If y=sin mx the value of the determinant `|{:(y,y_(1),y_(2)),(y_(3),y_(4),y_(5)),(y_(6),y_(7),y_98):}|` where `y_(n)=(d^(n)y)/(dx^(n))` is

if y= sin mx, them the value of the determinant |{:(y,,y_(1),,y_(2)),(y_(3),,y_(4),,y_(5)),(y_(6),,y_(7),,y_(8)):}|" Where " y_(n)=(d^(n)y)/(dx^(n)) " is "

If x_(1),x_(2) "and" y_(1),y_(2) are the roots of the equations 3x^(2) -18x+9=0 "and" y^(2)-4y+2=0 the value of the determinant |{:(x_(1)x_(2),y_(1)y_(2),1),(x_(1)+x_(2),y_(1)+y_(2),2),(sin(pix_(1)x_(2)),cos (pi//2y_(1)y_(2)),1):}| is

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

Show that the area of the triangle inscribed in the parabola y^2= 4ax is : (1)/(8a)|(y_(1)-y_(2))(y_(2)-y_(3))(y_(3)-y_(1))| , where y_(1),y_(2),y_(3) are the ordinates of the angular points.

If y =(ax +b)/( cx + d) , then prove that 2y_1y_3=3(y_2)^2