Binomial Theorem 21 min
        • Lecture7.2
          Binomial General Expansion and Their Derivations and its Related Sums 22 min
        • Lecture7.3
          Pascal’s Triangle Theorem, Addition of Two Expansion, NCERT Sums Ex-8.1 26 min
        • Lecture7.4
          Sums of Miscellaneous Exercise and Ex-8.1, Finding the Any Term from nth Term 42 min
        • Lecture7.5
          NCERT Sums Ex-8.1 14 min
        • Lecture7.6
          NCERT Sums Ex-8.1 04 min
        • Lecture7.7
          NCERT Sums Ex-8.2, Middle Term 21 min
        • Lecture7.8
          NCERT Sums Ex-8.2, Middle Term Related Sums 08 min
        • Lecture7.9
          To Find the Coefficient of X^r in the Expansion of (X+A)^n, NCERT Sums Ex-8.2 and Miscellaneous Ex. 40 min
        • Lecture7.10
          NCERT Sums Ex-8.2 10 min
        • Lecture7.11
          To Find the Sum of the Coefficients in the Expansion of (1+x)^n and its Related Sums 27 min
        • Lecture7.12
          Sums Related to Binomials Coefficients 24 min
        • Lecture7.13
          Binomial Theorem for any Index and its Related Sums 27 min
        • Lecture7.14
          Introduction to Binomial Theorem, General Term in the Expansion of (x+a)^n. 39 min
        • Lecture7.15
          NCERT Sums Ex-8.1 & 8.2, Pascals’ Triangle, pth Term from End 24 min
        • Lecture7.16
          Sums related to Finding the Coefficient, NCERT Sums Ex-8.2, Middle Term 40 min
        • Lecture7.17
          Sums Related to Middle Term 17 min
        • Lecture7.18
          Sums Related to Coefficient of the Any Term 31 min
        • Lecture7.19
          Chapter Notes – Binomial Theorem
      • 8.Sequences and Series
        14
        • Lecture8.1
          Introduction, A.P., nth Term and Sum of nth Term, P Arithmetic Mean B/w a and b, Sum Based on Fibonacci Sequence 27 min
        • Lecture8.2
          NCERT Sums Ex-9.2 37 min
        • Lecture8.3
          NCERT Sums Ex-9.2 18 min
        • Lecture8.4
          NCERT Sums Ex-9.2, Geometric Progression -Introduction, nth term, NCERT Sums Ex-9.3 39 min
        • Lecture8.5
          NCERT Sums Ex-9.3 16 min
        • Lecture8.6
          Sum of n term of G.P., NCERT Sums Ex-9.3 40 min
        • Lecture8.7
          NCERT Sums Ex-9.3 08 min
        • Lecture8.8
          NCERT Sums Ex-9.3, Insert P Geometrical Mean B/w a and b 36 min
        • Lecture8.9
          NCERT Sum Ex-9.3 17 min
        • Lecture8.10
          NCERT Sum Ex-9.3 09 min
        • Lecture8.11
          Some Special Series, NCERT Sum Ex-9.4 36 min
        • Lecture8.12
          NCERT Sum Ex-9.4 02 min
        • Lecture8.13
          NCERT Sum Ex-9.4 18 min
        • Lecture8.14
          Chapter Notes – Sequences and Series
      • 9.Properties of Triangles
        2
        • Lecture9.1
          Sine 7 Cosine Rule, Projection Formulae, Napier’s Analogy, Incircle, Some Sums 41 min
        • Lecture9.2
          Angle of Elevations and Depression. and Its Related Sums 13 min
      • 10.Straight Lines
        30
        • Lecture10.1
          Introduction, Equation of Line, Slope or Gradient of a line 24 min
        • Lecture10.2
          Sums Related to Finding the Slope, Angle Between two Lines 22 min
        • Lecture10.3
          Cases for Angle B/w two Lines, Different forms of Line Equation 23 min
        • Lecture10.4
          Sums Related Finding the Equation of Line 27 min
        • Lecture10.5
          Sums based on Previous Concepts of Straight line 32 min
        • Lecture10.6
