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PROGRAM - MCA
SEMESTER -
FIRST
SUBJECT CODE & NAME - MCA1030- FOUNDATION OF
MATHEMATICS
1. State Leibnitz’s theorem. Find
the nth derivative of 𝑦(𝑥) =𝑥2𝑒𝑎𝑥, using Leibnitz theorem.
Answer:
Let u(x) and v(x) be nth order
differentiable functions of x and let y = u(x)v(x). Then
2. Define Tautology and contradiction. Show that
a) 
is a tautology.
is a tautology.
b) 
is a contradiction
is a contradiction
Tautology:
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.
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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]
Let be
a continuous 
function
such that 
is differentiable. Then, there exists 
such that 
function
such that is differentiable. Then, there exists such that
Geometrical Interpretation
It states that given a line
l joining two points on the graph of a
differentiable function f, namely (a, f(a)) and (b, f(b)), there is a
mean value (i.e. c (a, b)) such
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.
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
5. Find the asymptote
parallel to the
coordinate axis of the
following curves
(i) (𝑥2+ 𝑦2) − 𝑎𝑦2= 0
Answer:
We have (𝑥2+ 𝑦2) − 𝑎𝑦2= 0
Or 𝑥3+ (𝑥 – 𝑎)2= 0
Asymptote parallel
to x – axis are
obtained by equating
to zero the
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(ii) 𝑥2𝑦2− 𝑎2 (𝑥2+ 𝑦2) = 0
Answer:
We have 𝑥2𝑦2− 𝑎2 (𝑥2+ 𝑦2) = 0
The coefficient of highest
power of 𝑥2 of x is (𝑦2 - a2)
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6. Define
(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.
ii) Sets are usually denoted by capital letters
A, B, C, X, Y, Z etc.
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