IPM Indore 2019


For the following questions answer them individually

Question 31

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

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Question 32

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

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Question 33

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

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Question 34

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

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Question 35

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

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Question 36

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

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Question 37

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

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Question 38

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

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Question 39

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

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Question 40

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

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