Let `G_1, G_2a n dG_3` be the centroids of the triangular faces `O B C ,O C Aa n dO A B ,` respectively, of a tetrahedron `O A B Cdot` If `V_1` denotes the volumes of the tetrahedron `O A B Ca n dV_2` that of the parallelepiped with `O G_1,O G_2a n dO G_3` as three concurrent edges, then prove that `4V_1=9V_2dot`

O is the circumcenter of A B Ca n dR_1, R_2, R_3 are respectively, the radii of the circumcircles of the triangle O B C ,O C A and OAB. Prove that a/(R_1)+b/(R_2)+c/(R_3) =(a b c)/(R_3)

Let vec O A= vec a , hat O B=10 vec a+2 vec ba n d vec O C= vec b ,w h e r eO ,Aa n dC are non-collinear points. Let p denotes the areaof quadrilateral O A C B , and let q denote the area of parallelogram with O Aa n dO C as adjacent sides. If p=k q , then find kdot

A plane passing through (1,1,1) cuts positive direction of coordinates axes at A ,Ba n dC , then the volume of tetrahedron O A B C satisfies a. Vlt=9/2 b. Vgeq9/2 c. V=9/2 d. none of these

O A B C is regular tetrahedron in which D is the circumcentre of O A B and E is the midpoint of edge A Cdot Prove that D E is equal to half the edge of tetrahedron.

(b) What is the total charge of 1.6g O^(2-) ion ?

O A ,O Ba n dO C ,w i t hO as the origin, are three mutually perpendicular lines whose direction cosines are l_rm_ra n dn_r(r=1,2a n d3)dot If the projection of O Aa n dO B on the plane z=0 make angles varphi_1a n dvarphi_2, respectively, with the x-axis, prove that tan(varphi_1-varphi_2)=+-n_3//n_1n_2dot

Let A_r ,r=1,2,3, , be the points on the number line such that O A_1,O A_2,O A_3dot are in G P , where O is the origin, and the common ratio of the G P be a positive proper fraction. Let M , be the middle point of the line segment A_r A_(r+1.) Then the value of sum_(r=1)^ooO M_r is equal to (a) (O A_1(O S A_1-O A_2))/(2(O A_1+O A_2)) (b) (O A_1(O A_2+O A_1))/(2(O A_1-O A_2) (c) (O A_1)/(2(O A_1-O A_2)) (d) oo

Find the value of lambda so that the points P ,Q ,R and S on the sides O A ,O B ,OC and A B , respectively, of a regular tetrahedron O A B C are coplanar. It is given that (O P)/(O A)=1/3,(O Q)/(O B)=1/2,(O R)/(O C)=1/3 and (O S)/(A B)=lambdadot (A) lambda=1/2 (B) lambda=-1 (C) lambda=0 (D) for no value of lambda

Three closed vessels A,B and C at the same temperature T and contain gases which obey the Maxwellian distribution of velocities. Vessel A contains only O_2 , B only N_2 and C a mixture of equal quantities of O_2 and N_2 . IF the average velocity of the O_2 molecules in vessel A is v_1 that of the N_2 molecules in vessel B is v_2 , the average velocity of the O_2 molecules in vessel C is

The equilibrium constant (K_c) for the reaction N_2(g) + O_2(g) rarr 2NO (g) at temperature T is 4 xx 10^-4 . The value of K_c for the reaction NO(g) rarr 1/2 N_2(g) + 1/2 O_2(g) at the same temperature is