# Mathematical Reasoning Questions

We provide mathematical reasoning practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on mathematical reasoning skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of mathematical reasoning Questions

Question NoQuestionsClass
1Given the true statement: If a
quadrilateral is a square, then it is a rectangle. It follows that, of the converse
and the inverse of this true statement.
A. Only the converse is true
B. only the inverse is true
c. Both are true
D. Neither is true
E. The inverse is true, but the converse is sometimes true
11
2Given ( P: 25 ) is a multiple of ( 5, q: 25 ) is a multiple of 8. Write the compound statement connecting these two
statements with “and”, “or” in ( 60^{t h} )
cases. Check the validity of the
statement.
11
3Which of the following statements are true and which are false? In each case
give a valid reason for saying so
(i) ( p: ) Each radius of a circle is a chord of the circle
(ii) ( mathrm{q}: ) The centre of a circle bisects each chord of the circle
(iii) ( r: ) Circle is a particular case of an ellipse
(iv) ( s: ) If ( x ) and ( y ) are integers such that
( boldsymbol{x}>boldsymbol{y} ) then ( -boldsymbol{x}<-boldsymbol{y} )
( (v) t: sqrt{11} ) is a rational number
11
4Write down the negations for the
following:
(a) If the diagonals of a parallelogram are perpendicular then it is a rhombus.
(b) Kanchanganga is in India and Everest is in Nepal.
(c) The Sun is a star or the Jupiter is a
planet.
11
5Which of the following statements is the inverse of “If you do not understand geometry, then you do not know how to reason deductively.”?
A. If you reason deductively, then you understand geometry.
B. If you understand geometry, then you reason deductively.
c. If the do not reason deductively, then you understand geometry.
D. None of these
11
6Assertion
STATEMENT-1: The minimum value of
the intercepts cut on tangent by a tangent to the ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 )
between the coordinate axes is ( a+b )
Reason

STATEMENT-2: For each pair of two negative real numbers a and b, nequality ( frac{boldsymbol{a}+boldsymbol{b}}{mathbf{2}} geq-sqrt{boldsymbol{a} boldsymbol{b}} ) holds.
A. Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-
B. Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-
c. Statement- – is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

11
7State the following statement is True or
False
The truth value of “4 is even and 8 is
odd” is True
A. True
B. False
11
8The contrapositive of “if ( x ) has courage
then ( x ) will win”, is
A. If ( x ) will in, then ( x ) has courage
B. If ( x ) has no courage, then ( x ) will not win.
c. If ( x ) will not win, then ( x ) has no courage
D. If ( x ) will not win, then ( x ) has courage
11
9The negation of the statement:”If
become a teacher, then I will open a
school” is
A. I will become a teacher and I will not open a school.
B. Either I will not become a teacher or I will not open a school
c. Neither I will become a teacher nor I will open a school
D. I will not become a teacher or I will open a school.
11
10Negation of the statement ( p: ) for every real number, either ( x>1 ) or ( x<1 ) is11
11Check the validity of the following
statement:
( boldsymbol{p}: 125 ) is a multiple of 5 and 7
A. True
B. False
11
12Assertion
( (A): P(n)=n^{2}+n+41 ) is a prime
( forall boldsymbol{n} in boldsymbol{N} )
Reason
(R): If a number is prime then it
contains only two factors; 1 and number itself.
A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
c. (A)is true but (R) is false.
D. (A)is false but (R) is true
11
13“No square of a real number is less than
zero” is equivalent to
A. for every real number a, ( a^{2} ) is non negative.
( mathbf{B} cdot forall a in R, a^{2} geq 0 )
C . either (1) or (2).
D. None of these
11
14The dual of the following statement “Reena is healthy and Meena is beautiful”‘ is
A. Reena is not beaufiful and Meena is not healthy
B. Reena is not beautiful or Meena is not healthy.
c. Reena is not healthy or Meena is not beautiful.
D. None of these.
11
15Find the quantifier which best describes the variable of the open
sentence ( x^{2}+2 geq 0 )
A. Universal.
B. Existential.
c. Neither (a) nor (b)
D. Does not exist.
11
16Which of the following sentences are
NOT a statement?
( A cdot 9 ) is less than 7
B. The sun is a star.
c. There is no rain without clouds
D. Mathematics is fun.
11
17Write the converse, inverse and
contrapositive of the following statements: “If a function is
differentiable then it is continuous”.
11
18( boldsymbol{p}: ) He is hard working.
( boldsymbol{q}: ) He will win
The symbolic form of “If he will not win then he is not hard working”, is
( mathbf{A} cdot p Rightarrow q )
в. ( (sim p) Rightarrow(sim q) )
c. ( (sim q) Rightarrow(sim p) )
D. ( (sim q) Rightarrow p )
11
19Show that the statement
( boldsymbol{p}: ) If ( boldsymbol{x} ) is a real number such hat ( boldsymbol{x}^{mathbf{3}}+ )
( 4 x=0, ) then ( x ) is 0 is true by
direct method
11
20Determine the contrapositve of each of the following statements:
If he has courage he will win.
11
21The converse of “If ( x ) has courage, then ( x )
will win”, is
A. If ( x ) wins, then ( x ) has courage.
B. If ( x ) has no courage, then ( x ) will not win.
C. If ( x ) will not win, then ( x ) has no courage.
D. If ( x ) will not win, then ( x ) has courage.
11
22( sim[(boldsymbol{p} wedge boldsymbol{q}) rightarrow(sim boldsymbol{p} vee boldsymbol{q})] ) is
A. Tautology
c. Neither (A) nor (B)
D. Either (A) or (B)
11
23If ( p, q, r ) have truth values ( T, F, T ) respectively, then which of the following
is True?
A ( cdot(p rightarrow q) wedge r )
в. ( (p rightarrow q) wedge sim r )
c. ( (p wedge q) wedge(p vee r) )
D. ( q rightarrow(p wedge r) )
11
24Negation of “A is in Class ( X^{t h} ) or ( B ) is in
( boldsymbol{X} boldsymbol{I} boldsymbol{I}^{t h^{prime prime}} ) is
A. A is not in class ( X^{t h} ) but ( B ) is in ( X I I^{t h}^{h} )
c. Either A is not in class ( X^{t h} ) or ( mathrm{B} ) is not in ( X I I^{t h} )
D. none of these
11
25Determine the contrapositive of the following statement:
If ( x ) is an integer and ( x^{2} ) is odd, then ( x ) is
odd.
11
26Given: All seniors are mature students.
Which statement expresses a
conclusion that logically follows from
the given statement?
A. All mature students are seniors
B. If Bill is a mature student, then he is a senior
C. If Bill is not a mature student, then he is not a senior
D. If Bill is not a senior, then he is not a mature student
E. All sophomores are not mature students.
11
27State whether the following statement
is True or False.

