1 MATHEMATICS Find the centre and radius of the circle formed by all the points represented by z x iy satisfyin k k 1 where a and are constant complex numbers given b the relation z BI a a B B iB 2 If 5 6 6 4 are four distinct vectors satisfying the conditions axb xd and ax bxd then prov that ab cd c bd 3 is an integer where n is a positive integer 4 If M is a 3x3 matrix where M M I and det M 1 then prove that det M 1 0 Using permutation or otherwise prove that 5 If y x 3 116 cosx cos 1 sin 8 that 0 de then find lin 7 If f 1 1 R and f 0 dy 5 T is a parallelopiped in which A B C and D are vertices of one face And the face just above has corresponding vertices A B C D T is now compressed to S with face ABCD remaining sam and A B C D shifted to A B C D in S The volume of parallelopiped S is reduced to 90 of T Prove that locus of A is a plane af and f 0 0 Find the value of 1 a 1 cos give n at x x EG If P x 51x 2323x45x 1035 using Rolle s theorem prove that atleast one root lie between 451 100 46 9 A plane is parallel to two lines whose direction ratios are 1 0 1 and 1 1 0 and it contain the point 1 1 1 If it cuts co ordinate axis at A B C Then find the volume of the tetrahedro OABC 10 If A and B are two independent events prove that P AUB P AB SP C where C is an ever defined that exactly one of A and B occurs x 1 y 3 11 A curve passes through 2 0 and the slope of tangent at point P x y equals the equation of the curve and area enclosed by the curve and the x axis in the fourth quadrant 12 A circle touches the line 2x 3y 1 0 at the point 1 1 and is orthogonal to the circle whic has the line segment having end points 0 1 and 2 3 as the diameter