IPM Indore 2019

A chord is drawn inside a circle, such that the length of the chord is equal to the radius of the circle. Now, two circles are drawn, one on each side of the chord, each touching the chord at its midpoint and the original circle. Let k be the ratio of the areas of the bigger inscribed circle and the smaller inscribed circle, then k equals

Points P, Q, R and S are taken on sides AB, BC, CD and DA of square ABCD respectively, so that AP : PB = BQ : QC = CR : RD = DS : SA = 1 : n . Then the ratio of the area of PQRS to the area of ABCD is

On a circular path of radius 6 m a boy starts from a point A on the circumference and walks along a chord AB of length 3 m. He then walks along another chord BC of length 2 m to reach point C. The point B lies on the minor arc AC. The distance between point C from point A is

The area enclosed by the curve 2|x| + 3|y| = 6 is

Two points on a ground are 1 m apart. If a cow moves in the field in such a way that it's distance from the two points is always in ratio 3: 2 then

Given that $$\cos x + \cos y = 1$$, the range of $$\sin x - \sin y$$ is

If $$\sin \theta + \cos \theta = m$$, then $$\sin^{6} \theta + \cos^{6}\theta$$ equals

If inverse of the matrix $$\begin{bmatrix}2 & -0.5 \\-1 & x \end{bmatrix}$$ is $$\begin{bmatrix}1 & 1 \\2 & 4 \end{bmatrix}$$, then the value of x is

The function $$f(x) = \frac{x^{3} + 5x^{2} - 8x}{3}$$ is

For a > b > c > 0, the minimum value of the function f(x) = |x - a| + |x - b| + |x - c| is