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ASSIGNMENT
| PROGRAM | MCA(REVISED FALL 2012) |
| SEMESTER | FIRST |
| SUBJECT CODE & NAME | MCA1030- FOUNDATION OF MATHEMATICS |
| CREDIT | 4 |
| BK ID | B1646 |
| MAX.MARKS | 60 |
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:
- 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) ).
- 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