Question

1. find the equation of set of all points whose distance from (0,4) are 2/3 of their distance from the line y=9.
2.find the number of 5 digit telephone numbers having atleast one of the digit repeated.
3.how many four-letter words can be formed using the letter of the word 'INEFFECTIVE'.
4.R is the relation on the set of natural numbers N defined by a R b<=>a/b is an integral power of
a) is (a,a) belongs to R, for all a belonging to N.
b) (a,b) belongs to R => (b,a) belongs to R
c) (a,b) belongs to R, (b,c) belongs to R => (a.c) belongs to R.
sir please help us with these questions because our exam is on 25th February.

Answer 1

1. P(x1,y1)
    (distance from (0,4) )² = (2/3)² (distance from y=0 )²
    x1² + (y1 - 4)² = 4/9 * (y1 - 9)²
   9 x1² + 9 y1² -9* 8 y1 + 9*16 = 4 y1² - 4* 18 y1 + 4* 81
   9 x² + 5 y² - 180 = 0     replace (x1, y1) by (x,y)
   It is an ellipse

2.   5 digit tel numbers  :   a b c d e 

   b,c,d,e  digits can be from 0 to 9.     a cannot be 0.  
   Let us choose a digit for a, then for b, then for c and so on....

  Number of 5 digit numbers with no repetitions at all = 9 * 9 * 8 * 7 * 6
       = 27216 or  9 * C(9, 5)

   Total number of 5 digit numbers abcde:  9* 10*10*10*10 = 90, 000
    Number of numbers having at least one digit repeated =                
       90, 000 - 27, 216 = 62, 784

3)
     Distinct letters of the word:  INEFFECTIVE:  I, N, E, F, C, T, V, 
     Number of 4 letter words:  P(7, 4) = 7!/3! = 7*6*5*4 = 840
     (assuming that each letter could be used only once).
     If each letter could be used any number of times, then:  7⁴

4)
     a R b    < = >  a/b  is an integer 
     (a , a) belongs to R   as    a/a = 1 (an integer):   TRUE

     Let  a R b, a≠b  ,  then  a/b = integer,   then b/a is not an integer. so (b,a) ∉ R
      False.

     (a, b) ∈ R,  and (b ,c) ∈ R.   So a/b is integer and  b/c is an integer.
            hence,   a/b * b/c = a/c is an integer.   
             hence,   (a, c) ∈ R      TRUE