upsc mathematics optional paper

T uple  =  A tuple is a finite ordered list of elements sequence of n-elements where n is a non-negative integer. Set of n-tuples of Real Numbers =>   R n Note:  O-tuple = Empty space  Eg. (0,0,0,0...........) A   particular n tuple in  R n  U =  (  a1,a2,a3,..................an  ) is called a point or vector. [   where, (a1,a2,a3,.............an) is called  Scalar  & U is called  Vector. ]                                            or, U = [a¡] ,where i is called as components/entries/co-ordinates/elements. Note:  Elements of  R =  S calar . 1.) Two vectors U & V are equal i.e U=V if they have same number of components and the corresponding components are equal .  e.g.  U =[a1,a2,a3]  V =[b1,b2,b3]  If U =V then    a1=b1...

When we have to calculate the factorial of a non-negative integer n, then it is denoted by n!, is the product of all positive integer less than or equal to n . so, 𝑛 ! = 𝑛 ∗ ( 𝑛 − 1 ) ∗ ( 𝑛 − 2 ) ∗ ( 𝑛 − 3 ) ∗ . . .3 ∗ 2 ∗ 1 n ! = n ∗ ( n − 1 ) ∗ ( n − 2 ) ∗ ( n − 3 ) ∗ . . .3 ∗ 2 ∗ 1 However, the recursive definition of factorial is of more use in this proof. 𝑛 ! = { 1 𝑛 ∗ ( 𝑛 − 1 ) ! 𝑛 = 0 𝑛 > 0 n ! = { 1 n = 0 n ∗ ( n − 1 ) ! n > 0 The recursive definition of  Factorial  leads to one interesting way of expressing factorial numbers . 𝑛 ! = ( 𝑛 + 1 ) ! ( 𝑛 + 1 ) n ! = ( n + 1 ) ! ( n + 1 ) This is valid since, as we expand  ( 𝑛 + 1 ) ! ( n + 1 ) !  from the recursive definition, we can cancel  ( 𝑛 + 1 ) ( n + 1 )  term from both numerator and denominator to get  𝑛 ! n ! . Or we can even calculate factorial in the numerator and then evaluate the division For example, 5 ! = 6 ! 6 = 720 6 = 4 ! = 5 ! 5 = 120 5 ...