r

Let ABC be a triangle with incentre I and inradius r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB, respectively, If r_(2)" and "r_(3) are the radii of circles inscribed in the quadrilaterls AFIE, BDIF and CEID respectively, then prove that r_(1)/(r-r_(1))+r_(2)/(r-r_(2))+r_(3)/(r-r_(3))=(r_(1)r_(2)r_(3))/((r-r_(1))(r-r_(2))(r-r_(3)))

Let there be a spherical symmetric charge density varying as p(r )=p_(0)(r )/(R ) upto r = R and rho(r )=0 for r gt R , where r is the distance from the origin. The electric field at on a distance r(r lt R) from the origin is given by -

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

Two circle of radii R and r ,R > r touch each other externally then the radius of circle which touches both of them externally and also their direct common tangent, is R (b) (R r)/(R+r) (R r)/((sqrt(R)+sqrt(r))^2) (d) (R r)/((sqrt(R)-sqrt(r))^2)

Two resistance R_(1) and R_(2) are made of different material. The temperature coefficient of the material of R_(1) is alpha and of the material of R_(2) is -beta . Then resistance of the series combination of R_(1) and R_(2) will not change with temperature, if R_(1)//R_(2) will not change with temperature if R_(1)//R_(2) equals

Two resistance R_(1) and R_(2) are made of different material. The temperature coefficient of the material of R_(1) is alpha and of the material of R_(2) is -beta . Then resistance of the series combination of R_(1) and R_(2) will not change with temperature, if R_(1)//R_(2) will not change with temperature if R_(1)//R_(2) equals

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

Two resistors of resistances R_(1)=100 pm 3 ohm and R_(2)=200 pm 4 ohm are connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation R=R_(1)+R_(2) and for (b) 1/(R')=1/R_(1)+1/R_(2) and (Delta R')/R'^(2)=(Delta R_(1))/R_(1)^(2)+(Delta R_(2))/R_(2)^(2)

The mass density of a spherical body is given by rho(r)=k/r for r le R and rho (r)=0 for r > R , where r is the distance from the centre. The correct graph that describes qualitatively the acceleration, a, of a test particle as a function of r is :

Three circles with radius r_(1), r_(2), r_(3) touch one another externally. The tangents at their point of contact meet at a point whose distance from a point of contact is 2 . The value of ((r_(1)r_(2)r_(3))/(r_(1)+r_(2)+r_(3))) is equal to