If `y=1+alpha/((1/x-alpha))+(beta//x)/((1/x-alpha)(1/x-beta))+(gamma//x^2)/((1/x-alpha)(1/x-beta)(1/x-gamma)),` find `(dy)/(dx)`

Evaluate: int1/((x-alpha)(beta-x))dx ,(beta>alpha)

Evaluate: int1/((x-alpha)(beta-x))dx ,(beta>alpha)

If alpha, beta and gamma the roots of the equation x^(3) + 3x^(2) - 4x - 2 = 0. then find the values of the following expressions: (i) alpha ^(2) + beta^(2) + gamma^(2) (ii) alpha ^(3) + beta^(3) + gamma^(3) (iii) (1)/(alpha)+(1)/(beta)+(1)/(gamma)

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

If alpha&beta are the roots of the quadratic equation a x^2+b x+c=0 , then the quadratic equation, a x^2-b x(x-1)+c(x-1)^2=0 has roots: alpha/(1-alpha),beta/(1-beta) (b) alpha-1,beta-1 alpha/(alpha+1),beta/(beta+1) (d) (1-alpha)/alpha,(1-beta)/beta

If alpha,beta,gamma are different from 1 and are the roots of a x^3+b x^2+c x+d=0a n d(beta-gamma)(gamma-alpha)(alpha-beta)=(25)/2 , then prove that |alpha/(1-alpha)beta/(1-beta)gamma/(1-gamma)alphabetagammaalpha^2beta^2gamma^2|=(25 d)/(2(a+b+c+d))

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

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

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

If (1+alpha)/(1-alpha),(1+beta)/(1-beta), (1+gamma)/(1-gamma) are the cubic equation f(x) = 0 where alpha,beta,gamma are the roots of the cubic equation 3x^3 - 2x + 5 =0 , then the number of negative real roots of the equation f(x) = 0 is :