CUSTOMER_CODE
SMUDE
DIVISION_CODE
SMUDE
EVENT_CODE
SMUAPR15
ASSESSMENT_CODE MCA1030_SMUAPR15
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
74003
QUESTION_TEXT
Let p: The south-west monsoon is very good this year.
q: Rivers are rising
Give the verbal translation of
i. ~ [(~p) v (~q)]
ii. p  q
iii. (p v q) v (~p)
iv. ~ (p ^ q)
v. (~p) ^ (~q)
SCHEME OF
EVALUATION
i. The south-west monsoon is very good this year and rivers are
rising.
ii. If the south-west monsoon is very good this year then the
rivers are rising.
iii. The south-west monsoon is very good this year or the rivers
are rising or the south-west monsoon is not very good this year.
iv. The south-west monsoon is not very good this year or the
rivers are not rising.
v. The south west monsoon is not very good and the rivers are
not rising.
(2  5 = 10 marks)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
74005
Explain the concept of Logical Equivalence with example.
QUESTION_TEXT
SCHEME OF
EVALUATION
Solution:
Two statements S1(p, q, r, …) and S2 (p, q, r, …) are said to be
logically equivalent, or simply equivalent if they have the same
truth values for all logical possibilities and is denoted by
S1(p, q, r, …) = S2 (p, q, r, …)
In other words, S1 and S2 are logically equivalent if they have
identical truth tables (by identical truth tables we mean the entries
in the last column of the truth tables are same.
(4 marks)
Example: Show that ~(p^q) is logically equivalent to (~p)v(~q)
(6 marks)
Solution: The truth tables for both the statements are
Truth table for ~(p^q)
p q (p^q) ~(p^q)
T T T
F
T F F
T
F T F
T
F F F
T
Truth table for (~p)v(~q)
p q ~p ~q (~p)v(~q)
T T F F
F
T F F T
T
F T T F
T
F F T T
T
Now, observe that the entries (truth values) in the last column of
both the tables are same. Hence, the statement ~(p^q) is
equivalent to the statement (~p)v(~q)
QUESTION_TYP
DESCRIPTIVE_QUESTION
E
QUESTION_ID
124969
Let A={1, 2, 3}, B={3, 4} and C={4, 5, 6}. Then find
i.
A×(B∩C)
QUESTION_TEX ii. (A×B)∩(A×C)
T
iii. A×(B∪C)
iv. (A×B)∪(A×C)
SCHEME OF
EVALUATION
i.
We have
ii.
We note that
and
. Therefore,
Therefore
iii.
Clearly
iv.
In view of (ii), we see that
. Thus
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
124970
QUESTION_TEXT
Explain the concepts of scalar and vector field with
example.
SCHEME OF
EVALUATION
(5 marks each)
QUESTION_TYPE
DESCRIPTIVE_QUESTION
QUESTION_ID
124974
QUESTION_TEXT
State the important results of consistency of system of equation
[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)
SCHEME OF
EVALUATION
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.
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)
SCHEME OF
EVALUATION
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.