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| PROGRAM | MCA(REVISED FALL 2012) |
| SEMESTER | FIRST |
| SUBJECT CODE & NAME | MCA1030- FOUNDATION OF MATHEMATICS |
| CREDIT | 4 |
| BK ID | B1646 |
| MAX. MARKS | 60 |
Q1. State Leibnitz’s theorem. Find the nthderivative of(𝑥)= 𝑥2𝑒𝑎𝑥, using Leibnitz theorem. [2+8] 10
Answer.
Leibnitz’s theorem
Statement: Let u(x) and v(x) be nthorder differentiable functions of x and
Q2. Define Tautology and contradiction. Show that
a) (pV q) V (~ p) is a tautology.
b) (p^ q) ^ (~ p) is a contradiction [2+2+3+3] 10 marks
Answer.
a) A statement is said to be a tautology if it is true for all logical possibilities. In other words, a statement is called tautology if its truth value is T and only T in the last column of its truth table.
Q3. State Lagrange’s Theorem. Verify Lagrange’s mean value theorem for the function f(x) = 3 x2 – 5x + 1 defined in interval [2, 5] [2+8] 10 marks
Answer.
Lagrange’s Theorem
Let f: [a,b] ->R be a continuous function such that f: (a,b)->R is differentiable. Then, there exists c ε (a,b) such that [f(b)-f(a)]/(b-a) =f’(c).
Geometrical Interpretation
Q4. 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. [2+8] 10 marks
Answer.
Negation
An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement is generally formed by introducing the word “not” at some proper place in the
Q5. Find the asymptote parallel to the coordinate axis of the following curves
(i) (𝑥2+𝑦2)x−𝑎𝑦2=0
(ii) 𝑥2𝑦2−𝑎2(𝑥2+𝑦2)=0 [5+5] 10 marks
Answer.
Solution: (i) We have (𝑥2+𝑦2)x−𝑎𝑦2=0
Or x3+(x-a)y2=0
Asymptote parallel to x – axis are obtained by equating to zero the coefficient of the highest power of x. Since the coefficient of highest power of x3 is 1, which is constant so there is no asymptote parallet to x – axis.
Asymptote parallel to y – axis are obtained by equating to zero the coefficient of the highest power of y. Since the coefficient of highest power of y3 is (x-a). which is constant so there is no asymptote parallel to x –axis.
Q6. Define (i) Set (ii) Null Set (iii) Subset (iv) Power set (v) Union set [5*2] 10 marks
Answer.
(i) Set
A set is a well defined collection of objects. The following points may be noted:
i) Objects, elements and members of a set are synonymous terms.
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