B.Sc. Mathematics Honours SEC-1 Question Paper 2018 (CBCS)

The figures in the right-hand margin indicate full marks.
Candidates are required to give their answers in their own words as far as practicable.
Illustrate the answers wherever necessary.


1. Answer any one question: 1 x 2
(a) Construct the truth table for $(p\rightarrow q)\rightarrow(q\rightarrow p)$
(b) Let $P(x)$ denotes the statement $x=x^2$. If the domain consists of the integers what is the truth values of
(i) $\exists x P(x)$ and (ii) $\forall x P(x)$

2. Answer any three questions: 3 x 5
(a) (i) Define conditional propositions with truth table. 2
(ii) What are the contra positive, converse and Inverse of the conditional proposition "If it is raining then the home team wins". 3
(b) Show that $p\vee(q\wedge r)$ and $(p\vee q)\wedge(p\vee r)$ are logically equivalent. 5
(c) Translate each of these statements into logical expressions using predicates quantifiers and logical connectivities:
(i) No Physics students know C++
(ii) All Mathematics students know C++
(iii) Not every Physics student knows C++
(iv) At least one Mathematics student know C++
(v) No Physics students nor Mathematics students know C++. 5
(d) Determine the truth value of these statements if the domain for all variables consists of all integers:
(i) $\forall n\exists m(n^2 < m)$
(ii) $\exists n\forall m(n < m^2)$
(iii) $\forall n\exists m(n + m = 0)$
(iv) $\exists n\exists m(n^2 + m^2 = 5)$
(v) $\forall n\exists m(n + m = 4 \wedge n - m = 1)$ 5
(e) What is tautology? Show that $(p\wedge q)\rightarrow(p\vee q)$ is a tautology. 1+4


3. Answer any one question: 1 x 2
(a) If $n(A)=5$ and $n(B)=3$ Then find the maximum and minimum value of $n(A\cup B)$.
(b) Find the numbers between 1 and 500 that are divisible by 2, 3 and 5.

4. Answer any one question: 1 x 5
(a) (i) If $aN=\{ax:x\in N\}$, then find $3N \cap 7N$ where $N$ is the set of natural numbers. 3
(ii) Show that $\phi$ is the subset of every set. 2
(b) (i) Define power set. If a finite set has $n$ elements then show that the power set has $2^n$ elements. 1+2
(ii) Differentiate between proper subset and subset with suitable examples. 2


5. Answer any one question: 1 x 10
(a) (i) For any three sets $A, B$ and $C$, prove that $A\times(B\cup C)=(A\times B)\cup(A\times C)$. 5
(ii) Define symmetric difference between two sets. 1
(iii) If $A$ and $B$ be two subsets of a set $X$, then prove that $A\subset B\Leftrightarrow X-B\subset X-A$. 4
(b) (i) A relation $\rho$ is defined on the set $Z$ by "$a\rho b$ iff $2a+3b$ is divisible by $5 \forall a, b \in Z$". Show that $\rho$ is an equivalence relation. 5
(ii) Define partial order relation. Show that the relation '$\subseteq$' (subset) defined on the power set $P(S)$ is a partial order relation. 1+4

6. Answer any three questions: 3 x 2
(a) Let $\rho$ and $\rho'$ be two equivalence relations then show that $\rho\cap\rho'$ also equivalence relation.
(b) Define partition of a set.
(c) Let $A$ be a set with 2 elements. How many reflexive relations can be defined on $A$?
(d) Give an example of a relation which is symmetric but not reflexive and transitive.


1. Answer any five questions: 5 x 2
(a) What are the different features of C++?
(b) Differentiate between pointer and reference variable.
(c) What are the different types of inheritance in C++?
(d) Explain Inline function.
(e) What do you mean by enumeration?
(f) What is implicit and explicit type conversion in C++?
(g) Differentiate between global and local object.
(h) What is friend function?

2. Answer any four questions: 4 x 5
(a) Discuss how data and functions are organized in an object oriented paradigm. List the major areas of application of OOP. 4+1
(b) What do you mean by member access modifiers in C++? Explain exception handling with example.
(c) Define copy constructor. Explain various types of constructors with examples. 1+4
(d) Explain Multi-level and Multiple inheritances with examples.
(e) Write different uses of scope resolution operator (::) in C++.
(f) Write a program to calculate area of rectangle using inline functions.

3. Answer any one question: 1 x 10
(a) Discuss the features of a function template. Write a C++ program to create a function template for finding minimum number out of given numbers. 5+5
(b) What is polymorphism? Elaborate the statement "Overloading is a type of polymorphism" with the help of suitable example and using the concept of function overloading. 2+8