To find all the natural numbers which gets reverse when multiplied by 9

To find all the natural numbers which gets reverse when multiplied by 9

1089 = 33×33

1089 ×9 = 9801 = 99×99

1089 is the smallest natural number which gets reverse when multiplied by 9.

There is interesting pattern for all the numbers.

 

1089 × 9 = 9801

10989 ×9 = 98901

109989 × 9 = 989901

1099989×9 = 9899901

10999989×9 = 98999901 

10891089×9 = 98019801

 ..............  

If number of digits in number is not  multiple of 4 then only one number is possible.

10999…(n-4) times 89 ×9 =98999…..(n-4) times 01

 Where n is number of digits.

If number of digits in the number  is multiple of 4 then 1 more number is possible.

10891089 × 9 = 98019801

10891089…….1089 × 9 = 98019801……….9801

To find N natural numbers not necessarily distinct such that their sum is equal to their product

To find N natural numbers not necessarily distinct such that their sum is equal to their product

2+2 =2×2

1+2+3=1×2×3

1+1+2=4=1×1×2×4

1+1+1+2+5=1×1×1×2×5

1+1+1+1+2+6=1×1×1×1×2×6

In general,

1+1+1+…..{(N-2) times}+2+N = 1×1×……{(N-2) times}×2×N

 

The above formula was published in Ramanujan Mathematical Society newsletter in June-2014 under problem and solution section.

 

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Linear Equation in Two Variables - Class 10 -MCQ - K Bhanumoorthy

Chapter Linear equations.     Class X, CBSE

1)    The given equation is x +2y = 6, the line meets the X axis at

a)  (6, 0)     b)  (0, 6)   c)   ( 3, 0 )

2)    The equation Y axis is   

a)  Y =0     b)  X =0   , c)    XY = 0

3)    The given equations  x + y = 2  ,  - x + y = 4; intersect at 

a) (3, 1)  b)    ( - 1, 3 )   c ) ( - 1, -3 )

4)    The given equations x + 2 y   = 3  ,   2x + 4 y = 5  are 

a)  Parallel    b)   Not parallel   c )  many  solutions

5)    The given  equations  x + y= 5, 2x + 2y = 5 are

                       a) Inconsistant       b) consistant       c) One solution 

6)    The line  2x + y  - 4 = 0,   meets the Y axis at

                       a) (4, 0)        b) ( 0, 4 )               c  ) ( 2,4)  

7)    The equations  3x + 2y = 2, 6x + 4y = 4 ,have

                        a) No solution   b) one solution c) infinitely many 

8)    The lines , x – 2y -2 = 0,  2x + y – 4 = 0  have

                       a) One solution b) No solution c) many solutions.

9)    The given equation is 2x – y – 2 = 0 , the line meets the Y axis at

                     a) ( - 2, 0 )    b ) ( 2, 0 )   c ( 0 , - 2 ) 

10)              The value of  k,  for which  kx + 3 y -  ( k – 3 ) = 0, and

                        16x  + ( 8 + K )y – k = 0 have infinite solutions

                         a)  2                   b ) 4             c)  8

Answers : 1) a , 2 ) b , 3)  b ,   4 ) a ,  5 ) a ,  6 )  b ,  7)  c ,  8)  a ,  9)  c ,   10 )  b

This MCQ paper is designed by Shri K Bhanumoorthy, a retired KV Prinicipal.

Mr. Bhanumoorthy is a winner of Ramanujan Award and Life time Achievement Award for his service in the field of mathematics. He lives in Banglore.