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` (c)`(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 sqrt(3)a^2 (b) (sqrt(3)a^2)/4 a^2 (d) (sqrt(3)a^2)/2

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 (a) (2+sqrt(3))r (b) ((2+sqrt(3)))/(sqrt(3))r (c) ((2-sqrt(3)))/(sqrt(3))r (d) (2-sqrt(3))r

The length of the chord of the parabola y^2=x which is bisected at the point (2, 1) is (a) 2sqrt(3) (b) 4sqrt(3) (c) 3sqrt(2) (d) 2sqrt(5)

An isosceles triangle A B C is inscribed in a circle x^2+y^2=a^2 with the vertex A at (a ,0) and the base angle Ba n dC each equal 75^0 . Then the coordinates of an endpoint of the base are. (a) (-(sqrt(3a))/2, a/2) (b) (-(sqrt(3a))/2, a) (c) (a/2,(sqrt(3a))/2) (d) ((sqrt(3a))/2,-a/2)

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

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

P is a point on the line y+2x=1, and Q and R two points on the line 3y+6x=6 such that triangle P Q R is an equilateral triangle. The length of the side of the triangle is (a) 2/(sqrt(5)) (b) 3/(sqrt(5)) (c) 4/(sqrt(5)) (d) none of these

Radius of the circle that passes through the origin and touches the parabola y^2=4a x at the point (a ,2a) is (a) 5/(sqrt(2))a (b) 2sqrt(2)a (c) sqrt(5/2)a (d) 3/(sqrt(2))a

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 also ("lim")_(h->0)A/(P^3)=______