Rajni invested ₹15000 at simple interest for 2 years at the rate of R% and gets an interest of ₹2100.He invested total amount (Principal+Interest) in a scheme.which offered compound interest at the rate of (R%+x%) for two years.→What are the possible integral values of 'x%' so that obtained compound interest is less than ₹3000.
(i) 1% (ii) 2% (iii) 3% (iv) 4% (v) 5%

Options

A. only (i)
B. only (i).(ii)
C. only (i).(ii)and(iii)
D. only (i).(ii).(iii) and (iv)

Explanation

To solve the problem step by step, we will follow the given information and apply the formulas for simple interest and compound interest.

### Step 1: Calculate the Rate of Interest (R%)
Rajni invested ₹15,000 at simple interest for 2 years and received ₹2,100 as interest. We can use the formula for simple interest:

\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount = ₹15,000
- \( R \) = Rate of interest (unknown)
- \( T \) = Time in years = 2

Substituting the known values into the formula, we have:

\[ 2100 = \frac{15000 \times R \times 2}{100} \]

### Step 2: Solve for R
Rearranging the equation to solve for \( R \):

\[ 2100 = \frac{30000R}{100} \]
\[ 2100 = 300R \]
\[ R = \frac{2100}{300} = 7\% \]

### Step 3: Calculate the Total Amount after 2 Years
Now, we will find the total amount after 2 years, which is the sum of the principal and the interest earned.

\[ \text{Total Amount} = P + \text{Simple Interest} = 15000 + 2100 = 17100 \]

### Step 4: Set Up the Compound Interest Calculation
Rajni then invests the total amount (₹17,100) in a scheme that offers compound interest at the rate of \( (R + x)\% \) for 2 years. Here, \( R = 7\% \), so the rate becomes \( (7 + x)\% \).

### Step 5: Write the Compound Interest Formula
The formula for compound interest is:

\[ \text{Compound Interest} = P \left(1 + \frac{r}{100}\right)^n - P \]

Where:
- \( P \) = Principal = ₹17,100
- \( r \) = Rate = \( 7 + x \)
- \( n \) = Number of years = 2

Substituting the values, we have:

\[ \text{Compound Interest} = 17100 \left(1 + \frac{7+x}{100}\right)^2 - 17100 \]

### Step 6: Set the Condition for Compound Interest
We want the compound interest to be less than ₹3,000:

\[ 17100 \left(1 + \frac{7+x}{100}\right)^2 - 17100 < 3000 \]

Adding ₹17,100 to both sides gives:

\[ 17100 \left(1 + \frac{7+x}{100}\right)^2 < 20100 \]

### Step 7: Simplify the Inequality
Dividing both sides by ₹17,100:

\[ \left(1 + \frac{7+x}{100}\right)^2 < \frac{20100}{17100} \]
\[ \left(1 + \frac{7+x}{100}\right)^2 < 1.176 \]

### Step 8: Take the Square Root
Taking the square root of both sides:

\[ 1 + \frac{7+x}{100} < \sqrt{1.176} \]

Calculating \( \sqrt{1.176} \approx 1.084 \):

\[ \frac{7+x}{100} < 0.084 \]

### Step 9: Solve for x
Rearranging gives:

\[ 7 + x < 8.4 \]
\[ x < 1.4 \]

### Step 10: Determine Integral Values of x
Since \( x \) must be an integral value, the possible values for \( x \) are:

- \( x = 1\% \)

### Conclusion
The only possible integral value of \( x \) that satisfies the condition is:

**1%**