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Quadratic Polynomial A polynomial, whose degree is 2, is called a quadratic polynomial. It is in the form of p(x) = ax 2  + bx + c,  where a ≠ 0 Quadratic Equation When we equate the quadratic polynomial to zero then it is called a Quadratic Equation i.e. if p(x) = 0 , then it is known as Quadratic Equation.  Standard form of Quadratic Equation where a, b, c are the real numbers and a≠0 Types of Quadratic Equations 1. Complete Quadratic Equation   ax 2  + bx + c = 0, where a ≠ 0, b ≠ 0, c ≠ 0 2. Pure Quadratic Equation    ax 2  = 0, where a ≠ 0, b = 0, c = 0 Roots of a Quadratic Equation Let x = α where α is a real number. If α satisfies the Quadratic Equation ax 2 + bx + c = 0 such that aα 2  + bα + c = 0, then  α is the root of the Quadratic Equation . As quadratic polynomials have degree 2, therefore Quadratic Equations can have two roots. So the zeros of quadratic polynomial p(x) =ax 2 +bx+c is same as the roots of the Quadratic Equa...

 

 

  CHAPTER 3 Pair of Linear Equations in two variables Linear Equations A Linear Equation is an equation of straight line. It is in the form of ax + by + c = 0 where a, b and c are the real numbers (a≠0 and b≠0) and x and y are the two variables, Here a and b are the coefficients and c is the constant of the equation. Pair of Linear Equations Two Linear Equations having two same variables are known as the pair of Linear Equations in two variables. a 1 x + b 1 y + c 1  = 0 a 2 x + b 2 y + c 2  = 0 Graphical method of solution of a pair of Linear Equations As we are showing two equations, there will be two lines on the graph. 1. If the two lines  intersect  each other at one particular point then that point will be the only solution of that pair of Linear Equations. It is said to be  a consistent  pair of equations. 2. If the two lines  coincide  with each other, then there will be infinite solutions as all the points on the line will be the sol...

  Chapter 2 Polynomials A polynomial is an expression consists of constants, variables and exponents. It’s mathematical form is- a­­­ n x n  + a n-1 x n-1  + a n-2 x n-2  + a 2 x 2  + a 1 x + a 0  = 0 where the (a i )’s are constant Degree of Polynomials Let P(y) is a polynomial in y, then the highest #ffffcc power of y in the P(y) will be the degree of polynomial P(y). Types of Polynomial according to their Degrees Type of polynomial Degree Form Constant 0 P(x) = a Linear 1 P(x) = ax + b Quadratic 2 P(x) = ax 2  + ax + b Cubic 3 P(x) = ax 3  + ax 2  + ax + b Bi-quadratic 4 P(x) = ax 4  + ax 3  + ax 2  + ax + b Value of Polynomial Let p(y) is a polynomial in y and α could be any real number, then the value calculated after putting the value y = α in p(y) is the final value of p(y) at y = α. This shows that p(y) at y = α is represented by p (α). Zero of a Polynomial If the value of p(y) at y = k is 0, that is p (k) = 0 then y = ...

  Chapter 1                                                                                        Real Numbers  Natural Numbers Non-negative counting numbers excluding zero are known as natural numbers. i.e. 5, 6, 7, 8,  ………. Whole numbers All non-negative counting numbers including zero are known as whole numbers. i.e. 0, 1, 2, 3, 4, 5, ……………. Integers All negative and non-negative numbers including zero altogether known as integers.  i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, ………….. The Fundamental Theorem of Arithmetic We can factorize each composite number as a product of some prime numbers and of course, this prime factorization of a natural number is unique as the order of the prime factors doesn’t matter. HCF of given numbers is the highest commo...

  Trigonometry To find the distances and heights we can use the mathematical techniques, which come under the  Trigonometry . It shows the relationship between the sides and the angles of the triangle. Generally, it is used in the case of a right angle triangle. Trigonometric Ratios In a right angle triangle, the ratio of its side and the acute angles is the trigonometric ratios of the angles. In this right angle triangle ∠B = 90°. If we take ∠A as acute angle then - AB is the base , as the side adjacent to the acute angle. BC is the perpendicular , as the side opposite to the acute angle. Ac is the hypotenuse , as the side opposite to the right angle. Trigonometric ratios with respect to  ∠A Ratio Formula Short form Value sin A P/H BC/AC cos A B/H AB/AC tan A P/B BC/AB cosec A H/P AC/BC sec A H/B AC/AB cot A B/P AB/BC Remark If we take ∠C as acute angle then BC will be base and AB will be perpendicular. Hypotenuse remains the same i.e. AC.So the ratios will be accor...