Puzzle 49

A rectangular sheet of paper ABCD is with width AD=1 and length AB more than the width but not more than double of it. The paper is folded through vertex A so that the edge along AD falls onto AB. Without unfolding, the paper is folded again along a line through B so that CB now lies on AB. Now the result is a triangular piece of paper. There is a region on this triangular sheet that has four layers of paper. Find the area of that region in terms of the length x, of the rectangle.

Puzzle 48 - Roots of a cubic

From Rolle’s theorem, we can conclude that between every pair of distinct roots of a polynomial P(x), there lies a root of its derivative P’(x). If P’(x) keeps its sign unchanged between any two distinct real numbers, then it is possible that P(x) does not have a root between them and hence may keep the same sign between those real numbers. Further if P(x) has two equal roots, then it will have a common root with its derivative. Consider with being an integer between and both inclusive, then find the number of values of if has to have
(i) one real root and two complex roots, (ii) only two equal roots

Puzzle 47

Semi circles are drawn on each side of a square. An elastic band is stretched tightly around the shape formed (see the figure). Find the length of the elastic band in terms of the length x, of the side of the square.

Puzzle 46 - Squares

Square ABCD, of side length 26, is centered at O. With D as center, draw another square EFGH with side length 22, none of its sides parallel to those of ABCD, so that OE is kept perpendicular to DA. Find the area of the square ABCD that is not inside the square EFGH.

Puzzle 45 - Similar Triangles

Let A is the set of all triangles in a plane. Define a function as implies that triangle is similar to with the dilation constant or scale factor ‘a’. Find all the possible triangles and such that

a. Lengths of the sides of and are integers
b. The perimeter and area of are same.

Puzzle 44

W(a, b) is the centre of the circle. TE is a tangent to the circle at E, which lies on the y-axis. The circle touches the x-axis at D. The equation of TE is. Calculate the values of a and b.

Puzzle 43 - Locus of midpoints

Two circles with different radii are given in the plane touching each other externally at . Consider points on and on , such that subtends right angle at . Find the locus of midpoints of all such segments .

Puzzle 42

Let is a function such that . Find the value of f(0).

Puzzlee 41 - Polynomials + Trigonometry

If are three real roots of the equation , show that .

Puzzle 40 - Matrices

Let and are two different matrices such that and . Does the matrix have inverse?

Puzzle 39 - 2012 Integration

Find the value of

Puzzle 38 - Point inside the Circle

Find the values of for which the point lies in the larger segment of the circle made by the chord whose equation is x + y – 2 = 0

Puzzle 37 - Extreme Values of Integral

Define the function , find the maximum and minimum values of f(x) in the interval

Puzzle 36 - Functional Equation

Find the function f(x) if f(x) is a non constant and satisfies the equation

Puzzle 35 - Complex system of equations

Find the complex roots of the system of equations

Puzzle 34 - Analytical Geometry

Vertex A of the triangle ABC is (14, 8). If the orthocenter of the triangle is H(8, 1) and the midpoint of the side BC is (3, -2), find the coordinates of the points B and C.

Puzzle 33 - Number Grid

In the grid shown, a number is to be placed in each smaller square so that the product of all the three numbers in any row, column, or diagonal is the same positive number.  Find the sum of x and y.

Puzzle 32 - Trigonometry

If and find the value of