MATHEMATICS II - Deneme Sınavı - 14

Since in ayn graph the sum of all the degrees is equal to twice the number of edges, we have (3 + 3 + 3 + 3 + 3 + 3 + 3 + 3) = 3 |H|. Hence number of edges is | H|=8. The correct answer is D.

Degrees of all vertices are given below

Vertex

a

b

c

d

e

f

Degree

2

4

2

2

2

2

Hence the largest vertex degree is 4. The correct answer is B.

Since only c, b, a, e are adjacent to d, the set of all neighbours of d is {c, b, a, e}. The correct answer is E.

We know from handshaking lemma that in any graph the sum of all the degrees is an even number but, 1+2+3+3+3+3=15 is not even. The correct answer is E.

Since only a, c, d and f are adjacent to b, the set of all neighbours of a is {a, c, d, f}. The correct answer is A.

All the graphs have the plane diagram except K_3,3 . The correct answer is B.

We know by Euler’s theorem on planar graphs that the number of edges is equal to m = f + n - 2. Hence, the number of edges is 9 + 7 - 2 = 14. The correct answer is A.

Cayley’s Theorem: The number of labelled trees on n vertices is n^n-2. So 4^4-2=4^1=16. The correct answer is C.

Number of faces + number of vertices = number of edges + 2

5 + number of vertices = 8 + 2. number of vertices = 5. The correct answer is D.

Vertex

a

b

c

d

e

Degree

3

2

2

3

2

So the largest vertex is 3. The correct answer is D.

Thus, the answer is A.

Linear programming is one of the branches of optimization and studies the minimization or maximization of a linear function under linear constraints. The source of linear programming is practical problems, where we are looking for the best way to allocate limited resources. In these problems, the goal may be maximizing profits or minimizing costs, or maximizing output or minimizing waste. The subject of linear programming was developed in the 1940’s in response to the war effort. Since that time, the subject has grown enormously and has found many commercial as well as non-commercial applications. The answer is C.

Let a, b, and c be given real numbers. It is well known that the equation ax + by = c defines a straight line in the plane. The graph of this line can be drawn by two points on it, for example, by the points of intersections with the coordinate axis. The straight-line ax + by = c separates the whole plane into two half-planes, one of the half-planes is represented by ax + by ≤ c,
the other is represented by ax + by ≥ c. Assume that b > 0. Then the half-plane ax + by ≤ c is a half-plane remaining below the graph of the straight line ax + by = c. The half-plane ax + by ≥ c is a half-plane remaining above the graph of the straight line ax + by = c. 

The answer is A.

A solution set of this system is the set of all points (x, y) satisfying all the inequalities, which is the intersection of the corresponding half-planes.

The answer is B.

The answer is C.