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Let \( a,b \in \mathbb{C} \). Let \( \alpha,\beta \) be the roots of the equation \( x^2 + \alpha x + b = 0 \). If \( \beta - \alpha = \sqrt{11} \) and \( \beta^2 - \alpha^2 = 3i\sqrt{11} \), then \( (\beta^3 - \alpha^3)^2 \) is equal to:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - 3x + r = 0 \), and \( \frac{\alpha}{2}, 2\beta \) be the roots of the equation \( x^2 + 3x + r = 0 \). If the roots of the equation \( x^2 + 6x = m \) are \( 2\alpha + \beta + 2r \) and \( \alpha - 2\beta - \frac{r}{2} \), then \( m \) is equal to:

If \( \alpha, \beta, \gamma \) are the roots of \( x^3 + 3x + 9 = 0 \), find the value of \( \left(\alpha^9 + \beta^9 + \gamma^9\right) \).

If \( \alpha, \beta \) are roots of equation \( x^2 - 2x - 1 = 0 \), then value of \( 5\alpha^4 + 12\beta^3 \) is ________.

If \( \alpha, \beta \), where \( \alpha < \beta \), are the roots of the equation \( \lambda x^2 - (\lambda + 3)x + 3 = 0 \) such that \( \frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3} \), then the sum of all possible values of \( \lambda \) is:

Suppose \( a,b,c \in \mathbb{R} \) and \( b \neq c \). If \( \alpha, \beta \) are roots of \( x^2 + ax + b = 0 \) and \( \gamma, \delta \) are roots of \( x^2 + ax + c = 0 \), then value of \( \dfrac{(\alpha - \gamma)(\alpha - \delta)}{(\beta - \gamma)(\beta - \delta)} \) is independent of:

Let one root of the quadratic equation in \( x \):
\( (k^{2}-15k+27)x^{2}+9(k-1)x+18=0 \) be twice the other. Then the length of the latus rectum of the parabola \( y^{2}=6kx \) is equal to:

The sum of all the real solutions of the equation \( \log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9) \) is equal to:

Let \( \alpha , \alpha+2 , \alpha \in \mathbb{Z} \), be the roots of the quadratic equation \( x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n \) for some \( n \in \mathbb{N} \). Then \( n+\alpha \) is equal to:

If \( x \in [5,7] \) the greatest value of expression \( -3x^2 + 54x - 200 \) is ________.

The sum of all the roots of the equation \( (x - 1)^2 - 5|x - 1| + 6 = 0 \), is:

Let \( x^2 - (m-3)x + m = 0 \) (\( m \in \mathbb{R} \)) be a quadratic equation. Find the values of \( m \) for which both the roots are negative?

If \( \alpha \) and \( \beta (\alpha < \beta) \) are the roots of the equation \( (-2 + \sqrt{3})\lvert \sqrt{x} - 3 \rvert + (x - 6\sqrt{x}) + (9 - 2\sqrt{3}) = 0, \ \ x \ge 0 \), then \( \sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha \beta} \) is equal to:

The smallest positive integer \( p \) for which expression \( x^2 - 2px + 3p + 4 \) is negative for atleast one real \( x \) is:

Consider the quadratic equation \( (n^2-2n+2)x^2-3x+(n^2-2n+2)^2=0,\, n\in R \). Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is: