Let $$\lambda $$ be a real number for which the system of linear equations x + y + z = 6, 4x + $$\lambda $$y – $$\lambda $$z = $$\lambda $$ – 2, 3x + 2y – 4z = – 5 has infinitely many solutions. Then $$\lambda $$ is a root of the quadratic equation:

The smallest natural number n, such that the coefficient of x in the expansion of $${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$$ is ⁿC₂₃, is :

If 5x + 9 = 0 is the directrix of the hyperbola 16x² – 9y² = 144, then its corresponding focus is :

The sum of the real roots of the equation
$$\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} & {2x} & {x + 2} \cr } } \right| = 0$$, is equal to :