Prove that `|[a_1alpha_1+b_1beta_1, a_1alpha_2+b_1beta_2, a_1alpha_3+b_1beta_3], [a_2alpha_1+b_2beta_1, a_2alpha_2+b_2beta_2, a_2alpha_3+b_2beta_3], [a_3alpha_1+b_3beta_1, a_3alpha_2+b_3beta_2, a_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.

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

If the roots of a_1x^2 + b_1x+ c_1 = 0 are 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 Largest integral value of N if alpha_1 =5 , alpha_2 = 3 and alpha_3=2 and beta_1, beta_2, beta_3=(0,1) (A)342. (D) 242 (C) 243 (B) 343 342 17 Difference of largest and smallest integral values of N if alpha_1 =5 , alpha_2 = 3 and alpha_3=2 and beta_1, beta_2, beta_3=(0,1) (D) 99 (C) 98 (B) 100 (A) 97

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 :

If alpha, beta are the roots of the quadratic equation x^2 + bx - c = 0 , the equation whose roots are b and c , is a. x^(2)+alpha x- beta=0 b. x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0 c. x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0 d. x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0

cos 2 alpha =(3 cos 2 beta -1)/( 3-cos 2 beta), then tan alpha=

If : sin (alpha + beta)=1 and sin (alpha-beta)=(1)/(2), "then" : tan (alpha+2beta)*tan(2alpha+beta)=

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) .