The minimum possible time period of a disc of radius R hanging from a wall through nail P is
(A) `2 pi sqrt((R)/(g))`
(B) `2 pi sqrt((2R)/(g))`
(C) `2 pi sqrt((sqrt(2)R)/(g))`
(D) None

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

The ratio of the areas of a circle and an equilateral triangle whose diameter and a side are respectively equal, is (a) pi\ :sqrt(2) (b) pi\ :sqrt(3) (c) sqrt(3)\ :pi (d) sqrt(2)\ :pi

The minimum value of cos theta+sintheta+ 2/(sin2theta) for theta in (0, pi/2) is (A) 2+sqrt(2) (B) 2 (C) 1+sqrt(2) (D) 2sqrt(2)

The value of the definite integral int_0^(pi/2)sqrt(tanx)dx is sqrt(2)pi (b) pi/(sqrt(2)) 2sqrt(2)pi (d) pi/(2sqrt(2))

If the area of a square is same as the area of the circle, then the ratio of their perimeters, in terms of pi , is (a) pi\ :sqrt(3) (b) 2\ :sqrt(pi) (c) 3\ :pi (d) pi\ :sqrt(2)

The solution for x of the equation int_(sqrt(2))^x(dt)/(tsqrt(t^2-1))=pi/2 is pi (b) (sqrt(3))/2 (c) 2sqrt(2) (d) none of these

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

The area of a circular path of uniform width h surrounding a circular region of radius r is (A) pi (h+2r)r (B) pi h(2r+h)dot (C) pi (h+r) r (D) pi (h+r) h

A satellite of mass m orbits the earth in an elliptical orbit having aphelion distance r_(a) and perihelion distance r_(p) . The period of the orbit is T. The semi-major and semi-minor axes of the ellipse are (r_(a) + r_(p))/(2) and sqrt(r_(p)r_(a)) respectively. The angular momentum of the satellite is: (A) (2m pi (r_(B)+r_(p)) sqrt(r_(a) r_(p)))/T (B) (m pi (r_(a)+r_(p)) sqrt(r_(a) r_(p)))/(2T) (C) (m pi (r_(a)+r_(p)) sqrt(r_(a)r_(p)))/(4T) (D) (mpi (r_(a)+r_(p)) sqrt(r_(a)r_(p)))/T

If n=4\ then B_n is equal to : R^2((4-pi))/2 (b) (R^2(4-pisqrt(2)))/2 R^2((4sqrt(2)-pi))/2 (d) none of these