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In a triangle ABC the angles are denoted by capital letters A, B and C and the length of the sides opposite to these angles are denoted by small letters a, b and c. Semi perimeter of the triangle is
given by s where  2s = a + b+c  and its area is denoted by Δ

SINE  RULE

Sine rule is an  important tool that relates   side lengths with angles of triangles  and circum radius of triangle NAPIER’S ANALOGY (TANGENT RULE) COSINE RULE
In a triangle ABC

PROJECTION RULE

(i) a = b cos C + c cos B (ii) b = c cos A + a cos C (iii) c = a cos B + b cos C

HALF ANGLE FORMULAE .m-n THEOREM  CENTROID AND MEDIANS OF A TRIANGLE
The line joining any vertex of a triangle to the mid point of the opposite side of the triangle is called the median of the triangle. The three medians of a triangle are concurrent and the point of concurrency of the medians of any triangle is called the centroid of the triangle. The centroid divides the median
in the ratio 2 : 1.

Circum circle The circle which passes through the angular points of a ABC, is called its circumcircle. The centre of this circle i.e., the point of concurrency of the perpendicular bisectors of the sides of the ABC, is called the circumcenter.

Radius of the circumcircle is given by the following formulae ORTHOCENTER AND PEDAL TRIANGLE OF A TRIANGLE. In a triangle the altitudes drawn from the three vertices to the opposite sides are concurrent and the point of cuncurrency of the altitudes of the triangle is called the orthocenter of the triangle. The triangle formed by joining the feet of these perpendiculars is called the pedal triangle i.e. DEF is the pedal triangle of ABC. BISECTORS OF THE ANGLES
If AD bisects the angle A and divide the base into portions x and y, we have, by Geometry, INCIRCLE
the circle which can be inscribed within the triangle so as to touch each of the sides of the triangle is called its incircle. The centre of this circle i.e., the point of concurrency of angle bisectors of the triangle is called the incentre of the ABC The distances Between the special points ESCRIBED CIRCLES
The circle which touches the side BC and the two sides AB and AC produced is called the escribed circle opposite the angle A. Its centre and radius will be denoted by I1 and r1 respectively. Excentral triangle
The triangle formed by joining the three excentres I1, I2 and I3 of
DABC is called the excentral or excentric triangle. Not that Inscribed & Circumscribed Polygons
(Important Formulae) SOLUTION OF TRIANGLES
When any three of the six elements (except all the three angles) of a triangle are given, the triangle is known completely. This process is called the solution of triangles.  ## Complex numbers for IIT-JEE MAINS /ADVANCED

Complex numbers basic concepts-Defining

A number in the form of a + ib, where a, b are real numbers and is called a complex
Number. A Complex Number can also be defined as an ordered pair of real numbers a and b any
may be written as (a, b), where the first number denotes the real part and the second number
denotes the imaginary part. If z = a + ib, then the real part of z is denoted by Re (z) and the
imaginary part by Im(z). A complex number is said to be purely real if Im(z)= 0, and is said to
be purely imaginary if Re(z) = 0. The complex number 0 = 0 + i0 is both purely real and imaginary.
Two complex numbers are said to be equal if and only if their real parts and imaginary parts are
separately equal i.e. a + ib = c + id implies a = c and b = d. However, there is no order relation
between complex numbers and the expressions of the type a + ib < (or > ) c + id are meaningless

Geometrical representation of complex numbers

A complex number z = x + iy, written as an ordered pair (x, y), can be represented by a point P whose Cartesian coordinates are (x, y) referred to axes OX and OY, usually called the real and the imaginary axes. The plane of OX and OY is called the Argand diagram or the complex plane. Since the origin O lies on both OX and OY, the corresponding complex number z = 0 is both purely real and purely imaginary.

