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ASSIGNMENT
|
PROGRAM
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MCA(REVISED
FALL 2012)
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|
SEMESTER
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FIRST
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SUBJECT CODE
& NAME
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MCA1030- FOUNDATION
OF MATHEMATICS
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|
CREDIT
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4
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BK ID
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B1646
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MAX.MARKS
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60
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Note:
Answer all questions. Kindly note that answers for 10 marks questions should be
approximately of 400 words. Each question is followed by evaluation scheme.
1
(i)State Cauchy’s Theorem.
Answer: Cauchy's theorem is
a theorem in the mathematics of group
theory, named after Augustin Louis Cauchy. It states that if G is a finite
group and p is a prime
number dividing
theorder of G (the number of elements in G), then G contains an element of order p. That is, there is x in G so that p is the lowest non-zero number with xp = e,
where e is the identity
element.
The theorem is related to Lagrange's theorem,
which states that the order of any subgroup of a finite group G divides the order of G. Cauchy's theorem implies
that for any prime divisor p of the order of G, there is a subgroup of G whose order is p—the
(ii)Verify Cauchy’s
Theorem for the following function
𝑆𝑖𝑛𝑥,𝑜𝑠𝑥 𝑖𝑛 [0,𝜋2]
Answer: Answer: - A
basic concept in the general Cauchy theory is that of winding number or index
of a point with respect to a closed curve not containing the point. In order to
make this precise, we need several preliminary results on logarithm and
argument
Q.2 Define Tautology and contradiction. Show that
a)
(pn q) n (~ p) is a
tautology.
b)
(pÙ q) Ù(~ p) is a
contradiction
Answer: - Tautology: - In logic, a tautology (from the Greek word ταυτολογία) is a formula
which is true in every possible interpretation. Philosopher Ludwig Wittgenstein
first applied the term to redundancies of propositional logic in 1921; (it had
been used earlier to refer to rhetorical tautologies, and continues to be used
in that alternate sense). A formula is satisfiable if it is true under at least
one interpretation, and thus a tautology is a formula whose negation is
unsatisfiable. Unsatisfiable statements, both through negation and affirmation,
are known
Q.3 State Lagrange’s Theorem. Verify Lagrange’s mean value theorem for
the function
f(x) = 3 x2 – 5x + 1 defined in interval [2, 5]
Answer: - Suppose f is a
function defined on a closed interval [a,b] (with a<b ) such that the
following two conditions hold:
1. f is a
continuous function on the closed interval
[a,b](i.e., it is right continuous at a , left continuous at b , and
two-sided continuous at all points in the open interval(a,b) ).
2. f is a differentiable
function on the open
Q.4 Define Negation. Write the negation of each of the following
conjunctions:
A) Paris is in France and London is in England.
B) 2 + 3 = 5 and 8 < 10.
Answer: - Negation: - In logic,
negation, also called logical complement, is an operation that takes a
proposition p to another proposition "not p", written ¬p, which is
interpreted intuitively as being true when p is false and false when p is true.
Negation is thus a unary (single-argument) logical connective. It may be
applied as an operation on propositions, truth values, or semantic values more
generally. The action or logical operation of negating or making negative b
: a negative statement, judgment, or
doctrine; especially : a logical
proposition formed
(b )𝑥2𝑦2−𝑎2(𝑥2+𝑦2)=0
Q.5 Find
the asymptote parallel to the coordinate axis of the following curves
(i) (𝑥2+𝑦2)𝑥−𝑎𝑦2=0
(ii) 𝑥2𝑦2−𝑎2(𝑥2+𝑦2)=0
Answer: -
(I) (𝑥2+𝑦2)𝑥−𝑎𝑦2=0
F(x) = (𝑥2+𝑦2)𝑥−𝑎𝑦2
(b )𝑥2𝑦2−𝑎2(𝑥2+𝑦2)=0
Q.6 Define (I) Set (ii) Null Set (iii) Subset (iv) Power set (v)Union
set
Answer: - Set: - In
mathematics, a set is a collection of distinct objects, considered as an object
in its own right. For example, the numbers 2, 4, and 6 are distinct objects
when considered separately, but when they are considered collectively they form
a single set of size three, written {2,4,6}. Sets are one of the most
fundamental concepts in mathematics. In everyday life, we have to deal with the
collections of objects of one kind or the other.
·
The
collection of even natural numbers less than 12 i.e., of the numbers 2,4,6,8,
and 10.
·
The
collection of vowels in the English alphabet, i.e., of the letters a ,e ,i ,o ,
u.
·
The
collection of all students of class
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