          Parametric Form of a Straight Line 16 min
        • Lecture10.7
          Sums Related to Parametric Form of a Straight Line 16 min
        • Lecture10.8
          Sums Based on Concurrent of lines, Angle b/w Two Lines 45 min
        • Lecture10.9
          Different condition for Angle b/w two lines 04 min
        • Lecture10.10
          Sums Based on Angle b/w Two Lines 36 min
        • Lecture10.11
          Equation of Straight line Passes Through a Point and Make an Angle with Another Line 09 min
        • Lecture10.12
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 15 min
        • Lecture10.13
          Sums Based on Equation of Straight line Passes Through a Point and Make an Angle with Another Line 17 min
        • Lecture10.14
          Finding the Distance of a point from the line 34 min
        • Lecture10.15
          Sum Based on Finding the Distance of a point from the line and B/w Two Parallel Lines 33 min
        • Lecture10.16
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 44 min
        • Lecture10.17
          Sums Based on Find the Equation of Bisector of Angle Between two intersecting Lines 02 min
        • Lecture10.18
          Introduction, Distance B/w Two Points, Slope, Equation of Line 32 min
        • Lecture10.19
          NCERT Sums Ex-10.1 43 min
        • Lecture10.20
          NCERT Sums Ex-10.1 29 min
        • Lecture10.21
          NCERT Sums Ex-10.1 & 10.2 43 min
        • Lecture10.22
          NCERT Sums Ex-10.2 30 min
        • Lecture10.23
          NCERT Sums Ex-10.2 41 min
        • Lecture10.24
          NCERT Sums Ex-10.2 & 10.3 21 min
        • Lecture10.25
          NCERT Sums Ex- 10.3 (Reduce the Equation into intercept Form, Normal form) 42 min
        • Lecture10.26
          NCERT Sums Ex-10.3 21 min
        • Lecture10.27
          NCERT Sums Ex-10.3 (Equation of Parallel line, Perpendicular Line of given line, Sums Based of Angle B/w Two Lines) 42 min
        • Lecture10.28
          NCERT Sums Ex-10.3 09 min
        • Lecture10.29
          NCERT Sums Ex-10.3 26 min
        • Lecture10.30
          Chapter Notes – Straight Lines
      • 11.Conic Sections
        21
        • Lecture11.1
          Introduction, General Equation of second Degree, Parabola, Sums based on Finding Equation of Parabola 41 min
        • Lecture11.2
          Sums Based on Equation of Parabola, Four Forms of Parabola-Form (i) 30 min
        • Lecture11.3
          Sums Based on Four Forms of Parabola-Form (i) 32 min
        • Lecture11.4
          Four Forms of Parabola-Form (ii), (iii) (iv) 13 min
        • Lecture11.5
          Sums Based on Four forms of Parabola 18 min
        • Lecture11.6
          Position of a Point with Respect to Parabola and its Sums 43 min
        • Lecture11.7
          Circles-Introduction, Different Cases for Circle Equations, NCERT Sums Ex-11.1 16 min
        • Lecture11.8
          NCERT Sums Ex-11.1 40 min
        • Lecture11.9
          Circle Important Point Revise, Intersection of Axes, NCERT Sums Ex-11.1 11 min
        • Lecture11.10
          NCERT Sums Ex-11.1 44 min
        • Lecture11.11
          Parabola- Introduction, General Equation , Sums, Some Important Concepts for Parabola 12 min
        • Lecture11.12
          Different Form of Parabola, NCERT Sum Ex-11.2 13 min
        • Lecture11.13
          NCERT Sum Ex-11.2 34 min
        • Lecture11.14
          Ellipse-Introduction, General Equation, NCERT Sums Ex-11.3 36 min
        • Lecture11.15
          NCERT Sums Ex-11.3 02 min
        • Lecture11.16
