Maths

Mathematics is the base of human civilization. From cutting vegetables to arranging books on the shelves, from tailoring clothes to motion of Planets - Mathematics applies everywhere.

Number SystemCalculus | Probability | Trigonometry | Geometry & Mensuration | Algebra & Arithmetic | Coordinate Geometry

Other Standard Form of Ellipse

If the major axis of the ellipse is along the y-axis, then the equation of the ellipse is

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Standard Equation of Ellipse

The locus of a point in a plane whose distance from a fixed point bears a constant ratio, less than one to its distance from a fixed line is called ellipse.

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Standard Parabolas

Parabolas can open rightward, open leftward, open upward and open downward.

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Important Definitions Regarding Parabola

Focus

The fixed point used to draw the parabola is called the focus (F). The focus is F(a, 0).

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Standard Equation of Parabola

The locus of a point whose distance from a fixed point is equal to its distance from a fixed line is called a parabola. A parabola is a conic whose eccentricity is 1.

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Classification of Conics

The equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 represents either a (non–degenerate) conic or a degenerate conic.

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General Equation of a Conic

Let F(x1, y1) be the focus, lx + my + n = 0, the equation of the directrix and e the eccentricity of the conic. Let P(x, y) be any point on the conic.

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Definition of a Conic

The curves obtained by slicing the cone with a plane not passing through the vertex are called conic sections or conics.

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Orthogonal Circles

Two circles are said to be orthogonal if the tangent at their point of intersection are at right angles.

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Circles Touching Each Other

Two circles may touch each other either internally or externally. Let C1, C2 be the centers of the circle and r1, r2 be their radii and P the point of contact.

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Equation of Chord of Contact of Tangents

The general equation of the circle is

x2 + y2 + 2gx + 2fy + c = 0

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Condition For Line To Be Tangent To Circle

Let the line y = mx + c be a tangent to the circle x2 + y2 = a2 at (x1, y1)

The equation of the tangent at (x1, y1) to the circle x2 + y2 = a2 is xx1 + yy1 = a2

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Length of Tangent to Circle

Let the equation of the circle be

x2 + y2 + 2gx + 2fy + c = 0

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Equation of Tangent to a Circle

A tangent to a circle is a straight line which intersects (touches) the circle inexactly one point.

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Parametric Form of Circle

Consider a circle with radius rand centre at the origin. Let P(x, y) be any point on the circle. Assume that OP makes an angle θwith the positive direction of x-axis. Draw the perpendicular PM to the x-axis.

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General Equation of Circle

The general equation of the circle is

x2 + y2 + 2gx + 2fy + c = 0

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Equation of a Circle

A circle is the locus of a point which moves in such a way that its distance from a fixed point is always constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

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Pair of Straight Lines

The equation of a pair of straight lines is in the form

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, c, f, g, h are constants.

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Condition for Three Lines To Be Concurrent

Let the three straight lines be given by

a1x + b1y + c1 = 0

a2x + b2y + c2 = 0

a3x + b3y + c3 = 0

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Angle Between Two Straight Lines

Let L1 and L2 be the two intersecting lines and assume that P be the point of intersection of the two straight lines which makes angle θ1 and θ2 with the positive direction of x-axis.

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Perpendicular Distance From Point to Straight Line

The length of the perpendicular from the point (x1, y1) to the line ax + by + c = 0 is

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Equation of Straight Line

A straight line is the simplest geometrical curve. Every straight line is associated with an equation. To determine the equation of a straight line, two conditions are required.

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Parametric Form of Straight Line

If two variables, say x and y, are functions of a third variable, say θ, then the functions expressing x and y in terms of θ are called the parametric representations of x and y. The variable θ is called the parameter of the function.

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Definite Integrals

The beauty and importance of the integral calculus is that it provides a systematic way for the exact calculations of many areas, volumes and other quantities.

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Integration by Parts

Integration by parts method is generally used to find the integral when the integrand is a product of two different types of functions or a single logarithmic function or a single inverse trigonometric function or a function which is not integrable directly.

