a trust fund has rs30000 that must be invested in two diferent types of bonds.the first bond pays 5% and second 7% interest per year. using matrix multiplication determine how to divide rs 30000 among two types of bonds if the trust must obtain na annual total interest of
(a)rs 1800 (b)rs2000 (c)rs1600
(A) $ 1800
To determine how to divide the Rs 30000 among the two types of bonds, we can set up a system of equations using matrix multiplication.
Let's denote the amount invested in the first bond as x and the amount invested in the second bond as y. Since we have two unknowns, we need two equations to solve for x and y.
The annual interest earned from the first bond is given by 5% of x, which is 0.05x. The annual interest earned from the second bond is given by 7% of y, which is 0.07y.
Therefore, our system of equations becomes:
0.05x + 0.07y = 1800 ...(1)
0.05x + 0.07y = 2000 ...(2)
0.05x + 0.07y = 1600 ...(3)
To solve this system of equations using matrix multiplication, we can represent it in matrix form as:
⎡0.05 0.07⎤ ⎡x⎤ ⎡1800⎤
⎢ ⎥ x ⎢ ⎥ = ⎢ ⎥
⎣0.05 0.07⎦ ⎣y⎦ ⎣2000⎦
⎣1600⎦
The left matrix represents the coefficients of x and y, the middle matrix represents the variables x and y, and the right matrix represents the constants on the right-hand side of the equations.
Now, we can solve this using matrix multiplication.
First, let's find the inverse of the left matrix:
⎡0.05 0.07⎤^-1 = 1/(0.05 * 0.07 - 0.05 * 0.07) ⎡ 0.07 -0.07⎤
⎣0.05 0.07⎦ ⎣ -0.05 0.05⎦
Simplifying this, we get:
⎡ 0.07 -0.07⎤
⎣-0.05 0.05⎦
Now, we can multiply the inverse of the left matrix with the right matrix:
⎡ 0.07 -0.07⎤ ⎡1800⎤ ⎡x⎤
⎣-0.05 0.05⎦ x ⎣2000⎦ = ⎣y⎦
⎣1600⎦
Using matrix multiplication, we can find the values of x and y.
⎡(0.07 * 1800) + (-0.07 * 2000)⎤ ⎡x⎤
⎣(-0.05 * 1800) + (0.05 * 2000)⎦ x ⎣y⎦
Simplifying this further, we get:
⎡-20⎤ ⎡x⎤
⎣ 20⎦ x ⎣y⎦
So, from the equation, we have:
-20x + 20y = 1600
Simplifying this, we get:
-1x + y = 80
x = y - 80
Now, we can solve for the values of x and y.
For part (a) where the trust fund must obtain an annual total interest of Rs 1800:
-20x + 20y = 1800
Substituting x = y - 80, we get:
-20(y - 80) + 20y = 1800
Simplifying this equation, we find:
60y = 2600
y = 43.33
Substituting the value of y back into x = y - 80, we get:
x = 43.33 - 80
x = -36.67
Since the amount invested cannot be negative, the solution is not valid. In this case, there is no way to divide Rs 30000 among the two types of bonds to obtain an annual total interest of Rs 1800.
Similarly, you can solve for parts (b) and (c) using the same method.