The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. Thecircles are tangent to one another and to the sides of therectangle as shown in the diagram. The ratio of the longerdimension of the rectangle to the shorter dimension can be written as `1/2(sqrtp - q)`, where p and q are positive integers. Find p + q.

Suppose that a parabola y=ax^(2)+bx+c where a>0 and (a+b+c) is an integer has vertex ((1)/(4),-(9)/(8)). If the minimum positive value of 'a' can be written as (p)/(q) where p and q are relatively prime positive integers, then find (p+q).

Two point charges q AND 2q are placed at (a,0) and (0,a) A point charge q^(1) is placed at a P on the quarter circle of radius a as shown in the diagram so that the electric field at the origin becomes zero

India and Australia play a series of 7 one-day matches. Each team has equal probability of winning a match. No match ends in a draw. If the probability that India wins atleast three consecutive matches can be expressed as (p)/(q) where p and q are relatively prime positive integers. Find the unit digit of p.

Circles C_1 and C_2 are externally tangent and they are both internally tangent to the circle C_3. The radii of C_1 and C_2 are 4 and 10, respectively and the centres of the three circles are collinear. A chord of C_3 is also a common internal tangent of C_1 and C_2. Given that the length of the chord is (msqrtn)/p where m,n and p are positive integers, m and p are relatively prime and n is not divisible by the square of any prime, find the value of (m + n + p).

Show that 1.272727 …..= 1. bar(2.7) can be expressed in the form (p)/(q) where p and q are integer and q ne 0

In the given diagram triangleABC is an isosceles right angled triangle, in which a rectangle is inscibed in such a way that the length of the rectangle in twice of breadth. Q and R lie on the hypotenuse and P, S lie on the two different smaller sides of the triangle. what is the ratio of the areas of the rectangle and that of triangle?

In the diagram as shown, a circle is drawn with centre C(1,1) and radius 1 and a line L. The line L is tangent to the circle at Q. Further L meets the y-axis at R and the x-axis at P in such a way that the angle OPQ equals theta where 0 lt theta lt (pi)/(2) . Coordinate of Q is