          NCERT Sums Ex-11.3 23 min
        • Lecture11.17
          Hyperbola-Introduction, NCERT Sums Ex-11.4 12 min
        • Lecture11.18
          NCERT Sums Ex-11.4 25 min
        • Lecture11.19
          Chapter Notes – Conic Sections Circles
        • Lecture11.20
          Chapter Notes – Conic Sections Ellipse
        • Lecture11.21
          Chapter Notes – Conic Sections Parabola
      • 12.Coordinate Geometry
        8
        • Lecture12.1
          Introduction to Rectangular Cartesian Coordinate Geometry (2D), Distance b/w two points 23 min
        • Lecture12.2
          Cartesian Coordinate of points 32 min
        • Lecture12.3
          Questions rel to cartesian coordinate of points 25 min
        • Lecture12.4
          Section Formula – Case 1, Case 2 24 min
        • Lecture12.5
          Problem Solving 26 min
        • Lecture12.6
          Centeroid, Incenter, Circumcenter of a triangle 30 min
        • Lecture12.7
          Locus Problems 17 min
        • Lecture12.8
          Problem Solving 21 min
      • 13.Three Dimensional Geometry
        3
        • Lecture13.1
          Introduction to 3D 18 min
        • Lecture13.2
          Numerical problems 14 min
        • Lecture13.3
          Chapter Notes – Three Dimensional Geometry
      • 14.Limits And Derivatives
        12
        • Lecture14.1
          Introduction to limits 42 min
        • Lecture14.2
          EX-13.1 16 min
        • Lecture14.3
          Questions based on algebra of limits 41 min
        • Lecture14.4
          Limits of a polynomial 12 min
        • Lecture14.5
          rational function 37 min
        • Lecture14.6
          trigo function 21 min
        • Lecture14.7
          Introduction to Derivatives 37 min
        • Lecture14.8
          Ex-13.2 22 min
        • Lecture14.9
          Algebra of derivatives 38 min
        • Lecture14.10
          Derivative of polynomial 13 min
        • Lecture14.11
          trigo function 11 min
        • Lecture14.12
          Chapter Notes – Limits And Derivatives
      • 15.Mathematical Reasoning
        3
        • Lecture15.1
          What is statement ? Special word and phrases, negation of statement , Compound statement , and & or in compound statement , truth table Solving the problems of Ex- 14.1 , 14.2 25 min
        • Lecture15.2
          Solving Ex-14.3, Ex-14.4, Implications, Validating statements, Ex-14.5, Direct method 24 min
        • Lecture15.3
          Chapter Notes – Mathematical Reasoning
      • 16.Statistics
        5
        • Lecture16.1
          Mean, Median, Mode, Range, Mean Deviation Solution of Ex-15.1 27 min
        • Lecture16.2
          Mean Deviation about Mean & Median, Ex-15.2, Mean and Variance, Standard deviation 35 min
        • Lecture16.3
          Ex-15.2 , Variance and Standard deviation 09 min
        • Lecture16.4
          Ex-15.3, Analysis of frequency distribution, comparison of two frequency distribution with same mean 23 min
        • Lecture16.5
          Chapter Notes – Statistics
      • 17.Probability
        3
        • Lecture17.1
          Outcomes & sample space, Ex. 16.3 19 min
        • Lecture17.2
          Ex.16.3, Probability of an event, Algebra of event 38 min
        • Lecture17.3
          Chapter Notes – Probability
      • 18.Binary Number
        2
        • Lecture18.1
          Binary numbers, Conversion of Binary to Decimal and Decimal to binary 45 min
        • Lecture18.2
          Addition, Subtraction, Multiplication, Division 02 min

