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To solve the problem step by step:

### Question (b):
We are tasked to show that the \( x \)-coordinate of the intersection of the line \( l \) with the curve \( C \) satisfies the equation:

\[
2x^2 - 4x + k^2 = 0
\]

#### 1. Curve Equation:
\[
y = \frac{k^2}{x} + 1
\]

#### 2. Line Equation:
\[
y = -2x + 5
\]

#### 3. Intersection of \( C \) and \( l \):
Equating the \( y \)-values:
\[
\frac{k^2}{x} + 1 = -2x + 5
\]

Simplify:
\[
\frac{k^2}{x} = -2x + 4
\]

Multiply through by \( x \) (valid since \( x \neq 0 \)):
\[
k^2 = -2x^2 + 4x
\]

Rearranging gives:
\[
2x^2 - 4x + k^2 = 0
\]

This verifies the required equation.

---

### Question (c):
We now find the values of \( k \) for which the line \( l \) is tangent to the curve \( C \). This means the quadratic equation:

\[
2x^2 - 4x + k^2 = 0
\]

has exactly one solution for \( x \). For a quadratic to have one solution, its discriminant must equal zero.

#### Discriminant:
The general quadratic equation \( ax^2 + bx + c = 0 \) has discriminant:
\[
\Delta = b^2 - 4ac
\]

For our equation \( 2x^2 - 4x + k^2 = 0 \):
- \( a = 2 \),
- \( b = -4 \),
- \( c = k^2 \).

The discriminant is:
\[
\Delta = (-4)^2 - 4(2)(k^2)
\]
\[
\Delta = 16 - 8k^2
\]

Set \( \Delta = 0 \) for tangency:
\[
16 - 8k^2 = 0
\]

Solve for \( k^2 \):
\[
8k^2 = 16 \quad \Rightarrow \quad k^2 = 2
\]

Thus, \( k = \pm\sqrt{2} \).

---

### Final Answers:
- (b) The \( x \)-coordinate of the intersection satisfies \( 2x^2 - 4x + k^2 = 0 \).
- (c) The values of \( k \) for which \( l \) is tangent to \( C \) are:
  \[
  k = \pm\sqrt{2}.
  \]

Let me know if you have further questions or want additional details!

---

### Related Questions:
1. What happens when \( k = 0 \)? Is the line \( l \) still tangent to the curve?
2. What is the geometric interpretation of the discriminant in this scenario?
3. How do the roots of the quadratic equation change as \( k \) varies?
4. Can the curve \( C \) intersect the line \( l \) at more than two points? Why or why not?
5. How would the solution change if the line \( l \) had a different slope?

---

### Tip:
Always check the discriminant when analyzing tangency conditions for quadratics, as it determines the number of real solutions (intersection points).

The curve C has equation y = k^2/x + 1, and the line l has equation y = -2x + 5. Show that the x-coordinate of any point of intersection of l with C is given by a solution of the equation 2x^2 - 4x + k^2 = 0, and find the exact values of k for which l is a tangent to C.

This problem explores the intersection of a line y = -2x + 5 with a curve y = k^2/x + 1, deriving the quadratic equation for the x-coordinates of intersection. It also determines the values of k for which the line is tangent to the curve, using discriminant analysis. Suitable for high school students in Grades 10-12.

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