A point `O` is the centre of a circle circumscribed about a triangle`A B Cdot` Then ` vec O Asin2A+ vec O Bsin2B+ vec O Csin2C` is equal to

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

Let D ,Ea n dF be the middle points of the sides B C ,C Aa n dA B , respectively of a triangle A B Cdot Then prove that vec A D+ vec B E+ vec C F= vec0 .

A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. If d_1 and d_2 are the distance of the tangent to the circle at the origin O from the points A and B, respectively, then the diameter of the circle is

If G is the centroid of triangle A B C ,t h e n vec G A+ vec G B+ vec G C is equal to a. vec0 b. 3 vec G A c. 3 vec G B d. 3 vec G C

A sector O A B O of central angle theta is constructed in a circle with centre O and of radius 6. The radius of the circle that is circumscribed about the triangle O A B , is 6costheta/2 (b) 6sectheta/2 3sectheta/2 (d) 3(costheta/2+2)

' I ' is the incentre of triangle A B C whose corresponding sides are a , b ,c , rspectively. a vec I A+b vec I B+c vec I C is always equal to a. vec0 b. (a+b+c) vec B C c. ( vec a+ vec b+ vec c) vec A C d. (a+b+c) vec A B

If O A B C is a tetrahedron where O is the origin and A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

A B C D is a quadrilateral. E is the point of intersection of the line joining the midpoints of the opposite sides. If O is any point and vec O A+ vec O B+ vec O C+ vec O D=x vec O E ,t h e nx is equal to a. 3 b. 9 c. 7 d. 4

Prove that the resultant of two forces acting at point O and represented by vec O B and vec O C is given by 2 vec O D ,where D is the midpoint of BC.

The volume of he parallelepiped whose sides are given by vec O A=2i-2j , vec O B=i+j-k a n d vec O C=3i-k is a. 4/13 b. 4 c. 2/7 d. 2