Modulus and Argument of a Complex Number

We define modulus of the complex number z = x + iy as Trigonometric ( or polar ) from a Complex Number  Remark : Method of finding the principal value of the argument of a complex number z = x + iy. Unimodular Complex Number Algebraic Operations with Complex Number Geometrical Meaning of Product and Division

Let z1 = x1 + i y1 and z2 = x2 + i y2 be two complex numbers represented by the points P1(x1, y1) and P2(x2, y2) respectively. By definition z1 + z2 should be represented by the point (x1 + x2 , y1 + y2 ). This point is the vertex which completes the parallelogram with the line segments joining the origin with OP1 and OP2 as the adjacent sides.
| z1 + z2 | = OP. lso by definition z1 – z2 should be represents by the point (x1 – x2, y1 – y2). This point is the vertex which completes the parallelogram with the line segments joining the origin with OP1 and (where the point represents ; the point can be obtained by producing the directed line P2O by length | z2 | ) as the adjacent sides. | z1 – z2 | = OQ = P2 P1. Remarks (a) In any triangle, sum of any two sides is greater than the third side and difference of any two Construction for the point representing the product z1 z2  .

Construction for the point representing the quotient z1/z2

Draw the triangle OQ1P directly similar to the triangle OQ2L
The P represents the quotient z1/z2 .  Square Root of a Complex Number

How we get the square root of complex number  Conjugate of a Complex Number

The conjugate of the complex number z = a + ib is defined to be a – ib and i denoted by . In other words is the mirror image of z in the real axis. Properties of Conjugate, Modulus, Argument  Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| .  Properties of Arguments  Complex Numbers Represented By Vectors It can be easily seen that multiplication by real numbers of a complex number is subjected to the same rule as the vectors . The addition or the subtraction of two complex numbers is also the same as the addition or the substraction of two vectors. This fact is fundamental in theory and very useful in practice. It should be noticed that if a number z is represented by points P and OP by a vector then| z | is the length OP and arg(z) is the angle which the directed line OP makes with directed OX. Please note that if z = x + iy and P is the point (x, y), a one–to–one correspondence exists between the number z and any of the following : (i) the point P; (ii) the displacement ; (iii) the vector (or directed length ) . Any one of these three things may therefore be said to represent z, or to be represented by z.

De-Moiver’s Theorem Application of De Moiver’s Theorem

This is a fundamental theorem and has various applications. Here we will discuss few of these
which are important from the examination point of view

The nth Roots of Unity Logarithm of Complex Number CONCEPT OF ROTATION   Geometrical Application

Section Formula Equation of a straight line

Equation of straight line with the help of coordinate geometry    Equation of Perpendicular Bisector arg (z – z0) =p  represents a line passing through z0 with slope (making angle p with the positive direction of x-axis).

Equation of a circle

Consider a fixed complex number z0 and let z be any complex number which moves in such a way
that it’s distance from z0 is always equals to ‘r’. This implies z would lie on a circle whose centre is
z0 and radius r. And its equation would | z – z0 | = r    Note :
(i) If we take ‘C’ to be mid–point of A2A1, it can be easily proved that CA.CB = (CA1)2 i.e. | z1 – z0 | | z2 – z0 | = r2, where the point C is denoted z0 and r is the radius of the circle.

(ii) If | z1 – z0 | = | z2 – z0 | hence P(z) would lie on the right bisector of the line A(z1)
and B(z2). Note that in this case z1 and z2 are the mirror images of each other with respect
to the right bisector.

Equation of tangent a given circle at a given point Some important results ENJOY LEARNING MATHS WITH VIDEO LECTURES OF KAMAL SIR kamal sir > Well – Known IIT JEE Maths mentor

BASIC RESULTS RELATED TO A QUADRATIC EQUATIONS The quantity D • The quadratic equation has real and equal roots if and only if D = 0

• The quadratic equation has real and distinct roots if and only if D > 0

• The quadratic equation has complex roots with non-zero imaginary parts if and only if D < 0 . If p + iq (p and q being real) is a root of the quadratic equation where i = , then p – iq is also a root of the quadratic equation.

