Quadratic Equation ⚫ What is Quadratic Equation ? ⚪ An equation in which the highest power of the unknown quantity is two is called quadratic equation. ⚫ Types of Roots of Quadratic Equation ⚪ The equation has real and distinct (different) roots if and only if D > 0 ⚪ The equation has real and equal roots if and only if D = b 2 -4ac = 0 ⚪ The equation has complex roots of the form α ± β , α , β ≠ 0 if and only if D = b 2 -4ac < 0 ⚪ The equation has rational roots if and if a, b ,c ∈ Q , and D = b 2 -4ac is a perfect square ⚪ The equation has (unequal) irrational roots if and only if D = b 2 -4ac > 0 and not a perfect square. In this case if p + √ q is an irrational root , then p - √ q is also a root ⚪ If α + iβ is a root of quadratic equation ,then α - iβ is also a root ⚫ Relation between roots and coefficients : ...

Logarithm The Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number 1. When base is 'e' , then the logarithmic function is called natural logarithmic function 2. When base is '10' , then the logarithmic function is called common logarithmic function Characteristic and Mantissa 1. The integer part of a Logarithm is called the characteristic and the fractional part is called mantissa. Log 10 N = integer(characteristic) + fraction(mantissa) 2. The mantissa part of a log of a number is always kept positive. 3. The characteristic of Log 10 N be n , then the number of digits in N is (n+1). Properties of Logarithms 1. log a a = 1 , log a 1 = 0 2. log a b. log b a = 1 ⇒ 3 log c a = log b a. log c b or 4. log a (mn) = log a m + log a n 5. 6. log a m n = n log a m 7. a log a m = m 8. 9. Practice set 1 Practice set 2

Permutation and Combination ⚪ What is Factorial ? ⚫ Let n be a positive integer. Then , the continued product of first n natural numbers is called factorial n , to be denoted by n! n! = n(n-1)(n-2)........3.2.1 ⚪ What is Permutation ? ⚫ The ways of arranging or selecting a smaller or an equal number of persons or objects at a time from a given group of persons or objects with due regard being paid to the order of arrangement or selection are called permutation for example: Three different things a, b and c are given , then different which can be made by taking two things from three given things are ab , ac ,bc, ba , ca , cb Therefore, the number of permutation is 6 ⚪ What is Combination ? ⚫ Each of the different groups or selections which can be formed by taking somne or all of a number of objects , irrespective of their arrangements , is called combination. The number of all combinations on n things , taken r at a time...

Binomial Theorem Question no. 31 The first 3 terms in the expansion of (1+ax) n are 1 , 6x , 16x 2 . then the value of a and n are respectively looks_one 2 and 9 looks_two 3 and 2 looks_3 2/3 and 9 looks_4 3/2 and 6 Answer Option looks_3 2/3 and 9 Solution Solution : First term = n C 0 (1) n (ax) 0 = 1 Second term = n C 1 (1) n-1 (ax) 1 = nax nax = 6x ⇒ na = 6 ⇒ a = 6/n Third term = n C 2 (1) n-2 (ax) 2 Question no. 32 Coefficients of x in the expansion of looks_one 9a 2 looks_two 10a 3 looks_3 10a 2 looks_4 10a Answer Option looks_two 10a 3 Solution Solution : Question no. 33 In the expansion of , the coefficient of x -10 will be looks_one 12a 11 looks_two 12b 11 a looks_3 12a 11 b looks_4 12a 11 b 11 Answer Option looks_3 12a 11 b Solution Solution : Question no. 34 The ratio of the coefficient of terms x n-r a r and x r a n-r in the bi...

Binomial Theorem What is Binomial Expression ? An algebraic expression consisting of two terms with +ve or -ve sign between them is called a binomial expression. For example : ( a + b) , ( 2x -3y ) Binomial theorem for positive integral index : Here n C 0 , n C 1 , n C 2 ,........ n C n are called binomial coefficients General term from the begining of the binomial expansion : The general term of the expansion is (r+1) th term usually denoted by T r+1 = n C r a n-r b r General term from the end of the binomial expansion : The general r th term from end of the expansion is usually denoted by Middle term : Middle term depends upon the value on n 1. When n is even : the total number of terms in the expansion of ( a + b ) n is n + 1 (odd) , So there is only one middle term : 2. When n is odd : the total number of terms in the expansion of ( a + b ) n is n + 1 (even) , So there is two middle term : Total number of terms ...