1.(a). If \(a,b,c\) are positive reals and \(a+b+c=50\) and \(3a+b-c=70\). If \(x=5a+4b+2c\), find the range of the values of \(x\).
(b). The sides of satisfy the equation :
Prove that is the arithmetic mean of .
2.In an isosceles , . The bisector of meets the side at . The line perpendicular to through meets at and produced at . Perpendiculars from and to are and respectively. If units, find the length of .
3.(a). Two circles with centres and and radii and respectively intersect each other at and and units. Chord is drawn to the bigger circle to cut it at and the smaller circle at such that is the midpoint of . Find the length of .
(b). Find the greatest common divisor of the numbers where
4.(a). A book contained problems on Algebra, Geometry and Number Theory. Mahadevan solved some of them. After checking the answers, we found that he answered correctly problems in Algebra, in Geometry and in Number Theory. He further found that he solved correctly of problems in Algebra and Number Theory put together, questions in Geometry and Number Theory put together. What is the percentage of correctly answered questions in all the three subjects?
(b). Find all pairs of positive integers such that .
5. are positive real numbers. Find the minimum value of
6.(a). Show that among any whole numbers, one can find two numbers such that their difference is divisible by .
(b). Show that for any natural number , there is a positive integer all of whose digits are or and is divisible by .