The sum of two odd numbers and one
even number is even.
A. True
B. False

11
28If ( boldsymbol{x}=mathbf{5} ) and ( boldsymbol{y}=-mathbf{2}, ) then ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{9} )
The contrapositive of this statement is/are
A. If ( x-2 y neq 9 ), then ( x neq 5 ) or ( y neq 2 )
B. If ( x-2 y neq 9 ), then ( x neq 5 ) and ( y neq-2 )
c. If ( x-2 y=9 ), then ( x=5 ) and ( y=-2 )
D. none of these.
11
2971. The Boolean expression
~ (pvqvp^q) is equivalent to :
(a) P (6) 9 (c) ~q
[JEE M 2018]
(d) ~P
11
30State whether the following statement
is True or False.
The product of two even numbers is always even.
A. True
B. False
11
31Prepare the truth table for the following.
( sim boldsymbol{p} wedge boldsymbol{q} )
11
32Which of the following is a statement?
( A ). I am Lion.
B. Logic is an interesting subject
C. A triangle is a circle and 10 is a prime number
D. None of these.
11
33The length ( L ) (in centimetre) of a copper
rod is a linear function of its
Celsius temperature ( C . ) In an experiment, if ( L=124.942 ) when ( C= ) 20 and ( L=125.134 ) when ( C=110 )
express ( L ) in terms of ( C )
11
34State the following statement is True or
False
If Shelly does not like John, then the truth value of the statement “Shelly likes Mike and she likes John” is True
A. True
B. False
11
35Prove that ( 3+sqrt{7} ) is irrational number.11
36( p: H e ) is hard working.
( boldsymbol{q}: ) He is intelligent.
Then ( sim boldsymbol{q} Rightarrow sim boldsymbol{p}, ) represents
A. If he is hard working, then he is not intelligent.
B. If he is not hard working, then he is intelligent
c. If he is not intelligent, then he is not had working
D. If he is not intelligent, then he is hard working
11
37Contrapositive of the statement ‘If two
number are not equal, then their squares are not equal’, is:
11
38Write the negation of the following
statements:
(i) Chennai is the capital of Tamil Nadu
(ii) ( sqrt{2} ) is not a complex number
(iii) All triangles are not equilateral triangle
(iv) The number 2 is greater than 7
(v) Every natural number is an integer
11
39Find the total surface area of a closed
cylindrical petrol storage tank whose diameter ( 4.2 mathrm{m} ) and height ( 4.5 mathrm{m} )
11
40Write the negation of the following
statements
i) ( sqrt{7} ) is a rational number.
ii) Length of both diagonals of any rectangle are equal
11
41The converse of ( boldsymbol{p} Rightarrow boldsymbol{q} ) is
( mathbf{A} cdot p Rightarrow q )
в. ( q Rightarrow p )
c. ( -p Rightarrow-q )
( mathbf{D} cdot-q Rightarrow-p )
11
42The negation of the statement “2 ( +3= )
( 5^{prime prime} ) and ” ( 810 )
c. ( 2+3 neq 5 ) or ( 8 nsupseteq 10 )
D. None of these
11
43Consider the statement, Given that
people who are in need of refuge and
consolation are apt to do odd things, it
is clear that people who are apt to do odd things are in need of refuge and consolation. This statement, of the form ( (P Rightarrow Q) Rightarrow(Q Rightarrow P), ) is logically
equivalent to
A. People who are in need of refuge and consolation are not apt to do odd things.
B. People are apt to do odd things if and only if they are in need of refuge and consolation.
C. People who are apt to do odd things are in need of refuge and consolation.
D. People who are in need of refuge and consolation are apt to do odd things
11
44Negation of the statement “Every natural number is an integer””
A. All natural numbers are whole numbers.
B. Every natural number is not an integer.
c. Every natural number is not a real number.
D. none of the above
11
45The statement form ( (boldsymbol{p} Leftrightarrow boldsymbol{r}) Rightarrow(boldsymbol{q} Leftrightarrow boldsymbol{r}) )
is equivalent to
( mathbf{A} cdot[(sim p vee r) wedge(p vee sim r)] vee sim[(sim q vee r) wedge(q vee sim r)] )
( mathbf{B} cdot sim[(sim p vee r) wedge(p vee sim r)] wedge[(sim q vee r) vee(q vee sim r)] )
( mathbf{c} cdot[(sim p vee r) wedge(sim p vee sim r)] wedge[(sim q vee r) wedge(q vee sim r)] )
( mathbf{D} cdot sim[(sim p vee r) wedge(p vee sim r)] wedge[(sim q vee r) wedge(q vee sim r)] )
11
46The compound statement, “If you won
the race; then you did not run faster
than others” is equivalent to
A. “If you won the race, then you ran faster than others””
B. “If you ran faster than others, then you won the race””
C. “If you did not win the race, then you did not run faster than others””
D. “If you ran faster than others, then you did not win the race”
11
47when does the current flow from A to B?
A. p is open, q is open, ris closed
B. p is closed, q is open, ris closed
c. p is closed, q is closed, ris open
D. p is open, q is closed, ris closed
11
48Determine the contrapositve of each of the following statements:

It is necessary to be strong in order to be a sailor.

11
49State whether the following statement
is True or False.

