We are given the following two equations:

1. The straight line $l$: $y = k(2x - 1)$ where $k$ is a constant.

2. The curve $C$: $y = x^2 + 2x + 11$

We are tasked with finding the set of values of $k$ for which the line $l$ does not cross or touch the curve $C$. This means we need to determine when the system of equations has no real solutions.

Step 1: Equating the two expressions for $y$

To find when the line and the curve do not intersect, we equate the two equations for $y$: $k(2x - 1) = x^2 + 2x + 11$

Step 2: Rearrange the equation into a quadratic form

Expand the left-hand side: $2kx - k = x^2 + 2x + 11$

Now, bring all terms to one side of the equation: $x^2 + (2 - 2k)x + (11 + k) = 0$ This is now a quadratic equation of the form: $Ax^2 + Bx + C = 0$ where $A = 1$, $B = 2 - 2k$, and $C = 11 + k$.

Step 3: Use the discriminant

For the line and the curve not to intersect, the quadratic equation must have no real solutions. This happens when the discriminant $\Delta$ is less than 0. The discriminant for a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = B^2 - 4AC$ Substitute the values for $A$, $B$, and $C$: $\Delta = (2 - 2k)^2 - 4(1)(11 + k)$

Simplify: $\Delta = (4 - 8k + 4k^2) - 4(11 + k)$ $\Delta = 4k^2 - 8k + 4 - 44 - 4k$ $\Delta = 4k^2 - 12k - 40$

Step 4: Solve $\Delta < 0$

For the line not to touch the curve, we require: $4k^2 - 12k - 40 < 0$

Divide through by 4 to simplify: $k^2 - 3k - 10 < 0$

Step 5: Solve the quadratic inequality

We first solve the equality $k^2 - 3k - 10 = 0$ using the quadratic formula: $k = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}$ $k = \frac{3 \pm \sqrt{9 + 40}}{2}$ $k = \frac{3 \pm \sqrt{49}}{2}$ $k = \frac{3 \pm 7}{2}$

Thus, the solutions are: $k = \frac{3 + 7}{2} = 5 \quad \text{or} \quad k = \frac{3 - 7}{2} = -2$

Step 6: Interpret the inequality

The quadratic inequality $k^2 - 3k - 10 < 0$ holds between the roots $k = -2$ and $k = 5$. Therefore, the set of values for which the line does not intersect the curve is: $-2 < k < 5$

Final Answer:

The line does not cross or touch the curve for $k \in (-2, 5)$.

Would you like a detailed explanation of any step?

Here are 5 questions to extend your understanding:

1. Why is the discriminant used to determine whether the line intersects the curve?
2. How would the solution change if we were asked for the values of $k$ where the line does intersect the curve?
3. What is the geometric interpretation of solving $k^2 - 3k - 10 < 0$?
4. How could we use a graphical method to solve this problem?
5. What would happen to the solution if the curve $C$ had different coefficients?

Tip: Always check the discriminant when solving for intersection conditions between a line and a curve, as it quickly tells you whether real solutions exist.