Let's solve both of these problems step by step:

2. Find the set of values of the constant $a$ for which the line $y = 2 - 5x$ intersects the curve $y = x^2 + ax + 18$ at two points.

Step 1: Set the equations equal

To find the points of intersection, set the line equation equal to the curve equation: $2 - 5x = x^2 + ax + 18$

Step 2: Rearrange the equation

Move all terms to one side to form a quadratic equation: $x^2 + ax + 18 - 5x - 2 = 0$ Simplify: $x^2 + (a - 5)x + 16 = 0$

Step 3: Analyze for two intersection points

For the line to intersect the curve at two points, the quadratic equation must have two distinct real roots. This means the discriminant must be positive. The discriminant $\Delta$ for a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = b^2 - 4ac$ For our quadratic equation $x^2 + (a - 5)x + 16 = 0$, the discriminant becomes: $\Delta = (a - 5)^2 - 4(1)(16)$ Simplify: $\Delta = (a - 5)^2 - 64$ For two distinct real roots, $\Delta > 0$: $(a - 5)^2 - 64 > 0$ $(a - 5)^2 > 64$ Taking square roots: $|a - 5| > 8$ This gives two inequalities: $a - 5 > 8 \quad \text{or} \quad a - 5 < -8$ Solve each inequality: $a > 13 \quad \text{or} \quad a < -3$

Thus, the set of values of $a$ is: $a < -3 \quad \text{or} \quad a > 13$

3. Find the range of values of $m$ for which the straight line $y = mx - 5$ intersects the curve $y = x^2 - 1$ in two distinct points.

Step 1: Set the equations equal

Set the equation of the line equal to the equation of the curve: $mx - 5 = x^2 - 1$

Step 2: Rearrange the equation

Move all terms to one side to form a quadratic equation: $x^2 - mx + (-1 + 5) = 0$ Simplify: $x^2 - mx + 4 = 0$

Step 3: Analyze for two distinct points

For two distinct points of intersection, the discriminant of this quadratic equation must be positive. The discriminant $\Delta$ for $x^2 - mx + 4 = 0$ is: $\Delta = (-m)^2 - 4(1)(4)$ Simplify: $\Delta = m^2 - 16$ For two distinct points, $\Delta > 0$: $m^2 - 16 > 0$ Solve: $m^2 > 16$ Taking square roots: $|m| > 4$ This gives two inequalities: $m > 4 \quad \text{or} \quad m < -4$

Thus, the range of values of $m$ is: $m < -4 \quad \text{or} \quad m > 4$

Would you like more details on any of these steps or further clarifications?

Here are 5 related questions you can explore:

1. What happens if the discriminant is zero in the quadratic equations?
2. How can you determine the points of intersection once you have the discriminant conditions?
3. How do you find the vertex of a parabola given in quadratic form?
4. What is the geometrical interpretation of discriminant conditions for curves and lines?
5. How do higher-degree curves (like cubic equations) intersect with lines?

Tip: When solving problems involving intersections between curves and lines, always start by simplifying the equations and analyzing the discriminant of the resulting quadratic.