CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
SMUAPR15
ASSESSMENT_CODE MCA1030_SMUAPR15
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10112
QUESTION_TEXT
Explain the concept of Conjunction?
SCHEME OF
EVALUATION
Solution:
If two simple statements p and q are connected by the word ‘and’,
then the resulting compound state “p and q” is called a conjunction
of p and q and is written in symbolic form as “p ^ q”
Example: Form the conjunction of the following simple statements:
p : Dinesh is a boy
q :Nagma is a girl
Solution: The conjunction of the statement p and q is given by
p ^ q: Dinesh is a boy and Nagma is a girl.
Example: Translate the following statement into symbolic form
“Jack and Jill went up the hill.”
Solution: The given statement can be rewritten as
“Jack went up the hill and Jill went up the hill”
Letp: Jack went up the hill and q: Jill went up the hill.
Then the given statement is symbolic form is p ^ q
Regarding truth value of conjunction of statements, we have
(D1):T ^ T=T
(D2): T^F=F^T=F^F=F
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
10114
QUESTION_TEXT
Define the following with an example: Empty set, finite and
infinite set, equal sets, subsets, powerset
SCHEME OF
EVALUATION
Solution: Empty set: A set which does not contain any element is
called an empty set or null set
Ex: P={x:1<x<2, x is a natural number}
Finite and infinite set: A set which is empty or consists of a definite
number of elements is called finite. Otherwise the set is called
infinite.
Ex: A={1, 2, 3, ….}infinite
B={1, 2, 3}finite
Equal sets: 2 sets A and B are said to be equal if they have exactly
the same elements and we write A=B
Ex: Let A ={1, 2, 3, 4}, B={3, 4, 1, 2}
Then A=B
Subsets: If every element of a set A is also an element of a set b,
then A is called a subset of B denoted by A C B
Ex: A={1, 2, 3, 4}, B={0, 1, 2, 3, 4, 5}
Then A C B
Power set: The collection of all subsets of a set A is called the
power set of A, denoted by P(A)
Ex: A={1, 2}
Then P(A)= {∅, {1}, {2}, {1, 2}}
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
74004
Explain the cartesian product of Sets.
QUESTION_TEXT
SCHEME OF
EVALUATION
Solution:
Let A, B be two sets. If a is in A, b is in B, then (a, b) denotes an
ordered pair whose first component is a and the second component
is b. Two ordered pairs (a, b) and (c, d) are said to be equal if and
only if a = c and b = d. (2 marks)
In the ordered pair (a, b), the order in which the elements a and b
appears in the bracket is important. Thus (a, b) and (b, a) are two
distinct ordered pairs if a not equal to b. Also, an ordered pair (a, b)
is not the same as the set {a, b}. (2 marks)
Definition: The set of all ordered pairs (a, b) of elements a is in A, b
is in B is called the Cartesian Product of sets A and B and is denoted
by A x B. Thus
A x B={(a, b): a is in A, b is in B}.
Let A={a1, a2}, B={b1, b2, b3. To write the elements of A x B, take a1
is in A and write all elements of B with a1, i.e., (a1, b1), (a1, b2), (a1,
b3). Now take a2 belongs to A and write all the elements of B with
a2, i.e., (a2, b1), (a2, b2), (a2, b3). Therefore, A x B will have six
elements, namely (a1, b1), (a1, b2), (a1, b3), (a2, b1), (a2, b2), (a2,
b3) (6 marks)
QUESTION_TYPE DESCRIPTIVE_QUESTION
QUESTION_ID
124973
Explain the following with example:
i.
Disjunction
QUESTION_TEXT ii. Negation
iii. Truth table
SCHEME OF
EVALUATION
Disjunction
If two simple statements p and q are connected by the word ‘or’,
then the resulting compound statement “p or q” is called
disjunction of statements p and q
Example: Form the disjunction of the following simple
statements:
p : The sun shines.
q :It rains.
Solution: The disjunction of the statements p and q is given by
pν q : The sun shines or it rains.
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
124974
QUESTION_TEXT
State the important results of consistency of system of equation
SCHEME OF
EVALUATION
[2+2+2+2+2]
1.
The system of equation A×=B is consistent if
and only if the matrices A and [A:b] are of the same rank
(2 Marks)
2.
unknowns n and
If the number of equations m ≥ number of
i.
If rank of A≠ to rank of [A:b], then equations are
consistent
ii.
If rank of A=to rank of [A:b]=n, the number of
unknowns, then equations are consistent & have a unique
solution
(2 Marks)
iii. If rank of A=to rank [A:b]<n, the number of unknowns,
then equations are consistent & have finite number of solutions
(2 Marks)
iv. Since for homogeneous systems of equations, rank of
A=rank of [A:b]
Hence homogeneous system of equation always have
solution and so consistent
(2 Marks)
v.
If the number of equations is same as number of
unknowns, then a nonzero solution exists only if the coefficient
matrix is non singular i.e |A|≠0
(2 Marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
124975
QUESTION_TEXT
Explain the concept of complex numbers. Briefly explain two
properties of complex numbers for addition and multiplication.
Let C denote the set of all ordered pairs of real numbers.
SCHEME OF
EVALUATION
i.e., C = {(x, y) : x, y  R}
on this Set C define addition ‘+’ and multiplication ‘.’ by
(x1, y1) + (x2, y2) = (x1 + x2 , y1 + y2) → (1)
(x1, y1) . (x2, y2) = (x1 x2 – y1y2, x1y2 + x2y1) → (2)
Then the elements of C which satisfy the above rules of addition
and multiplication are called complex numbers. (2 marks)
Properties of addition1.
Closure law : If z1 = (x1,y1), z2 (x2, y2)
Then from Equation (1) z1+ z2 = (x1,y1) + (x2, y2) = (x1 + x2,
y1+y2)
Which is also ordered pair of real numbers. Hence z1+z2  C,
therefore for every z1, z2 C, z1+z2  C (2 marks)
2.
Associate law :
z1+(z2 + z3) = (z1+ z2) + z3 for every z1, z2, z3C (2 marks)
Similarly other any 3 properties can be explained.
Properties of multiplication1.
Closure law : If z1 = (x1, y1), z2 = (x2, y2)C then from (2)
(2 marks)
z1, z2 = (x1, y1) (x2, y2) = (x1x2 – y1y2, x1y2 + x2y1)
Which is also an ordered pair of real number. Hence z1, z2 is also a
complex number. Thus for every z1, z2C, z1z2C
2.
Existence of identity element :
There exists (1, 0) C such that (x, y)(1, 0) = (x.1 –y.0, x.0 + 1.y) =
(x, y) for every (x, y) C. Here (1, 0) is called the
multiplicative identity element. (2 marks)
Similarly other any three properties can be explained.