MTH202

Question 1; Mark: 10Use Mathematical Induction to prove that.
Sol:

Induction given statement is true for all set of whole numbers.
Question 2; Mark: 5
If 3x+1 is odd then x is even (prove by contradiction).

Sol: Suppose 3x + 1 is odd and x is not evenx is not even à x is odd (x is either even or odd but not both)à x = 2k + 1 ( x is not divisible by 2)à 3x + 1 = 3 ( 2k +1) + 1 (substitute 2k +1 from x)
3x +1 = 6k + 3 + 1
= 6k + 4
= 2 ( 3k + 2) (Divisible by 2)
à 3x + 1 is even which is contradiction
Therefore 3x + 1 is odd then x is even

Question 3; Marks: 5

Find the number of arrangements of balls having colors Red, Yellow, Blue Green and Pink such that Red and Green balls must place next to each other.

Sol:

Red, Yellow, Blue, Green, Pink

Total numbers of balls = 5

Considering two balls Red and Green as a single ball ( due to restriction), we have to make the arrangements of 4 balls taken all together

No. of Arrangements = 4!

= 4 × 3 × 2 × 1= 24

But two balls (restricted) permuted among themselves in two ways

i.e. 2! = 2 × 1 = 2 ways

Therefore total numbers of arrangements = 4! × 2!= 24 × 2
= 48