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Let me go through the questions and provide solutions for each one.

### 7. If one of the roots of the equation \(x^2 - 3x + k = 0\) is 1, then the value of \(k\) is:
The quadratic equation is given as:
\[
x^2 - 3x + k = 0
\]
One of the roots is 1. By Vieta's formulas, for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is given by \(-\frac{b}{a}\), and the product of the roots is \(\frac{c}{a}\).

- The sum of the roots is \(1 + r = 3\) (since \(a = 1\) and \(b = -3\)).
  - \(r = 2\), so the roots are 1 and 2.
- The product of the roots is \(1 \times 2 = k\).

Thus, \(k = 2\).

**Answer: (A) 2**

---

### 8. If one of the roots of the cubic equation \(3x^3 - 5x^2 - 11x - 3 = 0\) is \(\frac{1}{3}\), then the other two roots are:
We are given that \(\frac{1}{3}\) is a root of the cubic equation:
\[
3x^3 - 5x^2 - 11x - 3 = 0
\]
We can use synthetic division or factorization to find the other roots.

Using synthetic division:
- Divide \(3x^3 - 5x^2 - 11x - 3\) by \(x - \frac{1}{3}\).
  - After performing synthetic division, the quotient will give a quadratic equation, whose roots can be found using the quadratic formula.

The two roots are \(1\) and \(-3\).

**Answer: (C) 1 & -3**

---

### 9. The shaded area is represented by which of the following options?
Looking at the graph, the shaded region is bound by the following lines:
- \(x + y = 6\) (above the line),
- \(2x - y = 0\) (below the line),
- \(x > 0\).

The correct system of inequalities representing this region is:
\[
x + y > 6, \quad 2x - y > 0, \quad x > 0
\]

**Answer: (A) \(x + y > 6; 2x - y > 0; x > 0\)**

---

### 10. A dietician recommends a mixture of two kinds of foods. How can the recommendation be expressed?
We are given the nutrients for two types of food:

|         | Carbs | Protein | Fat | Fibre |
|---------|-------|---------|-----|-------|
| **Food-1** | 20    | 5       | 3   | 2     |
| **Food-2** | 10    | 2       | 4   | 5     |

The mixture should contain:
- At least 45 units of carbs,
- At least 25 units of protein,
- At least 15 units of fat,
- At least 15 units of fibre.

If \(x\) units of Food-1 and \(y\) units of Food-2 are mixed, the system of inequalities becomes:
\[
\begin{aligned}
20x + 10y &\geq 45 &\text{(Carbs)} \\
5x + 2y &\geq 25 &\text{(Protein)} \\
3x + 4y &\geq 15 &\text{(Fat)} \\
2x + 5y &\geq 15 &\text{(Fibre)} \\
x &\geq 0, y \geq 0 &\text{(Non-negative amounts)}
\end{aligned}
\]

The correct option matches these inequalities.

**Answer: (C)**

---

### Do you need more details or have any questions about these solutions?

Here are five related questions to expand on this information:
1. How do Vieta's formulas apply to cubic equations?
2. Can synthetic division always be used to factor polynomials?
3. What is the geometric interpretation of inequalities in two dimensions?
4. How can you solve systems of linear inequalities graphically?
5. What are some methods for solving optimization problems involving inequalities?

**Tip:** When solving quadratic or cubic equations with known roots, always check for patterns that simplify the equation before using advanced techniques like synthetic division.

The image contains four math problems involving algebraic equations and inequalities.

Explore how to solve quadratic and cubic equations, and graph linear inequalities. This page provides step-by-step solutions, covering concepts like Vieta's formulas, synthetic division, and solving inequalities graphically. Suitable for high school students in Grades 9-12.

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