Given:
Simple Interest (S.I.) = Rs. 450
Compound Interest (C.I.) = Rs. 465
Time (T) = 2 years

To find:
Rate of Interest (R)

Solution:

We know that the formula for simple interest is:
S.I. = (P * R * T) / 100
where P is the principal amount, R is the rate of interest, and T is the time period.

Similarly, the formula for compound interest is:
C.I. = P * (1 + R/100)^T - P
where P is the principal amount, R is the rate of interest, and T is the time period.

Step 1: Finding the Principal amount
To find the principal amount, we can use the formula:
Principal (P) = (100 * S.I.) / (R * T)
Substituting the given values:
P = (100 * 450) / (R * 2)

Step 2: Calculating the Compound Interest
Using the formula for compound interest, we have:
465 = P * (1 + R/100)^2 - P
Simplifying the equation:
465 = P * ((1 + R/100)^2 - 1)
Dividing both sides by P:
465 / P = (1 + R/100)^2 - 1
Let x = (1 + R/100)^2
So, the equation becomes:
465 / P = x - 1

Step 3: Solving the equation
Now, we can substitute the value of P from Step 1 into the equation from Step 2:
465 / ((100 * 450) / (R * 2)) = x - 1
Simplifying the equation:
(465 * R * 2) / (100 * 450) = x - 1
Multiplying both sides by (100 * 450):
930 * R / 450 = 100 * x - 100

Step 4: Simplifying the equation
Dividing both sides by 930:
R / 450 = (100 * x - 100) / 930
Simplifying further:
R / 450 = (100x - 100) / 930

Step 5: Finding the value of R
Multiplying both sides by 450:
R = (450 * (100x - 100)) / 930
Simplifying:
R = 45 * (100x - 100) / 93

Step 6: Evaluating R
Now, we need to evaluate the value of R using the options given in the answer choices. We can substitute the values of R one by one and check which option satisfies the equation.

Once we find the correct value of R, we can calculate the principal amount (P) using the formula from Step 1.

Hence, the rate of interest and the principal amount can be determined using the given information.