If the points $(x + 1, 2)$,$(1, x + 2)$,and $\left( \frac{1}{x + 1}, \frac{2}{x + 1} \right)$ are collinear,then $x$ is

For $a \neq b \neq c$,if the lines $x+2ay+a=0$,$x+3by+b=0$ and $x+4cy+c=0$ are concurrent,then $a, b, c$ are in

The point of intersection of the lines $(a^3 + 3)x + ay + a - 3 = 0$ and $(a^5 + 2)x + (a + 2)y + 2a + 3 = 0$ (where $a$ is a real number) lies on the $y$-axis for:

If the system of equations $2x + 3y = -1$,$3x + y = 2$,and $\lambda x + 2y = \mu$ is consistent,then:

If $P \equiv \left( \frac{1}{x_p}, p \right), Q = \left( \frac{1}{x_q}, q \right), R = \left( \frac{1}{x_r}, r \right)$ where $x_k \neq 0$ denotes the $k^{th}$ term of an $H.P.$ for $k \in N$,then:

The number of values of $a$ for which the system of equations $a^2 x + (2 - a) y = 4 + a^2$ and $a x + (2 a - 1) y = a^5 - 2$ possesses no solution is: