Question

1. The mean and median of the data a,b,c are 50 and 35 respectively, where a 2. if sin theta = 3/5 belongs to 2nd quadrant , then cos theta ?
3.there are n Arithmetic mean's between 11 and 53 such that each each of them is an integer. how many distinct arithmetic progressions are possible from above data ?
4.four bells toll at intervals of 10 seconds , 15 seconds, 20 seconds , 30 seconds respectively. if they together at 10.00 A.M. at what time will they toll together for the first time after 10.00 A.M. ?
5. if a-b , b-c are roots of general form of quadratic equation , then (a-b)(b-c)/(c-a) ?

Answer 1

1)  let  a >= b >= c
       mean =  (a+b+c) /3  =  50
               a+b+c = 150
   median is the middle one ,  b = 35
     hence,    a + c =  115 .
   if we know  one quantity  we can find the other quantity.
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2.    Sin Ф  =  3/5      2nd quadrant.
         Cos² Ф = 1 -  sin² Ф = 1 - 9/25 =  16/25
           cos Ф =  - 4/5          we put a minus sign because in 2nd quadrant Cosine is negative.
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3.    AP  series:    11 , a₁ , a₂ , a₃ ..... a_n  , 53
          number of terms = n + 2                first term = 11
         common difference :  d
           53 = 11 + (n+1) d

            (n+1) d = 42  =  7 * 6 = 21 * 2  =  3 * 14  =  1 * 42

So there are 8 combinations possible.  then there could be 8  different series,

  all terms are integers,  means that  common difference d  is  an integer.
         n is an integer .
 
case 1:  d = 1   and  n+1 = 42    so n =41,          AP:  11,12,13,14....
case 2:  d = 42  and  n+1 = 1   so  n = 0,        AP:  11, 53, .....
case 3: d = 2  and n+1 = 21  , n = 20,      AP:  11,13,15,..51, 53
case 4: d = 21  and  n +1 = 2,  n = 1,    AP  :  11,  32,  53 , ....
case 5:  d = 3  and  n+1 = 14  , n = 13,    AP :    11,14, 17,...., 50, 53
case 6:  d= 14  and n+1 = 3      n = 2,    AP = 11, 25, 39, 53, ...
case 7 : d = 7  and  n+1 = 6      so n = 5,    AP =  11,18,25,32,39,46, 53,...
case 8:  d = 6,  n+1 = 7,...        so n = 6,    AP :  11, 17,23,29,35,41,47,53,
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4.    four bells: toll at  intervals of 10, 15 , 20 , 30 seconds ....
       the interval between two times that they will toll all togehter will be the LCM of the time intervals of each.

   LCM of 10, 15 , 20 , 30 seconds =  2 * 5 * 3  * 2 = 60
       So it is one minute.
   They will all together at  10 : 01 AM, 
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5.        a x² + b x + c = 0  is the general form of quadratic equation.
           sum of roots :  - b/a        product :  c/a    --- (3)
        
  [x - (a - b) ] [ x - (b - c) ] = 0  is the quadratic equation.
   =>     x² + (c-a) x + (a-b)(b-c) = 0        is the quadratic equation.
     
   It means that  product of roots is  (a - b) (b - c)
         sum of the roots = a - b + b - c =   - (c - a)
      
           - b / a  = - (c - a)    =>  b = a (c - a)            ---- (1)
           (a - b) (b - c)  = c/a
           a b - a c - b² + b c = c/a    --- (2)

 given (a-b)(b-c) / (c-a)  = - product / sum  =  - c / -b = c/b    ,,,,,   by using (3)