The sum of three odd numbers is even.
A. True
B. False

11
50Inverse of a statement can be explained
as
A. Negating both the hypothesis and conclusion of a conditional statement
B. Antecedent is the negation of the original antecedent and whose consequent is the negation of the original consequent.
c. both are correct
D. none is correct
11
51The negative of the statement “lf a
number is divisible by 15 then it is divisible by 5 or ( 3 ” )
A. if a number is divisible by 15 then it is not divisible by and 3
B. a number is divisible by 15 and it is not divisible by 5 o
3
c. a number is divisible by 15 or it is not divisible by 5 and
3
D. a number is divisible by 15 and it is not divisible by 5 and 3
11
52What is true about the statement “If two
angles are right angles the angles have equal measure” and its converse “If two angles have equal measure then the two angles are right angles”?
A. The statement is true but its converse is false
B. The statement is false but its converse is true
c. Both the statement and its converse are false
D. Both the statement and its converse are true
11
53In the above network, current flows from
( T ) to ( M, ) when
A. p closed, q closed and ropened
B. p closed, q opened and r closed
c. p opened; q closed and r closed
D. All the above
11
54The negative of the statement “If a
number is divisible by 15 then it is
divisible by 5 or ( 3^{prime prime} )
A. If a number is not divisible by ( 15, ) then it is not divisible by 5 and 3
B. A number is divisible by 15 and it is not divisible by 5 or 3
C. A number is not divisible by 15 or it is not divisible by 5 and 3
D. A number is divisible by 15 and it is not divisible by 5 and 3
11
55Check the validity of the following
statement:
( boldsymbol{p}: mathbf{1 0 0} ) is a multiple of ( mathbf{4} ) and ( mathbf{5} )
A. True
B. False
11
56Write the component statement of the following compounds statements and check whether the compound statement is true or false:
All rational number are real and all real
number are not complex.
11
5766. The Boolean Expression (pAqvqVP9
equivalent to:
JEE M 2016]
(a) pUq (b) PV~9 (c) ~paq (d) pUq
11
58“If ( p, ) then ( q^{prime prime} ) is logically equivalent to which of the following?
I. If ( q, ) then ( p )
II. If not ( p, ) then not ( q )
III. If not ( q, ) then not ( p )
A. None of the above
B. III Only
c. I and II only
D. I and III only
E. I, II and III
11
59Negate each of the following
statements:
There exists a number which is equal to
¡ts square.
11
60Tell whether the following is certain to happen, impossible can happen but not certain.
A tossed coin will land heads up.
11
61Which of the following connectives satisfy commutative law?
( A cdot wedge )
B.
( c cdot Leftrightarrow )
D. All the above
11
62Write converse of statement. In ( Delta A B C )
if ( A B=A C ) then ( C=B )
11
63When electric current is passed through acidified water for 1930 s, ( 1120 mathrm{mL} ), of ( H_{2} )
gas is collected (at STP) at the cathode.
What is the current passed in amperes?
A . 0.05
B. 0.50
( c . ) 5.
D. 50
11
64The inverse of the statement “lf a
number is divisible by 4 then it is also divisible by ( 2^{prime prime} ) is –
A. If a number is divisible by 4 , then it is always divisible by 2
B. If a number is not divisible by 4 , then it is divisible by
2.
c. If a number is not divisible by ( 4, ) then it is not divisible by 2.
D. None of the above
11
65Assertion
Triangles on the same base and between the same parallel lines are
equal in area.
Reason
The distance between two parallel lines
is same everywhere.
A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
c. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
66Tell if the following statement is true or false. In case give a valid reason for saying so
( boldsymbol{p}: ) Each radius of a circle is a chord of the circle.
A. True
B. False
11
67Which of the following statements is
the inverse of
“lf you do not understand geometry, then you do not know how to reason deductively.” ?
A. If you reason deductively, then you understand geometry.
B. If you understand geometry, then you reason deductively
c. If the do not reason deductively, then you understand geometry
D. None of the above
11
68In which of the following cases, ( boldsymbol{p} Leftrightarrow boldsymbol{q} ) is
true?
A . p is true, q is true
B. p is false, q is true
c. p is true, q is false.
D. None of these.
11
69the following
(a) 20.0 (U 101
47. Let S be a non-empty subset of R. Consider the follow
statement :
P: There is a rational number x € S such that x>0.
Which of the following statements is the negation of the
statement P?
[2010]
(a) There is no rational number x e S such than x<0.
(b) Every rational number x ES satisfies x<0.
(C) XE S and x<0 = x is not rational.
(d) There is a rational number x e S such that x<0.
following relations:
11
70In the above network, current flows from
( M ) to ( N, ) when
A. ( q ) closed, ropened and p closed
B. q opened, p opened and r closed.
C. q opened, p closed and r closed.
D. q closed, p closed and r opened
11
71If ( P(n) ) be the statement ( n(n+1), n in )
( N ) is even, then
A ( . P(2) ) is true
B. ( P(3) ) is true
c. ( P(4) ) is true
D. all of the above
11
72( boldsymbol{p} leftrightarrow boldsymbol{q} ) is equivalent to
( mathbf{A} cdot p rightarrow q )
( mathbf{B} cdot q rightarrow p )
( mathbf{c} cdot(p rightarrow q) vee(q rightarrow p) )
D. ( (p rightarrow q) wedge(q rightarrow p) )
11
73If ( P(n) ) be the statement ( n(n+1)+1 ) is odd, then which of the following is
false?
A ( . P(2) )
в. ( P(3) )
c. ( P(4) )
D. none of these
11
74Which of the following statements is
the inverse of

“Our pond floods whenever there is a
thunderstorm.”?
A. If there is a thunderstorm, then our pond floods
B. If we do not get a thunderstorm, then our pond does not flood.
c. If our pond does not flood, then we did not get a thunderstorm
D. None of the above

