Prove that `|a_1alpha_1+b_1beta_1a_1alpha_2+b_2beta_2a_1alpha_3+b_1beta_3a_2alpha_1+b_2beta_1a_2alpha_2+b_2beta_2a_2alpha_3+b_2beta_3a_ 3alpha_1+b_3beta_1a_3alpha _2+b_3beta_2a_3alpha_3+b_3beta_3|=0`

Prove that 2 sin^2 beta + 4 cos(alpha + beta) sin alpha sin beta + cos 2(alpha + beta) = cos 2alpha

If the roots of equation x^3+a x^2+b=0a r ealpha_1,alpha_2 and alpha_3(a ,b!=0) , then find the equation whose roots are (alpha_1alpha_2+alpha_2alpha_3)/(alpha_1alpha_2alpha_3),(alpha_2alpha_3+alpha_3alpha_1)/(alpha_1alpha_2alpha_3),(alpha_1alpha_3+alpha_1alpha_2)/(alpha_1alpha_2alpha_3)

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

If alpha,beta,gamma are such that alpha+beta+gamma=2,""alpha^2+beta^2+gamma^2=6,""alpha^3+beta^3+gamma^3=8,t h e n""alpha^4+beta^4+gamma^4 is a. 18 b. 10 c. 15 d. 36

If the roots of the equation a x^2-b x+c=0a r ealpha,beta, then the roots of the equation b^2 c x^2-a b^2x+a^3=0 are 1/(alpha^3+alphabeta),1/(beta^3+alphabeta) b. 1/(alpha^2+alphabeta),1/(beta^2+alphabeta) c. 1/(alpha^4+alphabeta),1/(beta^4+alphabeta) d. none of these

If cosalpha+cosbeta+cosgamma=0a n da l sosinalpha+sinbeta+singamma=0, then prove that cos2alpha+cos2beta+cos2gamma =sin2alpha+sin2beta+sin2gamma=0 sin3alpha+sin3beta+sin3gamma=3sin(alpha+beta+gamma) cos3alpha+cos3beta+cos3gamma=3cos(alpha+beta+gamma)

If alpha and beta are acute such that alpha+beta and alpha-beta satisfy the equation tan^(2)theta-4tan theta+1=0 , then (alpha, beta ) =

If sinalpha-sinbeta=1/3andcosbeta-cosalpha=1/2, show that cot(alpha+beta)/2=2/3

If the function f : [0,8] to R is differentiable, then for 0 < alpha <1 < beta < 2 , int_0^8 f(t) dt is equal to (a) 3[alpha^3f(alpha^2)+beta^2f(beta^2)] (b) 3[alpha^3f(alpha)+beta^2f(beta)] (c) 3[alpha^3f(alpha^2)+beta^2f(beta^3)] (d) 3[alpha^2f(alpha^3)+beta^2f(beta^3)]

Let alpha,beta be two real numbers satisfying the following relations alpha^2+beta^2=5, 3(alpha^5+beta^5)=11(alpha^3+beta^3)1. Possible value of alpha beta is