CONCEPT OF IDENTITY

If the quadratic equation is satisfied by more than two distinct numbers (real or complex), then it becomes an identity i.e. a = b = c = 0. For example is satisfied by three value of x which are a, b and c. Hence this is an identity in x. HOW TO SOlVE RATIONAL INEQUALITIES

In order to solve inequalities of the form we use the following method:

If x1 and x2 (x1 < x2) are two consecutive distinct roots of a polynomial equation, then within this
interval the polynomial itself takes on values having the same sign. Now find all the roots of the polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write Where a1, a2, . . . . . an, b1, b2, . . . . . , bm are distinct real numbers. Then f(x) = 0 for x = a1, a2, . . . . . , an and f(x) is not defined for x = b1, b2, . . . . . , bm. Apart from these (m + n) real numbers f(x) is either positive or negative. Now arrange a1, a2, . . . . . , an, b1, b2, . . . . . , bm in an increasing order say c1, c2, c3, c4, c5, . . . . . , cm+n. Plot them on the real line. Draw a curve starting from right of cm+n along the real line which alternately changes its position at these points. This curve is known as the wavy curve. The intervals in which the curve is above the real line will be the intervals for which f(x) is positive and the intervals in which the curve is below the real line will be the intervals in which f(x) is negative.

Other way of understanding the same concept

• Write the inequality in the correct form. One side must be zero and the other must be in product form
• Find the key or critical values. To find the key/critical values, set the numerator and denominator of the fraction equal to zero and change them in product of linear forms
• Make a sign analysis chart. To make a sign analysis chart, use the key/critical values found in Step 2 to and mention them on line in increasing form.
• Perform the sign analysis. and make wave that give the sign of product
• What is needed acording to that write final answers
• Problems related to rational inequalities with video solutions

CONDITIONS OF COMMON ROOTS FOR TWO QUADRATIC EQUATIONS

For two quadratic equations when one root is common / When both roots are common are given HOW TO FIND  RANGE OF A RATIONAL EXPRESSION IN X, WHERE X IS REAL

Put the given rational expression equal to y and form the quadratic equation in x.

Find the discriminant D of the quadratic equation obtained in step 1.

Since x is real, therefore, put D   GREATER THAN OR EQUAL TO 0. We get an inequation in y.

Solve the above inequation for y. The range of y so obtained determines the range attained by the given rational expression The expression ax2 + bx + c is said to be a real quadratic expression in x where a, b, c are real and
a 0. Let f(x) = ax2 + bx + c, where a, b, c R (a 0). f(x) can be re-written as

Therefore y = f(x) represents a parabola whose axis is parallel to the y-axis, with vertex at
A . Note that if a > 0, the parabola will be concave upwards and if a < 0 the parabola will be concave downwards and it depends on the sign of D that the parabola cuts the x-axis at two points (D > 0), touches the x-axis (D = 0) or never intersects with the x-axis (D< 0). This gives rise to the following cases: In this case the parabola always remains concave upwards and above the x-axis (iv) a < 0 and D< 0
f(x) < 0xR.
In this case the parabola remains concave downwards and always below the x-axis. RELATION BETWEEN THE ROOTS OF A POLYNOMIAL EQUATION OF DEGREE N AND coefficient  IMPORTANT POINTS

A polynomial equation of degree n has n roots (real or imaginary).

If all the coefficients are real then the imaginary roots occur in pairs i.e., number of imaginary roots is always even.

If the degree of a polynomial equation is odd then the number of real roots will also be odd. It follows that at least one of the roots will be real.

Polynomial in x of degree n can be factorized into a product of linear/quadratic form.

I

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## Probability for IIT-JEE MAINS /ADVANCED

Probability is very important topic for IIT JEE MAINS / ADVANCED.

Its the topic for which understanding is must .

For JEE MAINS its   weightage is 3 to 4%

and for  JEE ADVANCED its weightage is 11 TO 14 %