11
75Determine the contrapositive of the following statement:
It never rains when it is cold.
11
76the negation of
M 2019-9 April
76. For any two statements p and q, the ne
expression p v (p^q) is: [JEEM 2019_
(a) ~pa-q
(b) paq
(c) pHq
(d) “p V-9
11
77State whether the following sentence is always true, always false or ambiguous. Justify your answer February has only 28 days.11
78State whether the statement
( p: ) If ( x ) is a real number such that ( x^{3}+ )
( 19 x=0, ) then ( x ) is 0 is true / False
A. True
B. False
11
79State whether the following sentence is always true, always false or ambiguous. Justify your answer. Makarasankranthi falls on a Friday.11
80Which of the following statements is
the contrapositive of the statement, You
win the game if you know the rules but
are not overconfident.
A. If you lose the game then you dont know the rules or you are overconfident
B. A sufficient condition that you win the game is that you know the rules or you are not overconfident
c. If you dont know the rules or are overconfident you lose the game
D. If you know the rules and are overconfident then you win the game
11
81The contrapositive of ( sim boldsymbol{p} rightarrow(boldsymbol{q} rightarrow sim boldsymbol{r}) )
is
( mathbf{A} cdot(q wedge r) rightarrow p )
в. ( (q rightarrow r) rightarrow p )
c. ( (q vee r) rightarrow p )
D. None of these.
11
82The converse of “If in a triangle ( A B C, A B=A C, ) then ( angle B=angle C^{prime prime}, ) is
A. If in a triangle ( A B C, angle B=angle C ), then ( A B=A C ).
B. If in a triangle ( A B C, A B neq A C ), then ( angle B neq angle C ).
c. If in a triangle ( A B C, angle B neq angle C ), then ( A B neq A C ).
D. If in a triangle ( A B C, angle B neq angle C ), then ( A B=A C ).
11
83Which of the following is always true?
( mathbf{A} cdot sim(p rightarrow q) equiv sim p wedge q )
В . ( (p vee q) equiv sim p vee sim q )
c. ( sim(p Rightarrow q) equiv(p wedge sim q) )
D. ( (p wedge q) equiv sim p wedge sim q )
11
84is an irrational number”, q be the
dental number”, and r be the
[2008]
U=1, D-6
(d) a= 3, D = 4
Let p be the statement is an irrational number 19
statement “y is a transcendental number”, and
statement “x is a rational number iffy is a transcende
number”.
Statement-1 :ris equivalent to either q or p
Statement-2:ris equivalent to (p4-9).
(a) Statement -1 is false, Statement-2 is true
(b) Statement-1 is true, Statement-2 is true; Statement-
a correct explanation for Statement-1
(C) Statement -1 is true, Statement-2 is true; Statement-2
is not a correct explanation for Statement-1
(d) Statement -1 is true, Statement-2 is false
The statom
42.
11
85The negation of ( sim s vee(sim r wedge s) ) is
equivalent to
( mathbf{A} cdot S wedge r )
В. ( S wedge sim(r wedge sim s) )
c. ( S vee sim(r wedge sim s) )
D. None of These
11
86Show that the following statement is true by the method of contrapositive
( p: ) If ( x ) is an integer and ( x^{2} ) is even then ( x ) is also even
11
87The negation of ( boldsymbol{q} vee sim(boldsymbol{p} wedge boldsymbol{r}) ) is?11
8869. The following statement
(p →q) → [(p ) →q] is :
(a) a fallacy
(b) a tautology
(c) equivalent to – p > (d) equivalent to p–9
[JEEM 2017]
11
89The inverse of statement is ” If you grew
in Alaska, then you have seen snow.”
A. “If you did not grow up in Alaska, then you have not seen snow.”
B. “If you grow up in Alaska, then you have not seen snow.”
C. “If you did not grow up in Alaska, then you have seen snow.”
D. None of these
11
90“If we control population growth, then we prosper”. Negative of this proposition is:
A. If we do not control population growth, we prosper
B. If we control population, we do not prosper
c. we control population and we do not prosper
D. If we don’t control population, we do not prosper
11
91( sim(boldsymbol{p} wedge boldsymbol{q}) equiv )
( mathbf{A} cdot sim p vee sim q )
в. ( p vee sim q )
( mathrm{c} cdot sim p vee q )
D. None
11
92Determine whether the following compound statement are true of false:
Delhi is in England and ( 2+2=4 )
A . True
B. False
11
93When does the truth value of the
statement ( (boldsymbol{p} vee boldsymbol{r}) Leftrightarrow(boldsymbol{q} vee boldsymbol{r}) ) become
true?
A . p is true, q is true
B. p is false, q is false
c. p is true, r is true
D. Both (1) and (3)
11
94Let ( p: A ) triangle is equilateral, ( q: A ) triangle is equiangular then inverse of
( boldsymbol{q} rightarrow boldsymbol{p} ) is
A. If a triangle is not equilateral then it is not equiangular
B. If a triangle is not equiangular then it is not equilateral
C. If a triangle is equiangular then it is not equilateral
D. If a triangle is equiangular then it is equilateral
11
95Determine the contrapositve of each of the following statements:
If ( x ) is less than zero, then ( x ) is not
positive.
11
96( sim(p Leftrightarrow q) ) is equivalent to
( mathbf{A} cdot sim p wedge sim q )
B ( . sim p vee sim q )
( mathbf{C} cdot(p wedge sim q) vee(sim p wedge q) )
D. none of these
11
97( boldsymbol{p}: ) He is hard working.
( boldsymbol{q}: ) He will win
The symbolic form of “He is hard working then he will win”, is
A ( . p vee q )
в. ( p wedge q )
( mathbf{c} cdot p Rightarrow q )
D. ( q Rightarrow p )
11
98State whether true or false.
The sum of the interior angles of a
A. True
B. False
11
99Consider the following three
statements:
( P: 5 ) is a prime number
( Q: 7 ) is a factor of 192
( mathrm{R}: ) L.C.M. of 5 and 7 is 35
Then the truth value of which one of
thefollowing statements is true?
A ( cdot(P wedge Q) vee(sim R) )
в. ( (sim P) wedge(sim Q wedge R) )
c. ( (sim P) vee(Q wedge R) )
D. ( P vee(sim Q wedge R) )
11
100Given the following six statements:
(1) All women are good drivers
(2) Some women are good drivers
(3) No men are good drivers
(4) All men are bad drivers
(5) At least one man is a bad driver
(6) All men are good drivers. The statement that negates statement
(6) is:
A . (1)
B. (2)
( c cdot(3) )
( D cdot(4) )
E. (5)
11
101The contrapositive statement of
statement “If ( x ) is prime number, then ( x )
is odd” is
A. If ( x ) is not is prime number, then ( x ) is not odd
B. If ( x ) is not odd, then ( x ) is not a prime number
c. If ( x ) is a prime number, then ( x ) is not odd
D. If ( x ) is not a prime number, then ( x ) is odd
11
10242.
[2008]
(d) Statement -I is true, Statement-2 15 ans
The statement p → (qp) is equivalent to
(a) p (p9)
(b) p (pvq)
(c) p (p )
(d) p (p )
11
103Find the component statements of the
following compound statements and check whether they are true or false
(i) Number 3 is prime or it is odd
(ii) All integers are positive or negative
(iii)100 is divisible by 3,11 and 5
11
104The Boolean expression ( ((boldsymbol{p} wedge boldsymbol{q}) vee )
( (p vee sim q)) wedge(sim p wedge sim q) ) is equivalent
to :
( mathbf{A} cdot p wedge(sim q) )
B . ( p vee(sim q) )
c. ( (sim p) wedge(sim q) )
D. ( p wedge q )
11
105“If Deb and Sam go to the mall then it is
snowing” Which statement below is logically equivalent?
A. If Deb and Sam do not go to the mall then it is not snowing
B. If Deb and Sam do not go to the mall them it is snowing
C. If it is snowing then Deb and Sam go to the mall
D. If it is not snowing then Deb and Sam do not go to the mal
11
106Consider the statement, if ( n ) is divisible
by 30 then ( n ) is divisible by 2,3 and by 5 Which of the following statements is equivalent to this statement??
A. If ( n ) is not divisible by 30 then ( n ) is divisible by 2 or divisible by 3 or divisible by 5
B. If ( n ) is not divisible by 30 then ( n ) is not divisible by 2 or not divisible by 3 or not divisible by 5
c. If ( n ) is divisible by 2 and divisible by 3 and divisible by 5 then ( n ) is divisible by 30
D. If ( n ) is not divisible by 2 or not divisible by 3 or not divisible by 5 then ( n ) is not divisible by 30
11
107The contrapositive of ( (boldsymbol{p} vee boldsymbol{q}) Rightarrow boldsymbol{r} ) is
A ( cdot r Rightarrow(p vee q) )
в. ( r Rightarrow(p vee q) )
c. ( r Rightarrow sim p wedge sim q )
D. ( R Rightarrow(q vee r) )
11
108State true or false.
A rhombus is a parallelogram.
A. True
B. False
11
109If ( boldsymbol{x}=mathbf{5} ) and ( boldsymbol{y}=mathbf{2} ) then ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{9 . T h e} )
contrapositive of this statement is
A. If ( x-2 y neq 9 ) then ( x neq 5 ) or ( y neq 2 )
B. If ( x-2 y neq 9 ) then ( x neq 5 ) and ( y neq-2 )
c. If ( x-2 y=9 ) then ( x=5 ) and ( y=-2 )
D. none of these.
11
110Which of the following statements is the converse of “If the moon is full, then
the vampires are prowling.”?
A. If the vampires are prowling, then the moon is full.
B. If the moon is not full, then the vampires are prowling
C. If the vampires are not prowling, then the moon is not full
D. None of these
11
111Here are some words translated from an
artificial language mie pie is blue light
mie tie is blue berry
aie tie is raspberry
Which words could possibly mean “light fly”?
A. pie zie
B. pie mie
c. aie zie
D. aie mie
11
112The negation of the statement, “l go to school everyday”, is
A. I never go to school.
B. Some days, I do not go to school.
c. Not all the days I do not go to school.
D. All days I go to school.
11
113The converse of ( boldsymbol{p} Rightarrow boldsymbol{q}, ) is
( mathbf{A} cdot p Rightarrow q )
в. ( q Rightarrow p )
c. ( sim p Rightarrow sim q )
D. ( q Rightarrow sim p )
11
114The converse of: “If two triangles are
congruent then they are similar” is
A. If two triangles are similar then they are congruent.
B. If two triangles are not congruent then they are not similar.
C. If two triangles are not similar then they are not congruent
D. None
11
115The inverse of: “If two triangle are congruent then they are similar” is
A. If two triangles are similar then they are congruent.
B. If two triangles are not congruent then they are not ( operatorname{similar} )
c. If two triangles are not similar then they are not congruent.
D. None
11
116Write the negations of the following
statements:
(a) All students of this college live in the
hostel.
(b) 6 is an even number or 36 is a
perfect square
11
117Write opposite of the following:
a)30 ( k m ) north
b)Increase in weight
c) Loss of ( R s .700 )
d) ( 100 m ) above sea level
11
118Find the inverse of the statement,” If
( triangle A B C ) is equilateral, then it is
isosceles”.
( mathbf{A} cdot ) If ( triangle A B C ) is isosceles, then it is equilateral
B. If ( triangle A B C ) is not equilateral, then it is isosceles.
C. If ( triangle A B C ) is not equilateral, then it is not isosceles.
D. If ( triangle A B C ) is not isosceles, then it is not equilateral.
11
119The converse of “if in a triangle
( A B C, A B>A C, ) then ( angle C=angle B^{prime prime}, ) is
( mathbf{A} cdot ) If in a triangle ( A B C, angle C=angle B, ) then ( A B>A C )
B. If in a triangle ( A B C, A B not=A C, ) then ( angle C not=angle B ).
C. If in a triangle ( A B C, angle C not=angle B, ) then ( A B not=A C )
D. If in a triangle ( A B C, angle C not=angle B, ) then ( A B>A C ).
11
120Consider statement “If I do not work, I
will sleep. If I am worried, I will not sleep. Therefore if I am worried, I will work”.
This statement is
A. valid
B. invalid
c. a fallacy
D. none of these
11
121The inverse of statement” If a
quadrilateral is a rectangle,then it has two pairs of parallel sides.”
A. If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides.
B. If a quadrilateral is a rectangle, then it has two pairs of parallel sides.
C. If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle.
D. None of these
11
122Are the following pairs of statements negations of each other?
(i) The number ( x ) is not a rational
number

