A man wants to invest at most 12000 in savings certificate and N.S.C. He has to invest at leat Rs. 2000 in saving certificate and at least Rs. 4000 in N.S.C. If the rate of interest on saving certificate is 8% per annum and rate of interest on N.S.C. is 10% per annum, how much money he should invest to earn maximum yearly income? Find also the maximum yearly income.

Answer

To solve the problem, we need to determine how much money the man should invest in savings certificates (denoted as \(X\)) and in National Savings Certificates (NSC, denoted as \(Y\)) to maximize his yearly income, given the constraints.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let \(X\) be the amount invested in savings certificates.
- Let \(Y\) be the amount invested in NSC.

2. **Set Up the Constraints**:
- The total investment should not exceed Rs. 12,000:
\[ X + Y \leq 12000 \quad \text{(1)} \]
- The minimum investment in savings certificates is Rs. 2,000:
\[ X \geq 2000 \quad \text{(2)} \]
- The minimum investment in NSC is Rs. 4,000:
\[ Y \geq 4000 \quad \text{(3)} \]
- Both \(X\) and \(Y\) must be non-negative:
\[ X \geq 0, \quad Y \geq 0 \quad \text{(4)} \]

3. **Objective Function**:
- The yearly income from the investments can be expressed as:
\[ Z = 0.08X + 0.10Y \quad \text{(5)} \]

4. **Graph the Constraints**:
- We will graph the inequalities defined by (1), (2), (3), and (4) to find the feasible region.
- The line \(X + Y = 12000\) intersects the axes at points (12000, 0) and (0, 12000).
- The line \(X = 2000\) is a vertical line at \(X = 2000\).
- The line \(Y = 4000\) is a horizontal line at \(Y = 4000\).

5. **Identify Corner Points**:
- The feasible region is bounded by the lines defined by the constraints. The corner points of the feasible region can be found by solving the equations:
- Intersection of \(X + Y = 12000\) and \(X = 2000\):
\[ 2000 + Y = 12000 \implies Y = 10000 \quad \text{(Point A: (2000, 10000))} \]
- Intersection of \(X + Y = 12000\) and \(Y = 4000\):
\[ X + 4000 = 12000 \implies X = 8000 \quad \text{(Point B: (8000, 4000))} \]
- Intersection of \(X = 2000\) and \(Y = 4000\):
\[ (2000, 4000) \quad \text{(Point C: (2000, 4000))} \]

6. **Evaluate the Objective Function at Each Corner Point**:
- For Point A: \(Z = 0.08(2000) + 0.10(10000) = 160 + 1000 = 1160\)
- For Point B: \(Z = 0.08(8000) + 0.10(4000) = 640 + 400 = 1040\)
- For Point C: \(Z = 0.08(2000) + 0.10(4000) = 160 + 400 = 560\)

7. **Determine the Maximum Income**:
- The maximum income occurs at Point A (2000, 10000):
\[ \text{Maximum Yearly Income} = 1160 \]

8. **Final Investment Amounts**:
- The man should invest Rs. 2000 in savings certificates and Rs. 10000 in NSC.

### Conclusion:
- The optimal investment strategy is to invest Rs. 2000 in savings certificates and Rs. 10000 in NSC to achieve a maximum yearly income of Rs. 1160.