If `P` is any arbitrary point on the circumcircle of the equilateral triangle of side length `l` units, then `| vec P A|^2+| vec P B|^2+| vec P C|^2` is always equal to `2l^2` b. `2sqrt(3)l^2` c. `l^2` d. `3l^2`

If vec aa n d vec b are any two unit vectors, then find the greatest positive integer in the range of (3| vec a+ vec b|)/2+2| vec a- vec b| .

If P is a point on the altitude AD of the triangle ABC such the /_C B P=B/3, then AP is equal to 2asinC/3 (b) 2bsinC/3 (c) 2csinB/3 (d) 2csinC/3

The scalars l and m such that l vec a+m vec b= vec c ,where vec a , vec b and vec c are given vectors, are equal to

The tangent at any point P on the circle x^(2)+y^(2)=2 , cuts the axes at L and M. Find the equation of the locus of the midpoint L.M.

In triangle A B C ,/_A=60^0,/_B=40^0,a n d/_C=80^0dot If P is the center of the circumcircle of triangle A B C with radius unity, then the radius of the circumcircle of triangle B P C is (a) 1 (b) sqrt(3) (c) 2 (d) sqrt(3) 2

If C_1: x^2+y^2=(3+2sqrt(2))^2 is a circle and P A and P B are a pair of tangents on C_1, where P is any point on the director circle of C_1, then the radius of the smallest circle which touches c_1 externally and also the two tangents P A and P B is (a) 2sqrt(3)-3 (b) 2sqrt(2)-1 (c) 2sqrt(2)-1 (d) 1

If the angles A,B,C of a triangle are in A.P. and sides a,b,c, are in G.P., then prove that a^2, b^2,c^2 are in A.P.

If in a right-angled triangle A B C , the hypotenuse A B=p ,then vec(AB).vec(AC)+ vec(BC). vec(BA)+ vec(CA).vec(CB) is equal to 2p^2 b. (p^2)/2 c. p^2 d. none of these

If vec a , vec ba n d vec c are unit coplanar vectors, then the scalar triple product [2 vec a- vec b2 vec b- vec c2 vec c- vec a] is 0 b. 1 c. -sqrt(3) d. sqrt(3)

If any tangent to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 intercepts equal lengths l on the axes, then find l .