The number ( x ) is not an irrational
number
(ii) The number ( x ) is a rational number
The number ( x ) is an irrational number

11
123The contrapositive of ( boldsymbol{p} rightarrow(sim boldsymbol{q} rightarrow sim boldsymbol{r}) )
is
A ( cdot(sim q wedge r) rightarrow sim p )
в. ( (q rightarrow r) rightarrow sim p )
c. ( (q vee sim r) rightarrow sim p )
D. none of these.
11
124( sim(p vee q) vee(sim p wedge q) ) is logically
equivalent to
( mathbf{A} cdot sim p )
B.
( c cdot q )
( mathrm{D} cdot sim q )
11
125If ( p ) and ( q ) are two statement then ( (p leftrightarrow )
( sim q) ) is true when
A. ( p ) and ( q ) both are true
B. ( p ) and ( q ) both are false
C. ( p ) is false and ( q ) is true
D. None of these
11
126( (p wedge sim q) wedge(sim p wedge q) ) is a:
A. A tautology
C. Both a tautology and a contradiction
D. Neither a tautology nor a contradiction
11
127Determine the contrapositive of the following statements:
If it snows, then they do not drive the
car.
11
128The negation of ( boldsymbol{q} vee sim(boldsymbol{p} wedge boldsymbol{r}) ) is?
( mathbf{A} cdot sim q wedge sim(p vee r) )
В ( . sim q wedge(p wedge r) )
( mathbf{c} cdot sim q vee(p wedge r) )
D. ( r vee(p wedge r) )
11
129Write the component statement of the following compounds statements and check whether the compound statement is true or false:
The sand heats up quickly in the sun and does not cool down fast at night
11
130If a sentence can be judged to be true or false, but not both then it is called
A. an open sentence
B. a statement
c. a tautology.
11
131Find the truth value of 14 is a
composite number or 15 is a prime no.
11
132( (sim p wedge q) wedge q ) is
A. a tautology
C. neither a tautology nor a contradiction
D. none of these
11
133When ( y=3 ) which of the following is
FALSE?
A. ( y ) is prime and ( y ) is odd
B. ( y ) is odd or ( y ) is even
c. ( y ) is not prime and ( y ) is odd
D. ( y ) is odd and ( 2 y ) is even
11
134Which of the following is not a
statement?
A. Every set is a finite set.
B. Every square is a rectangle.
c. The sun is a star.
D. Shut the window.
11
135The converse of “if ( boldsymbol{x} in boldsymbol{A} cap boldsymbol{B} ) then ( boldsymbol{x} in )
( A ) and ( x in B ” ), is
A. If ( x in A ) and ( x in B ), then ( x in A cap B )
B. If ( x notin A cap B ), then ( x notin A ) or ( x notin B ).
c. If ( x notin A ) or ( x notin B, ) then ( x notin A cap B )
D. If ( x notin A ) or ( x notin B ), then ( x in A cap B ).
11
136The inverse of “If two angles are
congruent, then they have the same measure” is
A. If two angles have the same measure, then they are congruent
B. If two angles are not congruent, then they do not have the same measure
C. If two angles do not have the same measure, then they are not congruent
D. None of these
11
137Let
( boldsymbol{p}: mathbf{5 7} ) is an odd prime number ( q: 4 ) is a divisor of 12 ( boldsymbol{r}: mathbf{1 5} ) is the LCM of ( boldsymbol{3} ) and ( mathbf{5} )
be three simple logical statements. Which one of the following is true?
A ( . p vee(sim q wedge r) )
В ( . sim p vee(q wedge r) )
c. ( (p wedge q) vee sim r )
D. ( (p vee(q wedge r) )
E . ( (p vee q) wedge r )
11
138Determine the contrapositive of the following statement:

Only if he does not tire will he win.

