Consider square `A B C D` of side length 1. Let `P` be the set of all segments of length 1 with endpoints on the adjacent sides of square `A B C D` . The midpoints of segments in `P` enclose a region with area `Adot` The value of `A` is

Line segments A C and B D are diameters of the circle of radius one. If /_B D C=60^0 , the length of line segment A B is_________

Let G be the centroid of the DeltaABC , whose sides are of lengths a,b,c. If P be a point in the plane of triangleABC , such that PA=1,PB=3, PC=4 and PG=2 , then the value of a^(2)+b^(2)+c^(2) is

If a,b,c are in H.P , b,c,d are in G.P and c,d,e are in A.P. , then the value of e is

Let ABCD is a square with sides of unit length. Points E and F are taken om sides AB and AD respectively so that AE= AF. Let P be a point inside the square ABCD.The maximum possible area of quadrilateral CDFE is-

Consider the set A of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1 . Then

Let a,b,c be the sides of a triangle ABC, a=2c,cos(A-C)+cos B=1. then the value of C is

Let A B C be a triangle with /_B=90^0 . Let AD be the bisector of /_A with D on BC. Suppose AC=6cm and the area of the triangle ADC is 10c m^2dot Find the length of BD.

Let S_1, S_2, be squares such that for each ngeq1, the length of a side of S_n equals the length of a diagonal of S_(n+1)dot If the length of a side of S_1i s10c m , then for which of the following value of n is the area of S_n less than 1 sq. cm? a. 5 b. 7 c. 9 d. 10

Consider a triangle A B C and let a , ba n dc denote the lengths of the sides opposite to vertices A , B ,a n dC , respectively. Suppose a=6,b=10 , and the area of triangle is 15sqrt(3)dot If /_A C B is obtuse and if r denotes the radius of the incircle of the triangle, then the value of r^2 is

Let P be any point on any directrix of an ellipse. Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at (a)some point on the major axis depending upon the position of point Pdot (b)the midpoint of the line segment joining the center to the corresponding focus (c)the corresponding focus (d)none of these