        Chapter Notes – Complex Numbers and Quadratic Equation

        Imaginary Numbers
        The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc.
        The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota.

        Integral Power of IOTA (i)
        i = √-1, i2 = -1, i3 = -i, i4 = 1
        So, i4n+1 = i, i4n+2 = -1, i4n+3 = -i, i4n = 1

        Note:

        • For any two real numbers a and b, the result √a × √b : √ab is true only, when atleast one of the given numbers i.e. either zero or positive.
          √-a × √-b ≠ √ab
          So, i2 = √-1 × √-1 ≠ 1
        • ‘i’ is neither positive, zero nor negative.
        • in + in+1 + in+2 + in+3 = 0

        Complex Number
        A number of the form x + iy, where x and y are real numbers, is called a complex number, x is called real part and y is called imaginary part of the complex number i.e. Re(Z) = x and Im(Z) = y.

        Purely Real and Purely Imaginary Complex Number
        A complex number Z = x + iy is a purely real if its imaginary part is 0, i.e. Im(z) = 0 and purely imaginary if its real part is 0 i.e. Re (z) = 0.

        Equality of Complex Number
        Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2)
        Note: Order relation “greater than’’ and “less than” are not defined for complex number.

        Algebra of Complex Numbers
        Addition of complex numbers
        Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their sum defined as
        z1 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)

        Properties of Addition

        • Commutative: z1 + z2 = z2 + z1
        • Associative: z1 + (z2 + z3) = (z1 + z2) + z3
        • Additive identity z + 0 = z = 0 + z
          Here, 0 is additive identity.

        Subtraction of complex numbers
        Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their difference is defined as
        z1 – z2 = (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1 – y2)

        Multiplication of complex numbers
        Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their multiplication is defined as
        z1z2 = (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1)

         

        Properties of Multiplication

        • Commutative: z1z2 = z2z1
        • Associative: z1(z2z3) = (z1z2)z3
        • Multiplicative identity: z . 1 = z = 1 . z
          Here, 1 is multiplicative identity of an element z.
        • Multiplicative inverse: For every non-zero complex number z, there exists a complex number z1 such that z . z1 = 1 = z1 . z
        • Distributive law: z1(z2 + z3) = z1z2 + z1z3

        Division of Complex Numbers
        Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their division is defined as
        Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

        Conjugate of Complex Number
        Let z = x + iy, if ‘i’ is replaced by (-i), then said to be conjugate of the complex number z and it is denoted by \bar { z }, i.e. \bar { z } = x – iy

        Properties of Conjugate
        Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

        Modulus of a Complex Number
        Let z = x + iy be a complex number. Then, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e. |z| = \sqrt { { x }^{ 2 }+{ y }^{ 2 } }
        It represents a distance of z from origin in the set of complex number c, the order relation is not defined
        i.e. z1 > z2 or z1 < z2 has no meaning but |z1| > |z2| or |z1|<|z2| has got its meaning, since |z1| and |z2| are real numbers.

        Properties of Modulus of a Complex number
        Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

         

        Argand Plane
        Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called argand plane or gaussian plane. A purely number x, i.e. (x + 0i) is represented by the point (x, 0) on X-axis. Therefore, X-axis is called real axis. A purely imaginary number iy i.e. (0 + iy) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is called the imaginary axis.

        Argument of a complex Number
        The angle made by line joining point z to the origin, with the positive direction of X-axis in an anti-clockwise sense is called argument or amplitude of complex number. It is denoted by the symbol arg(z) or amp(z).
        arg(z) = θ = tan-1(\frac { y }{ x })
        Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

        Argument of z is not unique, general value of the argument of z is 2nπ + θ, but arg(0) is not defined. The unique value of θ such that -π < θ ≤ π is called the principal value of the amplitude or principal argument.

        Principal Value of Argument

        • if x > 0 and y > 0, then arg(z) = θ
        • if x < 0 and y > 0, then arg(z) = π – θ
        • if x < 0 and y < 0, then arg(z) = -(π – θ)
        • if x > 0 and y < 0, then arg(z) = -θ

        Polar Form of a Complex Number
        If z = x + iy is a complex number, then z can be written as z = |z| (cosθ + isinθ), where θ = arg(z). This is called polar form. If the general value of the argument is θ, then the polar form of z is z = |z| [cos (2nπ + θ) + isin(2nπ + θ)], where n is an integer.

        Square Root of a Complex Number

        Complex Numbers and Quadratic Equations Class 11 Notes Maths Chapter 5

        Solution of a Quadratic Equation
        The equation ax2 + bx + c = 0, where a, b and c are numbers (real or complex, a ≠ 0) is called the general quadratic equation in variable x. The values of the variable satisfying the given equation are called roots of the equation.

        The quadratic equation ax2 + bx + c = 0 with real coefficients has two roots given by \frac { -b+\surd D }{ 2a } and \frac { -b-\surd D }{ 2a }, where D = b2 – 4ac, called the discriminant of the equation.

        Note:
        (i) When D = 0, roots ore real and equal. When D > 0 roots are real and unequal. Further If a,b, c ∈ Q and D is perfect square, then the roots of quadratic equation are real and unequal and if a, b, c ∈ Q and D is not perfect square, then the roots are irrational and occur in pair. When D < 0, roots of the equation are non real (or complex).

        (ii) Let α, β be the roots of quadratic equation ax2 + bx + c = 0, then sum of roots α + β = \frac { -b }{ a } and the product of roots αβ = \frac { c }{ a }.

        Prev Sums Based on Roots of Quadratic Equations, Sums of Polar form
        Next Introduction, Solve some Linear Inequalities and its Graph

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