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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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Q1. State Leibnitz’s theorem. Find the nth
derivative of(𝑥)= 𝑥2𝑒𝑎𝑥, using Leibnitz theorem. [2+8] 10
Answer.
Leibnitz’s theorem
Statement: Let u(x) and v(x) be nth
order 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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