1. How
many committees of five people can be chosen from 20 men and 12 women such that
each committee contains at least three women?

(A) 75240 (B) 52492

(C) 41800 (D) 9900

Answer:
B

**Explanation:**

We must
choose at least 3 women, so we calculate the case of 3 women, 4 women and 5
women and by addition rule add the results.

^{12}C

_{3}x

^{20}C

_{2}+

^{12}C

_{4}x

^{20}C

_{1}+

^{12}C

_{5}x

^{20}C

_{0}= (

^{12x11x10}/

_{3x2x1}) x (

^{20x19}/

_{2x1})

+
(

^{12x11x10x9}/_{4x3x2x1}) x 20
+
(

^{12x11x10x9x8}/_{5x4x3x2x1}) x 1
=
220 x 190 + 495 x 20 + 792

=
52492

2. Which
of the following statement(s) is/are false?

(a) A
connected multigraph has an Euler Circuit if and only if each of its vertices
has even degree.

(b) A
connected multigraph has an Euler Path but not an Euler Circuit if and only if
it has exactly two vertices of odd degree.

(c) A
complete graph (K

_{n}) has a Hamilton Circuit whenever n≥3
(d) A cycle
over six vertices (C

_{6}) is not a bipartite graph but a complete graph over 3 vertices is bipartite.**Codes:**

(A) (a) only (B) (b) and (c)

(C) (c) only (D) (d) only

Answer:
D

**Explanation:**

From the
above definitions, we can see that (d) is false. So answer is (D).

3. Which
of the following is/are not true?

(a) The set
of negative integers is countable.

(b) The set
of integers that are multiples of 7 is countable.

(c) The set
of even integers is countable.

(d) The set
of real numbers between 0 and

^{1}/_{2}is countable.
(A) (a) and
(c) (B) (b) and (d)

(C) (b) only (D) (d) only

Answer:
D

4. Consider
the graph given below:

(A) (v

_{1}, v_{4}, v_{6}); (v_{2}, v_{3}, v_{5}, v_{7}, v_{8}) (B) (v_{1}, v_{7}, v_{8}); (v_{2}, v_{3}, v_{5}, v_{6})
(C) (v

_{1}, v_{4}, v_{6}, v_{7}); (v_{2}, v_{3}, v_{5}, v_{8}) (D) (v_{1}, v_{4}, v_{6}, v_{7}, v_{8}); (v_{2}, v_{3}, v_{5})
Answer:
C

**Explanation:**

A simple graph
G=(V,E) is called bipartite if its vertex set can be partitioned into two disjoint
subsets V=V

_{1}⋃V_{2}, such that every edge has the form e=(a,b) where aÏµV_{1}and bÏµV_{2}.
1. Assign Red color to the source
vertex (putting into set V

_{1}).
2. Color all the neighbours with Black
color (putting into set V

_{2}).
3. Color all neighbour’s neighbour
with Red color (putting into set V

_{1}).
4. This way, assign color to all
vertices such that it satisfies all the constraints of m way coloring problem
where m = 2.

5. While
assigning colors, if we find a neighbour which is colored with same color as
current vertex, then the graph cannot be colored with 2 colors (ie., graph is
not Bipartite).

So answer is
option (C).

5. A
tree with n vertices is called graceful, if its vertices can be labelled with
integers 1, 2, ...,n such that the absolute value of the difference of the
labels of adjacent vertices are all different. Which of the following trees are
graceful?

(A) (a) and
(b) (B) (b) and (c)

(C) (a) and
(c) (D) (a), (b) and (c)

6. Which
of the following arguments are not valid?

(a) “If Gora
gets the job and works hard, then he will be promoted. If Gora gets promotion,
then he will be happy. He will not be happy, therefore, either he will not get
the job or he will not work hard”.

(b) “Either
Puneet is not guilty or Pankaj is telling the truth. Pankaj is not telling the
truth, therefore, Puneet is not guilty”.

(c) If n is
a real number such that n>1, then n

^{2}>1. Suppose that n^{2}>1, then n>1.**Codes:**

(A) (a) and
(c) (B) (b) and (c)

(C) (a), (b)
and (c) (D) (a) and (b)

Answer: Marks to all

7. Let
P(m,n) be the statement “m divides n” where the Universe of discourse for both
the variables is the set of positive integers. Determine the truth values of
the following propositions.

(a) ∃m ∀n
P(m,n) (b) ∀n P(1,n) (c)
∀m ∀n P(m,n)

(A) (a)-True;
(b)-True; (c)-False (B) (a)-True;
(b)-False; (c)-False

(C) (a)-False;
(b)-False; (c)-False (D) (a)-True;
(b)-True; (c)-True

Answer:
A

8. Match
the following terms:

**List - I List - II**

(a) Vacuous
proof (i) A proof that the
implication p→q is true

based on the
fact that p is false

(b) Trivial
proof (ii) A proof that the
implication p→q is true

based
on the fact that q is true

(c) Direct
proof (iii) A proof that the
implication p→q is true

that
proceeds by showing that q must be true when p is true.

(d) Indirect
proof (iv) A proof that the
implication p→q is true

that
proceeds by showing that p must be false when q is false.

**Codes:**

(a) (b) (c)
(d)

(A) (i) (ii)
(iii) (iv)

(B) (ii) (iii)
(i) (iv)

(C) (iii) (ii)
(iv) (i)

(D) (iv) (iii)
(ii) (i)

Answer:
A

9. Consider
the compound propositions given below as:

(a) p˅~(p˄q) (b) (p˄~q)˅~(p˄q) (c) p˄(q˅r)

Which of the
above propositions are tautologies?

(A) (a) and
(c) (B) (b) and (c)

(C) (a) and
(b) (D) (a), (b) and (c)

Answer:
Marks to all

Only (a)

10. Which
of the following property/ies a Group G must hold, in order to be an Abelian
group?

(a) The
distributive property

(b) The
commutative property

(c) The
symmetric property

**Codes:**

(A) (a) and
(b) (B) (b) and (c)

(C) (a) only (D) (b) only

Answer:
D

## 1 Comments

answer of q.6 is A

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