The determinant
`Delta = |(b,c,b alpha +c),(c,d,c alpha + d),(b alpha + c,c alpha + d,a a^(3) - c alpha)|`
is equal to zero, if

In Delta ABC , prove that c cos (A - alpha) + a cos (C + alpha) = b cos alpha

If a sec alpha-c tan alpha=d and b sec alpha + d tan alpha=c , then eliminate alpha from above equations.

Let alpha be a repeated root of a quadratic equation f(x)=0 and A(x),B(x),C(x) be polynomials of degrees 3, 4, and 5, respectively, then show that |(A(x),B(x),C(x)) ,(A(alpha),B(alpha),C(alpha)),(A '(alpha),B '(alpha),C '(alpha))| is divisible by f(x) , where prime (') denotes the derivatives.

For determinant |((a^2+b^2)/c,c,c),(a,(b^2+c^2)/a,a),(b,b,(c^2+a^2)/b)|=alpha abc, then the value of alpha is -

If n=pi/(4alpha), then tan alpha tan 2alpha tan 3 alpha ... tan(2n-1)alpha is equal to (a) 1 (b) 1/2 (c) 2 (d) 1/3

The value of the determinant |(alpha,beta,l),(alpha,x, n),(alpha,beta,x)| is independent of l b. independent of n c. alpha(x-l)(x-beta) d. alphabeta(x-l)(x-n)

If alpha_r=(cos2rpi+isin2rpi)^(1/10) , then |(alpha_1, alpha_2, alpha_4),(alpha_2, alpha_3, alpha_5),(alpha_3, alpha_4, alpha_6)|= (A) alpha_5 (B) alpha_7 (C) 0 (D) none of these

The value of int_0^(pi/2) sin|2x-alpha|dx, where alpha in [0,pi], is (a) 1-cos alpha (b) 1+cos alpha (c) 1 (d) cos alpha

Let alpha be a repeated root of a quadratic equation f(x)=0a n dA(x),B(x),C(x) be polynomials of degrees 3, 4, and 5, respectively, then show that |A(x)B(x)C(x)A(alpha)B(alpha)C(alpha)A '(alpha)B '(alpha)C '(alpha)| is divisible by f(x) , where prime (') denotes the derivatives.

If alpha=\ \ ^m C_2,\ then find the value of \ ^(alpha)C_2dot