Discrete Mathematics Questions & Answers for Exams on “Counting – Number of Equations Solution”.
1. The linear system Cx = d is known as _________ if d! = 0.
a) homogeneous
b) heterogeneous
c) nonhomogeneous
d) augmented system
Answer: c
Clarification: A linear system Cx = d is known as a homogeneous system if d! = 0. The homogeneous linear system Ax = 0 is called its corresponding homogeneous linear system.
2. Every linear equation determines a _______ in n-dimensional space for n variables.
a) shipshape
b) hyperplane
c) cone
d) pyramid
Answer: b
Clarification: In an m-dimensional space, every linear equation produces a hyperplane for n variables. The solution set is the intersection of these hyperplanes and is planar which may have any dimension smaller than m.
3. Determine all possibilities for the number of solutions of the system of 7 equations in 5 unknowns and it has x1 = 0, x2 = −6, and x3 = 4 as a solution.
a) unique or infinitely many
b) unique
c) finitely many
d) zero
Answer: a
Clarification: Let i be the number of equations and j be the number of unknowns in the given system. Since i> j, the system has at least one solution x1 = 0, x2 = −6, and x3 = 4 and so it is consistent. Thus, it results in either a unique solution or infinitely many solutions.
4. Determine all possibilities for the solution set of the homogeneous system that has y1 = 6, y2 = −4, y3 = 0 as a solution.
a) zero
b) infinitely many
c) finitely many
d) only one
Answer: b
Clarification: Since m<n, the system is either inconsistent or has infinitely many solutions. Since y1 = 6, y2 = −4, y3 = 0 is a solution of the system, the system is not inconsistent. Thus the only possibility is infinitely many solutions.
5. Determine all possibilities for the solution set of the homogeneous system of 5 equations in 3 unknowns and the rank of the system is 3.
a) more than two
b) only one
c) zero
d) infinite
Answer: c
Clarification: Since the rank of this homogeneous system(which is always consistent) and the number of unknowns are equal, the only possible solution is zero and it is a unique solution.
6. Determine all possibilities for the solution set of a homogeneous system that has y1 = 5, y2 = −3, y3 = 2 as a solution.
a) one
b) finitely many
c) infinitely many
d) either one or infinitely many
Answer: c
Clarification: The possibilities for the solution set for any homogeneous system is either a unique solution or infinitely many solutions. Since the homogeneous system has the zero solution and y1 = 5, y2 = −3, y3 = 2 is another solution, it has at least two distinct solution. Thus the only possibility is infinitely many solutions.
7. Determine all possibilities for the solution set of the system of 2 equations in 3 unknowns that has x1 = 4, x2 = −7, x3 = 0 as a solution.
a) one or finitely many
b) infinite
c) finite
d) zero
Answer: b
Clarification: Since m1 = 4, x2 = −7, x3 = 0 is a solution of the system, the system is not inconsistent. Thus the only possibility is infinitely many solutions.
8. Determine all possibilities for the solution set of a homogeneous system of 4 equations in 4 unknowns.
a) only one
b) finitely many or zero
c) zero
d) one or infinitely many
Answer: b
Clarification: Here the number of equations and the number of unknowns are equal and the system is homogeneous, so it may have the zero solution or infinitely many solutions.
9. Determine all possibilities for the solution set of a homogeneous system of 6 equations in 5 unknowns.
a) only one
b) zero
c) one or infinitely many
d) finitely many
Answer: c
Clarification: Since the system is homogeneous and there are more equations than the number of unknowns, so the possibilities are either a unique solution or infinitely many solutions. However, if the rank r of the system is 5, then it can be a unique solution as well as if r<5, then there are infinitely many solutions.
10. Determine all possibilities for the solution set of a homogeneous system of 5 equations in 4 unknowns and the rank of the system is 3.
a) finite
b) zero or finitely many
c) only one
d) infinite
Answer: d
Clarification: A homogeneous system is consistent. The rank is r = 3 and the number of variables is n = 4. Hence there is n – r = 1 free variable. Thus there are infinitely many solutions.