Quadratic functions

The diagram shows part of the curve $y = 3 + \frac{1}{1+x}$ and part of the curve $y = 4 - \frac{12}{(1+x)^2}$.

The curve $y = 3 + \frac{1}{1+x}$ meets the $y$-axis at the point $A$.

The curve $y = 4 - \frac{12}{(1+x)^2}$ meets the $x$-axis at the point $C$.

The curves intersect at the point $B$.

Find the area of the shaded region $OABC$.
Give your answer in exact form.

Show that there is no real value of $k$ for which the equation

has two equal roots.

In this question, $k$ is a constant.

It is given that $2x^2 + (3k - 2)x + k = 0$ has roots that are real and distinct.

Find the set of possible values of $k$.

Use algebra to show that the equation $5x(x - 3) = 5x - 26$ has no real solutions.

It is given that $y = mx^2 + \frac{x}{2} + n$, where $m$ and $n$ are non-zero constants. It is also given that

for all values of $x$. Find the values of $m$ and $n$.

The line with equation $x + 3y = k$, where $k$ is a positive constant, is a tangent to the curve with equation $x^2 + y^2 + 2y - 9 = 0$. Find the value of $k$ and hence find the coordinates of the point where the line touches the curve.

Find the non-zero value of $k$ for which the line $y = -2x - 6k - 1$ is a tangent to the curve $y = x(x + 2k)$.

Find the values of $k$ for which the curve $y = x^2 + kx + (4k - 15)$ is completely above the $x$-axis.

Find the possible values of $k$ for which the equation $kx^2 + (k + 5)x - 4 = 0$ has real roots.