Pre Rmo Sample Questions

Pre-RMO Sample Questions

 Find the range of values of the abscissa of a point moving on the line y = 1 and always remaining in the interior of the triangle formed by the lines x = y, x-axis and the line 2x+y = 4.  f(x, y) = x2 + y2 + 2ax + 2by + c = 0 is a circle. If f(x, 0) has equal roots, each equal to 2 and f (0, y) = 0 has -4, -1 as its roots then find the centre of the circle.  A circle touches all the four sides of a quadrilateral ABCD. If AB = 6 and CD = 4 find the sum of the other two sides of the quadrilateral.  In  ABC, B = 90. Lengths of the medians drawn from A and C are 5 and 40 respectively. Find the length of the hypotenuse.  ‘n’ numbers of the sequence in the form 9, 99, 999 …, have the sum equal to 1111104. Find n.  If the sides of a triangle are given by m2 + n2, m2 – n2 and 2mn and if one of its angles is  + 90, find the other angles of the triangle.  If the equation cot4x – 2cosec2x + a2 = 0 has at least one solution, then find the possible positive integral value of a.  Find the number of terms in the series 3 + 3 + 33 + …, when the sum of the series is 39 + 133.  Find the reflection of the line 2x – y = 2 about the y-axis.  U = (a2cos2x + b2sin2x) + (a2sin2x + b2cos2x).If f(x) = U2, find the difference of the Max and Min values of f(x).  The number of ordered triples (a, b, c) of positive integers which satisfy the simultaneous equations ab + bc = 44, ac + bc = 33 is___________.