Show that the lines `(x-a+d)/(alpha-delta)=(y-a)/alpha=(z-a-d)/(alpha+delta)` and `(x-b+c)/(beta-gamma)=(y-b)/beta=(z-b-c)/(beta+gamma)` are coplanar.

Show that lines. (x-a+d)/( alpha- delta)=(y-a)/( alpha)=(z-a-d)/( alpha+ delta) and (x-b+c)/( beta- lambda)=(y-b)/(beta)=(z-b-c)/( beta + lambda) are coplanar.

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|= -64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)(gamma-beta)

|{:(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

If alpha,beta,gamma are the angles of a triangle and system of equations cos(alpha-beta)x+cos(beta-gamma)y+cos(gamma-alpha)z=0 cos(alpha+beta)x+cos(beta+gamma)y+cos(gamma+alpha)z=0 sin(alpha+beta)x+sin(beta+gamma)y+sin(gamma+alpha)z=0 has non-trivial solutions, then triangle is necessarily a. equilateral b. isosceles c. right angled "" d. acute angled

If alpha,beta be the roots x^2+px-q=0 and gamma,delta be the roots of x^2+px+r=0, p+rphi0 ,then ((alpha-gamma)(alpha-delta))/((beta-gamma)(beta-delta)) is equal to

Simplify: cot (beta - gamma) cot (gamma - alpha) + cot (gamma - alpha) cot (alpha - beta) + cot (alpha - beta) cot (beta - gamma)

|{:(1,alpha,alpha^2),(1,beta,beta^2),(1,gamma,gamma^2):}|=(alpha-beta)(beta-gamma)(gamma-alpha)

If b sin alpha = alphasin (alpha+2beta),"show that"(b+a) tan beta=(b-a)tan (alpha+beta)

If alpha,beta,gamma,delta are four complex numbers such that gamma/delta is real and alpha delta - beta gamma !=0 then z = (alpha + beta t)/(gamma+ deltat) where t is a rational number, then it represents:

If alpha,beta,gamma are different from 1 and are the roots of a x^3+b x^2+c x+d=0a n d(beta-gamma)(gamma-alpha)(alpha-beta)=(25)/2 , then prove that |alpha/(1-alpha)beta/(1-beta)gamma/(1-gamma)alphabetagammaalpha^2beta^2gamma^2|=(25 d)/(2(a+b+c+d))