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For what values of $a \in \mathbb{Z}$,the quadratic expression $(x+a)(x+1991)+1$ can be factorised as $(x+b)(x+c)$,where $b, c \in \mathbb{Z}$?

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is:

The midpoint of the diagonal of a rectangle formed by $x^2+5x-6=0$ and $y^2-8y-20=0$ is

If $\alpha, \beta, \gamma$ are roots of $x^3 + x^2 - 5x - 1 = 0$,then the value of $[\alpha] + [\beta] + [\gamma]$ is (where $[.]$ denotes the greatest integer function):

The equation whose roots are $\frac{1}{3 + \sqrt{2}}$ and $\frac{1}{3 - \sqrt{2}}$ is

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