The largest area of the trapezium inscribed in a semi-circle or radius `R ,` if the lower base is on the diameter, is (a) `(3sqrt(3))/4R^2` (b) `(sqrt(3))/2R^2` `(3sqrt(3))/8R^2` (d) `R^2`

A rectangle is inscribed in an equilateral triangle of side length 2a units. The maximum area of this rectangle can be (a) sqrt(3)a^2 (b) (sqrt(3)a^2)/4 a^2 (d) (sqrt(3)a^2)/2

Find the derivatives w.r.t. x : log(sqrt(x-2)+sqrt(x-3))

The three sides of a trapezium are equal, each being 8cm. The area of the trapezium, where it is maximum, is (a) 24sqrt(3) c m^2 (b) 48sqrt(3)c m^2 (c)72 sqrt(3)c m^2 (d) none of these

Three equal circles each of radius r touch one another. The radius of the circle touching all the three given circles internally is (2+sqrt(3))r (b) ((2+sqrt(3)))/(sqrt(3))r ((2-sqrt(3)))/(sqrt(3))r (d) (2-sqrt(3))r

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6sqrt3r .

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is (2R)/(sqrt(3))

If a circle of radius r is touching the lines x^2-4x y+y^2=0 in the first quadrant at points Aa n dB , then the area of triangle O A B(O being the origin) is (a) 3sqrt(3)(r^2)/4 (b) (sqrt(3)r^2)/4 (c) (3r^2)/4 (d) r^2

ABC is an isosceles triangle inscribed in a circle of radius rdot If A B=A C and h is the altitude from A to B C , then triangle A B C has perimeter P=2(sqrt(2h r-h^2)+sqrt(2h r)) and area A= ____________ and = __________ and also ("lim")_(h vec 0) A/(P^3)=______

Find the area of the region in the first quadrant enclosed by x-axis, line x = sqrt3y and the circle r^(2)+ y^(2) = 4 .