MATHEMATICS II - Deneme Sınavı - 6

The lowest element among the 2nd row elements {-4, -10, -7} is -10 and the lowest element among the 4th row elements {-2, 3, 0} is -2. Therefore the guaranteed values/wins for Player I is -10 and -2, summation of them is -12. The correct answer is E.

Player I, by choosing a row, maximizes the obtained element. The lowest elements of the rows are -2, -1 and 0 respectively. The greatest among those values is 0 on the 3^rd row which will be the maximin strategy for Player I. The correct answer is C.

If the Player II chooses the 3rd column, the guaranteed value for him will be the maximal element among {2, -8, -2, 0, -7} which is 2. The correct answer is D.

If Player I chooses 2d row he guarantees win -8, which is minimal among {-8, -2, 5}. Hence the correct answer is A.

For each row write the lowest element to the right, and for each column write the greatest element below the matrix

maximum {2,1} = 2, minimum {4, 2, 7} = 2. There exists an equilibrium pair (1st row, 2nd column). The number 2, which lies in the intersection of 1st row and 2nd column, is the lowest in the 1st row, is the greatest in the 2nd column.

Using the formula for the present value of money we write

So the correct answer is A.

Let us put

and substitute it into the differential equation. Then the characteristic equation is

We can write the solution like

Using initial conditions

we have

Hence

The correct answer is D.

The coefficient matrix of the given system of linear equations is

We find the determinant of the coefficient matrix using Sarrus’ rule:

and we find  

The coefficient matrix of the system of linear equations is

and the detreminant of this matrix is

The other matrices we use in the Cramer’s rule are

and their determinats are

Then the solution is

We can calculate the matrices 

as follows:

Then

We multiply the second equation by 2, then the linear equations become

Now we add the equations to eliminate the term

then one gets

If we back-subsitute

into the equations

we obtain

Then these lines intersect at the point

Since the inverse of a matrix

the inverse of

Since

one has

Since

then

is obtained.  Since

is given, we  should have

From this equation one has

Then answer is  

The coefficient matrix of the system of linear equations is

The other matrices we use in the Cramer’s rule are

The determinant of these matrices are

Hence the solution of this system of linear equations is

Because the intersection point of these lines is

the point

must also provide both  of the equations. Hence we obtain

Then one gets

Since

we obtain

Hence

Considering

The coefficient matrix of the linear system is

and the determinant of

is 6.