Well ,let's make mincemeats of these --- :)
1 > Do try to find the remainder when 21990 is divided by 1990 .
I have found a way without using binomial but , still it would good to see the other ways too ---- maybe an easier way is there , and I am not sure of my way :)
2 > For each natural number K , let Sk denote the the circle with radius K centimeters and center at the origin O . On the circle Sk , a particle moves K centimeters in the counter - clockwise direction . After completing its motion on Sk , it moves to Sk+1 in the radial direction . The motion of the particle starts at ( K , 0 ) .
The motion of the particle continues in this manner . If the particle , for the first time , crosses the starting position K times on the circle Sn , then locate that circle .
3 > In a certain test , there are n questions . In this test 2n-i students give wrong answers to at least i question ( 1 ≤ i < n ) .
If total number of wrong answers given is 2047 , then find number of questions .
4 > Show that any positive integral power ( greater than 1 ) of a positive integer mr , is the sum of m consecutive odd positive integers . Find the first such integer in terms of m .
5 > Find the number of words that can be made from the letters of the word “ TRIANGLE “ such that the realative order of vowels do not change ?
1