Which of the following values of `alpha` satisfy the equation
`|{:((1+alpha)^(2),,(1+2alpha)^(2),,(1+3alpha)^(2)),((2+alpha)^(2),,(2+2alpha)^(2),,(2+3alpha)^(2)),((3+alph)^(2),,(2+2alpha)^(2),,(2+3alpha)^(2)):}|=648alpha ?`

Which of the following values of alpha satisfying the equation |(1+alpha)^2(1+2alpha)^2(1+3alpha)^2(2+alpha)^2(2+2alpha)^2(2+3alpha)^2(3+alpha)^2(3+2alpha)^2(3+3alpha)^2|=-648alpha? -4 b. 9 c. -9 d. 4

Which of the following is independent of alpha in the hyperbola (0 < alpha < pi/2)x^2/cos^2 alpha-y^2/sin^2alpha=1

Prove that sin 4alpha = 4 tan alpha (1 - tan^(2) alpha)/((1 + tan^(2) alpha)^(2))

Write each of the following expressions in terms of alpha+beta and alphabeta . (1)/(alpha^(2)beta)+(1)/(beta^(2)alpha)

If a variable takes the discrete values alpha-4 , alpha -(7)/(2), alpha-(5)/(2), alpha-2,alpha+(1)/(2), alpha-(1)/(2), alpha+5(alpha gt 0) , then the median is

Prove that sin4 alpha=4tan alpha (1- tan^(2)alpha)/((1+tan^2 alpha)^2

If sinx+cos e cx+tany+coty=4 where x and y in [0,pi/2] ,then tan(y/2) is a root of the equation (a) alpha^2+2alpha+1=0 (b) 2 alpha^2-2alpha-1=0 (c) 2alpha^2-2alpha-1=0 (d) alpha^2-alpha-1=0

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)) .

The minimum integral value of alpha for which the quadratic equation (cot^(-1)alpha)x^(2)-(tan^(-1)alpha)^(3//2)x+2(cot^(-1)alpha)^(2)=0 has both positive roots