11
139Assertion
STATEMENT ( 1: sim(p leftrightarrow sim q) ) is equivalent
to ( (boldsymbol{p} vee sim boldsymbol{q}) wedge(sim boldsymbol{p} vee boldsymbol{q}) )
Reason
STATEMENT ( 2: sim(p leftrightarrow q) ) is equivalent
to ( (boldsymbol{p} wedge sim boldsymbol{q}) vee(sim boldsymbol{p} wedge boldsymbol{q}) )
A. Statement 1 is True, Statement 2 is True; Statement 2 i a correct explanation for Statement 1
B. Statement 1 is True, Statement 2 is True ; Statement 2 is NOT a correct explanation for Statement 1
c. statement 1 is True, Statement 2 is False
D. Statement 1 is False, Statement 2 is True
11
140Find truth value of compound
statement ‘All natural number are even
or odd”
11
141The converse of the contrapositive of the
conditional ( boldsymbol{p} rightarrow sim boldsymbol{q} ) is :
A ( . p rightarrow q )
в. ( p rightarrow sim q )
( mathbf{c} . sim q rightarrow p )
( mathbf{D} cdot sim p rightarrow q )
11
142Which of the following statements is the inverse of “Our pond floods whenever there is a thunderstorm.”?
A. If there is a thunderstorm, then our pond floods
B. If we do not get a thunderstorm, then our pond does not flood.
c. If our pond does not flood, then we did not get a thunderstorm
D. None of these
11
143Negation of ” ( 2+3=5 ) and ( 8<10 " ) is
A ( .2+3 neq 5 ) and ( <10 )
B. ( 2+3=5 ) and ( 8 Varangle 10 )
( c cdot 2+3 neq 5 ) or ( 8 Varangle 10 )
D. None of the above
11
144The cost of an article including the sales tax is ( R s .616 . ) The rate of sales
( operatorname{tax} ) is ( 10 %, ) if the shopkeeper has made a profit of ( 12 %, ) then the cost price of the article is?
A. Rs .350
в. ( R s .400 )
c. ( R s .500 )
D. Rs .800
11
145The contrapositive of the statement ‘If am not feeling well, then I will go to the
doctor’ is:
A. If I am feeling well, then I will not go to the doctor
B. If / will go to the doctor, then I am felling well
c. If 1 will not go to the doctor, then I am feeling well
D. If I will go to the doctor, then I am not feeling well
11
146The converse of the statement “if ( boldsymbol{p}<boldsymbol{q} )
then ( p-x<q-x^{prime prime} ) is :
A. If ( pq-x )
B. If ( p>q ), then ( p-x>q-x )
C. If ( p-x>q-x ), then ( p>q )
D. If ( p-x<q-x ), then ( p<q )
11
147The negation of the compound
proposition ( boldsymbol{p} vee(sim boldsymbol{p} vee boldsymbol{q}) ) is
A ( . p wedge q )
B.
( c cdot f )
D. ( (p wedge q) wedge sim p )
11
148State whether the following statement
is True or False.

Sum of two prime numbers is always
even

11
149The converse of ( boldsymbol{p} rightarrow(boldsymbol{q} rightarrow boldsymbol{r}) ) is
( mathbf{A} cdot(q wedge sim r) vee p )
в. ( (sim q vee r) vee p )
( c cdot(q wedge sim r) wedge sim p )
( p )
D. ( (q wedge sim r) wedge p )
11
150The converse of the contrapositive of the
conditional ( boldsymbol{p} rightarrow sim boldsymbol{q} ) is.
A ( . p rightarrow q )
в. ( p rightarrow sim q )
( mathbf{c} . sim q rightarrow p )
( mathbf{D} cdot sim p rightarrow q )
11
151Assertion
Two distinct lines cannot have more
than one point in common.
Reason
Any number of lines can be drawn
through one point.
A. Both Assertion and Reason are correct and Reason is
the correct explanation for Assertion
B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
C. Assertion is correct but Reason is incorrect
D. Assertion is incorrect but Reason is correct
11
152State whether the following statement
is True or False.

Prime numbers do not have any factors.
A. True
B. False

11
153(d) 2
53. The negation of the statement
[2012]
“If I become a teacher, then I will open a school”, is :
I will become a teacher and I will not open a school.
(b) Either I will not become a teacher or I will not open a
school.
(c) Neither I will become a teacher nor I will open a school.
(d) I will not become a teacher or I will open a school.
BA Letr
ben obseruation
11
154If a compound statement is made up of three simple statements, then the number of rows in the truth table is
( mathbf{A} cdot mathbf{8} )
B. 6
( c cdot 4 )
D.
11
155Negation of “A is in Class ( X^{t h} ) or ( mathrm{B} ) is in
( X I I^{t h prime prime} ) is
A. A is not in class ( X^{t h} ) but ( B ) is in ( X I I^{t h}^{h} )
C. Either A is not in class ( X^{t h} ) or ( B ) is not in ( X I I^{t^{t}} )
D. None of these.
11
156( sim[boldsymbol{p} wedge(sim boldsymbol{q})]= )
( mathbf{A} cdot sim p wedge sim q )
в. ( sim p vee sim q )
( mathbf{c} . sim p wedge q )
( mathrm{D} cdot sim p vee q )
11
157Which of the following is a statement?
( A ). I am Lion.
B. Logic is an interesting subject
C. A triangle is a circle and 10 is a prime number
D. None of these.
11
158If a compound statement is made up of
three simple statements, then the
number of rows in the truth table is
11
159Write the following statement in five different ways conveying the same meaning
( mathbf{p}: ) If triangle is equiangular then it is an obtuse angled triangle
11
160Identify the quantifier in the following
statements and write the negation of
the statements
(i) There exists a number which is equal
to its square
(ii) For every real number ( x, x ) is less
( operatorname{than} x+1 )
(iii) There exists a capital for every state in India
11
161State whether the following statement is True or False.

