What is the coefficient of $x^{4}$ in the expansion of $(1 + x + x^{2})^{5}$ ?

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The quadratic $x^{2} + p x + q = 0$ has roots $r$ and $r + 1$ , and the sum of the squares of the roots is $13$ . What is the value of $q$ ?
For how many integer values of $k$ does the equation $x^{2} - k x + 12 = 0$ have two distinct integer roots?
A circle passes through the points $(0, 0)$ , $(4, 0)$ and $(0, 6)$ . What is its radius?
How many tangents to the curve $y = x^{2}$ pass through the point $(1, - 3)$ ?
What is the value of $\int_{0}^{1} (x^{2} + 1)^{2} d x$ ?
A sequence is defined by $u_{1} = 3$ and $u_{n + 1} = 3 u_{n} - 2$ . What is the smallest value of $n$ for which $u_{n} > 1000$ ?