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Integration by Substitution

Sometimes, the given functions may not be in an integration form and the  variable of integration (x in dx) can be suitably changed into a new variable by substitution so that the new function can be integrated. Integration by Substitution is also called u-substitution.

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Integrals of Linear Functions of x

This rule applies to functions of ax+b. It does not apply in other cases.

If ∫ f(x) dx = F(x) + C, then

∫ f(ax + b) dx = 1/a × F(ax + b) + C

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Integration

Integration is the inverse process of differentiation. A function F(x) is called an anti derivative or integral of a function f(x) on an interval I if

F′(x) = f(x) for every value of x in I

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Derivative of Composite Function (Chain Rule)

If u = f(x) and y = F(u), then y = F(f(x)) is the composition of f and F. u is called the intermediate argument.

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Quotient Rule For Differentiation

If u and v are differentiable function and if v(x) ≠ 0, then

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Product Rule For Differentiation

Let u and v be differentiable functions of x. Then the product function

y = u(x) v(x) is differentiable.

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Concept of Differentiation

Consider a function y = f(x) of a variable x. Suppose x changes from an initial value x0 to a final value x1. Then, the increment in x is defined to be the amount of change in x. It is denoted by ∆x.

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Derivative of a Function

The derivative of a given function y = f(x) is defined as the limit of the ratio of the increment ∆y of the function to the corresponding increment ∆x of the independent variable, when the latter tends to zero.

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Continuous Functions

Let f be a function defined on an interval I = [a, b]. A continuous function on I is a function whose graph y = f(x) can be described by the motion of a particle travelling along it from the point (a, f(a)) to the point (b, f(b)) without moving off the curve.

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Limit of a Function

The notion of limit is related to the intuitive idea of nearness or closeness. Degree of such closeness cannot be described in terms of basic algebraic operations of addition and multiplication and their inverse operations subtraction and division respectively.

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Odd and Even Functions

If f(x) = f(−x) for all x in the domain then the function is called an even function.

If f(x) = − f(−x) for all x in the domain then the function is called an odd function.

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Signum Function

The sign function or signum function extracts the sign of a real number.

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Step Functions

Greatest Integer Function

The function whose value at any real number x is the greatest integer less than or equal to x is called the greatest integer function (floor function). It is denoted by [x].

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Absolute Value Function (Modulus Function)

If f : R → R defined by f(x) = |x| then the function is called absolute value function of x.

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Exponential Functions

For any number a > 0, a ≠ 1, the function f : R → R defined by f(x) = ax is called an exponential function.

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Linear Function

If a function f : R → R is defined in the form f(x) = ax + b then the function is called a linear function. Here, a and b are constants.

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Constant Function

If the range of a function is a singleton set then the function is called a constant function.

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Inverse of a Function

To define the inverse of a function f i.e. f−1 (f inverse), the function f must be one-to-one and onto.

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Composition of Functions

Let A, B and C be any three sets and let f : A → B and g : B → C be any two functions. The domain of g is the co-domain of f.

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Identity Function

A function f from a set A to the same set A is said to be an identity function if f(x) = x for all x ∈ A i.e. f : A → A is defined by f(x) = x for all x ∈ A. Identity function is denoted by IA or I. Therefore, I(x) = x always.

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One-to-One Function

A function is said to be one-to-one if each element of the range is associated with exactly one element of the domain. Two different elements in the domain (A) have different images in the co-domain (B).

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Onto and Into Functions

If the range of a function is equal to the co-domain then the function is called an onto function. Otherwise it is called an into function.

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Functions

A function is a special type of relation. In a function, no two ordered pairs can have the same first element and a different second element. That is, for a function, corresponding to each first element of the ordered pairs, there must be a different second element.

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Constants and Variables

A quantity which retains the same value throughout a mathematical process is called a constant. A variable is a quantity which can have different values in a particular mathematical process.

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