The product of three odd numbers is odd.
A. True
B. False

11
162State the following statement is True or
False
f ( p, q, r ) are statements, with truth
values ( mathrm{T} ), ( mathrm{F} ), ( mathrm{T} ) respectively, then the truth value of ( (sim p vee q) wedge sim r Rightarrow p ) is ( T )
A. True
B. False
11
163Determine whether the argument used to check the validity of the following statement is correct.
( p: ) If ( x^{2} ) is irrational, then ( x ) is rational
The statement is true because the
number ( x^{2}=pi^{2} ) is irrational, therefore
( x=pi ) irrational.
A. True
B. False
11
164Which statement represents the inverse of the statement “If it is snowing then
Skeeter wears a sweater”?
A. If Skeeter wears a sweater then it is snowing
B. If Skeeter does not wear a sweater then it is not snowing
C. If it is not snowing then Skeeter does not wear a sweater
D. If it is not snowing then Skeeter wears a sweater
11
165The converse of converse of the
statement ( boldsymbol{p} Longrightarrow sim boldsymbol{q} ) is
( mathbf{A} cdot sim q Longrightarrow p )
в. ( p Longrightarrow q )
( mathbf{c} cdot p Longrightarrow sim q )
D. ( q Longrightarrow sim p )
11
166Earth is a planet. Choose the option that is a negation of this statement.
A. Earth is round
B. Earth is not round
c. Earth revolves round the sun
D. Earth is not a planet
11
167Write the truth values of following
statements
(i) 9 is a perfect square but 11 is a prime number.
(ii) Moscow is in Russia.
(iii) London is in France.
11
168when ( boldsymbol{x} in boldsymbol{R} )
Are the following pairs of the
statements are negative of each other:
The number ( x ) is a rational number.
The number ( x ) is not an irrational
number.
If Yes put 1 else 0.
11
169STATEMENT: All cabbages are red. Which of the following statements shows that the statement above is
FALSE?
A. David is eating a red apple
B. Bill is eating a green apple
c. Alice is not eating a red cabbage
D. Ted is eating a red cabbage
E. Keisha is eating green cabbage
11
170Find the converse of the statement,”If
ABCD is square, then it is a rectangle”.
( A . ) If ( A B C D ) is a square, then it is .not a rectangle.
B. If ( A B C D ) is not a square, then i is a rectangle.
C. if ABCD is a rectangle, then it is square.
D. If ( A B C D ) is not a square, then it is not a rectangle.
11
171The contrapositive of ( (sim p wedge q) rightarrow sim r ) is
equivalent to
A. ( (p wedge q) rightarrow r )
в. ( (p wedge q) vee r )
c. ( r rightarrow(p vee sim q) )
D. none of these
11
172The inverse of “If ( x ) has courage, then ( x )
will win”, is
( mathbf{A} cdot ) If ( x ) will win, then ( x ) has courage.
B. If ( x ) has no courage, then ( x ) will not win.
C. If ( x ) will not win, then ( x ) has no courage.
D. If ( x ) will not win, then ( x ) has courage.
11
173The negation of the statement ( boldsymbol{q} vee ) ( (p wedge sim r) ) is equivalent to
( mathbf{A} cdot sim q wedge(p rightarrow r) )
в. ( q vee sim(p rightarrow r) )
c ( cdot q wedge(sim p wedge r) )
D. None of these.
11
17443.
[2009]
Statement-1: -(pH-q) is equivalent
Statement-2 : – ( – 9) is a tautology
(a) Statement-1 is true, Statement-2 is true;
Statement-2 is not a correct explanation for Statens
(6) Statement-1 is true, Statement-2 is false.
(c) Statement-1 is false, Statement-2 is true.
(d) Statement-1 is true, Statement-2 is true,
Statement-2 is a correct explanation for statement-
on for Statement-1.
11
175The statement pattern ( (boldsymbol{p} wedge boldsymbol{q}) wedge[sim boldsymbol{r} vee )
( (p wedge q)] vee(sim p wedge q) ) is equivalent to
( A )
B. 9
( c cdot p wedge q )
D.
11
176The disjunction of the statements, “lt is raining; The sun is shining” is
A. It is raining and the sun is shining
B. It is raining or the sun is shining
c. It is raining and the sun is not shining
D. It is not raining or the sun is not shining
11
177The contrapositive of the statement ‘I go to school if it does not rain’ is:
A. If it rains, I do not go to school.
B. If I do not go to school, it rains.
c. If it rains, I go to school.
D. If i go to school, it rains
11
178If ( boldsymbol{x}+mathbf{4}=mathbf{8}, ) then ( boldsymbol{x}=mathbf{4}, ) Inverse of the
statement is-
A. If ( x+4=8 ), then ( x neq 4 )
B. If ( x+4 neq 8 ), then ( x neq 4 )
c. If ( x+4 neq 8 ), then ( x=4 )
D. none of the above
11
179Mary says “The number I am thinking is divisible by 2 or it is divisible by ( 3 ” ). This statement is false if the number Mary is
thinking of is
( A cdot 6 )
B. 8
( c cdot 11 )
D. 15
11
180Write the negation of the following statement and check whether the
resulting statement is true.
The sum of 2 and 5 is 9
11
181State whether the following sentences
are always true, always false or
11
18261. The statement ~(po~q) is :
(JEEM 20141
(a) a tautology (b) a fallacy
© eqivalent to p q (d) equivalent to – p a
11
183Which of the following is logically equivalent to ( sim(sim p Rightarrow q) ? )
A ( cdot p wedge q )
В . ( p wedge sim q )
( mathrm{c} cdot sim p wedge q )
D ( . sim p wedge sim q )
11
184Write the compound statement, “If ( mathrm{p} ) then ( q ) and if ( q, ) then ( p^{prime prime} ) in symbolic form.
A ( (p wedge q) wedge(q wedge p) )
a ( (p wedge q) wedge(q wedge p) )
в. ( (p Longrightarrow q) vee(q Longrightarrow p) )
c. ( (q Longrightarrow p) wedge(p Longrightarrow q) )
D. ( (p wedge q) vee(q wedge p) )
11
185Tell whether the following is certain to happen, impossible can happen but not certain.
You are older today than yesterday.
11
186Assertion
( P(n)=n^{2}+n+1 ) is an odd natural
number ( forall n in N )
Reason
If 1 is added to an even number then it
becomes an odd number
A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
c. (A)is true but (R) is false
D. (A)is false but (R) is true
11
187Write the negative of the following
statement:

All the students completed their homework.

11
188Which of the following is a statement?
A. I like lions
B. Logic is an interesting subject
C. A triangle is a circle and 10 is a prime number
D. None of these.
11
189What is the truth value of the statement
( mathbf{2} times mathbf{3}=mathbf{6} ) or ( mathbf{5}+mathbf{8}=mathbf{1 0} ? )
A. True
B. False
C. Neither True nor False
D. Cannot be determined
11
190Construct the truth table for the
statement ( (boldsymbol{p} wedge boldsymbol{q}) vee boldsymbol{r} )
11
191State whether the following statements are true or false. Give reasons for your

Square numbers can be written as the sum of two odd numbers.

