Let `x-ysinalpha-zs inbeta=0,xs inalpha=zs ingamma-y=0a n dxsinbeta+ysingamma-z=0` be the equations of the planes such that `alpha+beta+gamma=pi//2(w h e r ealpha,betaa n dgamma!=0)dot` Then show that there is a common line of intersection of the three given planes.

If alpha,betaa n dgamma are the roots of x^3+8=0 then find the equation whose roots are alpha^2,beta^2a n dgamma^2 .

Consider the planes 3x-6y-2z-15 =0 and 2x+y-2z - 5=0 Statement 1:The parametric equations of the line intersection of the given planes are x=3+14 t ,y=2t ,z=15 tdot Statement 2: The vector 14 hat i+2 hat j+15 hat k is parallel to the line of intersection of the given planes.

Let a ,b ,c be real. If a x^2+b x+c=0 has two real roots alphaa n dbeta,w h e r ealpha >1 , then show that 1+c/a+|b/a|<0

Let alpha,beta,gamma>0a n dalpha+beta+gamma=pi/2dot Then prove that sqrt(tanalphatanbeta) + sqrt(t a nbeta"tan"gamma)+sqrt(tanalphatangamma)lt=sqrt(3)

Find the equations of the bisectors of the angles between the planes 2x-y+2z+3=0a n d3x-2y+6z+8=0 and specify the plane which bisects the acute angle and the plane which bisects the obtuse angle.

Find the equations of the bisectors of the angles between the planes 2x-y+2z+3=0a n d3x-2y+6z+8=0 and specify the plane which bisects the acute angle and the plane which bisects the obtuse angle.

If alpha,betaa n dgamma are roots of 2x^3+x^2-7=0 , then find the value of sum(alpha/beta+beta/alpha) .

If alpha,beta are roots of x^2+p x+1=0a n dgamma,delta are the roots of x^2+q x+1=0 , then prove that q^2-p^2=(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta) .

Let alpha ,beta and gamma be the roots of the equations x^(3) +ax^(2)+bx+ c=0,(a ne 0) . If the system of equations alphax + betay+gammaz=0 beta x +gamma y+ alphaz =0 and gamma x =alpha y + betaz =0 has non-trivial solution then

If alpha, beta and gamma are three consecutive terms of a non-constant G.P. such that the equations ax^(2) + 2beta x + gamma = 0 and x^(2) + x - 1= 0 have a common root, then alpha (beta + gamma) is equal to