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 |{:(a_(1)alpha_(1)+b_(1)beta_(1),a_(1)alpha_(2)+b_(1)beta_(2),a_(1)alpha_(3)+b_(1)beta_(3)),(a_(2)alpha_(1)+b_(2)beta_(1),a_(2)alpha_(2)+b_(2)beta_(2),a_(2)alpha_(3)+b_(2)beta_(3)),(a_(3)alpha_(1)+b_(3)beta_(1),a_(3)alpha_(2)+b_(3)beta_(2),a_(3)alpha_(3)+b_(3)beta_(3)):}| =0.

Prove that if alpha, beta, gamma !=0 then |(alpha+a_1b_1, a_1b_2, a_1b_3), (a_2b_1, beta+a_2b_2, a_2b_3), (a_3b_1, a_3b_2, gamma+a_3b_3)|=alpha beta gamma [1+(a_1b_1)/alpha + (a_2b_2)/beta+(a_3b_3)/gamma]

If Delta = |(cos (alpha_(1) - beta_(1)),cos (alpha_(1) - beta_(2)),cos (alpha_(1) - beta_(3))),(cos (alpha_(2) - beta_(1)),cos (alpha_(2) - beta_(2)),cos (alpha_(2) - beta_(3))),(cos (alpha_(3) - beta_(1)),cos (alpha_(3) - beta_(2)),cos (alpha_(3) - beta_(3)))|" then " Delta equals

Let log_(3)N=alpha_(1)+beta_(1) log_(5)N=alpha_(2)+beta_(2) log_(7)N=alpha_(3)+beta_(3) where alpha_(1), alpha_(2) and alpha_(3) are integers and beta_(1), beta_(2), beta_(3) in [0,1) . Q. Largest integral value of N if alpha_(1)=5, alpha_(2)=3 and alpha_(3)=2 .

Let log_(3)N=alpha_(1)+beta_(1) log_(5)N=alpha_(2)+beta_(2) log_(7)N=alpha_(3)+beta_(3) where alpha_(1), alpha_(2) and alpha_(3) are integers and beta_(1), beta_(2), beta_(3) in [0,1) . Q. Difference of largest and smallest values of N if alpha_(1)=5, alpha_(2)=3 and alpha_(3)=2 .

Prove that: (alpha^3)/2cos e c^2(1/2tan^(-1)(alpha/beta))+(beta^3)/2s e c^2(1/2tan^(-1)(beta/alpha))=(alpha+beta)(alpha^2+beta^2) .

If sin beta = 1/5 sin(2alpha +beta) , show that tan(alpha +beta) = 3/2 tan alpha .

If the roots of a_1x^(2)+b_1x+c_1=0 and alpha_1 , beta_1 and those of a_2x^(2)+b_2x+c_2=0 are alpha_2 , beta_2 such that alpha_1alpha_2=beta_1beta_2=1 then :

Let log_(3)N=alpha_(1)+beta_(1) log_(5)N=alpha_(2)+beta_(2) log_(7)N=alpha_(3)+beta_(3) where alpha_(1), alpha_(2) and alpha_(3) are integers and beta_(1), beta_(2), beta_(3) in [0,1) . Q. Number of integral values of N if alpha_(1)=4 and alpha_(2)=2 :

If tan(alpha-beta)=(sin2 beta)/(3-cos2 beta) then (a) tan alpha=2tan beta(b)tan beta=2tan alpha(c)2tan alpha=3tan beta(d)3tan alpha=2tan beta