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.

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-gamma)(beta-delta)(gamma-delta)

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

Let the line (x-2)/(3)=(y-1)/(-5)=(z+2)/(2) lies in the plane x+3y-alphaz+beta=0 . Then, (alpha, beta) equals

Show that the determinant Delta (x) is given by Delta (x) = |{:(sin(x+alpha),cos(x+alpha),a+xsinalpha),(sin(x+beta),cos(x+beta),b+xsinbeta),(sin(x+gamma),cos(x+gamma),c+xsingamma):}| is independent of x.

Two lines L_(1) : x=5, (y)/(3-alpha)=(z)/(-2) and L_(2) : x=alpha, (y)/(-1)=(z)/(2-alpha) are coplanar. Then, alpha can take value(s)

Using properties of determinants in Exercise 11 to 15 prove that |{:(alpha,alpha^2,beta+gamma),(beta,beta^2,gamma+alpha),(gamma,gamma^2,alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha+beta+gamma)(alpha-beta)

If alpha,beta are the roots of x^2+p x+q=0a n dgamma,delta are the roots of x^2+r x+s=0, evaluate (alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta) in lterms of p ,q ,r ,a n dsdot Deduce the condition that the equation has a common root.

Explain alpha beta and gamma rays

The value of the determinant |(1,(alpha^(2x)-alpha^(-2x))^2,(alpha^(2x)+alpha^(-2x))^2),(1,(beta^(2x)-beta^(-2x))^2,(beta^(2x)+beta^(-2x))^2),(1,(gamma^(2x)-gamma^(-2x))^2,(gamma^(2x)+gamma^(-2x))^2)| is (a) 0 (b) (alphabeta gamma)^(2x) (c) (alpha beta gamma)^(-2x) (d) None of these

If 102!=2^(alpha)*3^(beta)*5^(gamma)*7^(delta) …, then