Prove that, `|[(a+1)(a+2),a+2,1],[(a+2)(a+3),a+3,1],[(a+3)(a+4),a+4,1]|=-2`

prove that , |{:(1,1+a,1+a+b),(2,3+2a,4+3a+2b),(3,6+3a,10+6a+3b):}|=1

Given A=[(5, 0, 4),( 2, 3, 2),( 1, 2, 1)] , B^(-1)=[(1, 3, 3),( 1, 4, 3),( 1, 3, 4)] . Compute (A B)^(-1) .

Prove that log_(10) 2lies between 1/3 and 1/4 .

Prove that r_(1) r_(2) + r_(2) r_(3) + r_(3) r_(1) = (1)/(4) (a + b + c)^(2)

Prove that, tan^(-1)((3 sin 2x)/(5+3 cos 2x))+tan^(-1)((1)/(4) tan x)=x .

Prove that the points ( 4 , 7 , 6) , ( 2 , 3 , 2 ) , ( - 1, -2, -1) and ( 1, 2 , 3) taken in order are the vertices of a parallelogram . Is the the parallelogram a rectangle?

If the determinant of the matrix [(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3))] is denoted by D, then the determinant of the matrix [(a_(1)+3b_(1)-4c_(1),b_(1),4c_(1)),(a_(2)+3b_(2)-4c_(2),b_(2),4c_(2)),(a_(3)+3b_(3)-4c_(3),b_(3),4c_(3))] will be -

Prove that the points ( 2 , 3, 5) , ( - 4 , 7 , - 7) , ( - 2 , 1 , -10) and ( 4 , - 3 , 2) are the vertices of a rectangle .

|{:(1,-3,2),(4,-1,2),(3," "5,2):}|

Show that (i) [(5,-1),(6,7)][(2,1),(3,4)] ne [(2,1),(3,4)][(5,-1),(6,7)] (ii) [(1,2,3),(0,1,0),(1,1,0)][(-1,1,0),(0,-1,1),(2,3,4)] ne [(-1,1,0),(0,-1,1),(2,3,4)][(1,2,3),(0,1,0),(1,1,0)]