The polynomial $4x ^ {10} - x ^ 9 + 3x ^ {11} - 5x ^ 7 + c x ^ 6 + 2x ^ 5 - x ^ 4 + x ^ 3 - 4x ^ 2 + 6x - 2$ when divided by $x - 1$ leaves a remainder $2.$ Then the value of $c + 6$ is

If the polynomial $ax^2 + bx + 5$ leaves a remainder $3$ when divided by $x - 1$, and a remainder $2$ when divided by $x + 1$, then $2b - 4a$ equals

A polynomial $P(x)$ leaves a remainder $2$ when divided by $(x - 1)$ and a remainder $1$ when divided by $(x - 2)$ The remainder when $P(x)$ is divided by $(x - 1)(x - 2)$ is

The remainder when $11^{1011} + 1011^{11}$ is divided by $9$ is

If $m$ is a positive integer then the values of $k$ for which $6m + k$ cannot be a perfect square are

The remainder when $(29^{29})^{29}$ is divided by $9$ is

The remainder when $7^{103} + 7^{101}$ is divided by $9$ is ___