11
192The negation of the statement ( (boldsymbol{p} rightarrow )
( boldsymbol{q}) wedge boldsymbol{r} ) is
( mathbf{A} cdot p wedge sim q vee sim r )
В ( .(sim p wedge q) wedge(sim r) )
c. ( (p wedge sim q) wedge(r) )
D. ( (p wedge sim q) wedge(sim r) )
11
193The statement ” ( x>5 ) or ( x<3 " ) is true,
if x equals
( mathbf{A} cdot mathbf{1} )
B. 3
( c cdot 4 )
D.
11
194The compound statement, “If you want to top the school, then you do not study hard” is equivalent to
A. “If you want to top the school, then you need to study hard”
B. “If you will not top in the school, then you study hard””
c. “If you study hard, then you will not top the school”.
D. “If you do not study hard, then you will top in the school”
11
195Negation of ( boldsymbol{q} vee sim(boldsymbol{p} wedge boldsymbol{r}) ) is
A ( cdot sim q wedge sim(p wedge r) )
В . ( q wedge(p wedge r) )
( mathbf{c} cdot sim q vee(p wedge r) )
D. None of these
11
196The truth value of the statement, “We
celebrate our Independence day on 15 August”, is
A . T
B. F
C. neither T nor F
D. Cannot be determined
11
197Write the inverse of the statement-
If you do not drink your milk, you will not be strong.
A. If you are strong, then you drink your milk
B. If you do not drink your milk, then you are strong
C. If you drink your milk, then you are strong
D. None of the above
11
198If ( p rightarrow(q vee r) ) is false, then the truth
values of ( p, q, r ) are respectively
( mathbf{A} cdot mathbf{T}, mathbf{F}, mathbf{F} )
B. F, F, F
c. F, T, T
D. Т, т, ( F )
11
199Let ( P(n) ) denote the statement that
( boldsymbol{n}^{2}+boldsymbol{n} ) is odd. It is seen that ( boldsymbol{P}(boldsymbol{n}) Rightarrow )
( P(n+1), P(n) ) is true for all
( mathbf{A} cdot n>1 )
B.
c ( . n>2 )
D. None of these
11
200If ( p, q, r ) are simple proportions with
truth values ( T, F, T, ) then the truth
value of ( (sim p vee q) wedge sim r Rightarrow p ) is
A. True
B. False
c. True, if ( r ) is false
D. True, if ( q ) is true
11
201Which of the following is a statement?
A. Open the door.
c. Hurrah! We have won the match.
D. Two plus two is five
11
202The contrapositive of “if in a triangle
( A B C, A B=A C, ) then ( angle B=angle C^{prime prime}, ) is
( mathbf{A} cdot ) If in a triangle ( A B C, angle B=angle C, ) then ( A B=A C )
B. If in a triangle ( A B C, A B neq A C ), then ( angle B neq angle C ).
C. If in a triangle ( A B C, angle B neq angle C, ) then ( A B neq A C ).
D. If in a triangle ( A B C, angle B neq angle C, ) then ( A B=A C ).
11
203Write the converse and contrapositive of
the statement
“If two traingles are congruent, then their areas are equal.”
11
204Determine the contrapositive of the following statement:
If she works, she will earn money
11
205Re write each of the following
statements in the form “p if and only if
( q^{prime prime} )
(i) ( p: ) If you watch television then your mind is free and if your mind is free then you watch television
(ii) ( q: ) For you to get an A grade it is necessary and sufficient that you do all the homework regularly
(iii) ( r: ) If a quadrilateral is equiangular then it is a rectangle and if a quadrilateral is a rectangle then it is equiangular
11
206Consider the following two statements:
( P: ) If 7 is an odd number, then 7 is
divisible by 2.
Q: If 7 is a prime number, then 7 is an odd number
If ( V_{1} ) is the truth value of the
contrapositive of ( mathrm{P} ) and ( V_{2} ) is the truth value of contrapositive of ( Q, ) then the ordered pair ( left(V_{1}, V_{2}right) ) equals:
A. ( (F, F) )
в. ( (T, T) )
c. ( (T, F) )
D. ( (F, T) )
11
207Inverse of the statement “If the triangle
with side lengths ( a, b, c ) is a right
triangle, then ( a^{2}+b^{2}=c^{2} )
A ( cdot ) If ( a^{2}+b^{2} neq c^{2} ) then the triangle with side lengths ( a, b, c ) is not a right triangle.
B. If ( a^{2}+b^{2}=c^{2} ) then the triangle with side lengths ( a, b, c ) is a right triangle
C. If the triangle with side lengths ( a, b, c ) is a not right triangle, then ( a^{2}+b^{2} neq c^{2} )
D. None of the above.
11
208If ( p ) ‘s truth value is ( T ) and ( q^{prime} ) s truth value
is ( F, ) then which of the following have
the truth value ( T ) ?
(i) ( p vee q )
(ii) ( sim boldsymbol{p} vee boldsymbol{q} )
( (text { iii) } p vee(sim q) )
( (text { iv }) p wedge(sim q) )
A ( cdot(i),(text { ii) }, ) (iii)
B. (i), (iii), (iv)
c. (i), (ii), (iv)
D. (ii), (iii), (iv)
11
209The contrapositive of ( boldsymbol{p} rightarrow(sim boldsymbol{q} rightarrow sim boldsymbol{r}) )
is equivalent to
( mathbf{A} cdot(sim q wedge r) rightarrow sim p )
В ( cdot(q wedge sim r) rightarrow sim p )
c. ( p rightarrow(sim r vee q) )
D. ( p wedge(q vee r) )
11
210The contrapositive of the following statement, “If the side of a square doubles, then its area increases four
times” is:
A. If the area of a square increases four times, then its side is not doubled
B. If the area of a square increases four times, then its side is doubled
C. If the area of a square does not increase four times then its side is not doubled
D. If the side of a square is not doubled, then its area does not increase four times
11
211Using truth tables, examine whether
the statement pattern ( (boldsymbol{p} wedge boldsymbol{q}) vee(boldsymbol{p} wedge boldsymbol{r}) )
is a tautology, contradiction or contingency
11
212Show that the statement
( p: ) “If ( x ) is a real number such that ( x^{3}+ )
( 4 x=0 ) then ( x ) is ( 0^{prime prime} ) is true by
(i) direct method
(iii) method of contrapositive
11
213The contrapositive of “if in a triangle
( A B C, A B>A C, ) then ( angle C>angle B^{prime prime}, ) is
( mathbf{A} cdot ) If in a triangle ( A B C, angle C>angle B, ) then ( A B>A C )
B. If in a triangle ( A B C, A B not ) # A then ( angle C Varangle angle B ).
C. If in a triangle ( A B C, angle C Varangle angle B, ) then ( A B>A C ).
D. If in a triangle ( A B C, angle C Varangle angle B, ) then ( A B Varangle A C ).
11
214Write the given statement using numbers, literals and signs of basic
operations. State what each letter
represents. The selling price equals the sum of the cost price and the profit.
11
215Check the validity of the following
statement:
( boldsymbol{p}: boldsymbol{6} boldsymbol{0} ) is a multiple of ( boldsymbol{3} ) and ( boldsymbol{5} )
A. True
B. False
11
216Assertion
( (A): ) Let ( n in N )
( boldsymbol{p}(boldsymbol{n})=boldsymbol{n}(boldsymbol{n}+1) ) is an even number
Reason
(R): Product of two consecutive natural
numbers is even.
A. Both (A) & (R) are individually true &
(R) is correct explanation of (A),
B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
c. (A)is true but (R) is false
D. (A)is false but (R) is true
11
217In the following letter sequence, some of the letters are missing. These are given in order as one of the alternatives below
Choose the correct alternative.
( boldsymbol{alpha} boldsymbol{beta}_{-} boldsymbol{alpha} boldsymbol{alpha}_{-} boldsymbol{beta} boldsymbol{beta} boldsymbol{beta}_{-} boldsymbol{alpha} boldsymbol{alpha} boldsymbol{alpha} boldsymbol{a}_{-} boldsymbol{beta} boldsymbol{beta} boldsymbol{beta} ldots )
( mathbf{A} cdot alpha beta beta alpha )
B. ( beta alpha beta alpha )
( mathrm{c} cdot alpha alpha alpha beta )
( mathbf{D} cdot alpha beta alpha beta )
11
218Which of the following is the negation of the statement, For all odd primes ( boldsymbol{p}<boldsymbol{q} )
there exists positive non-primes ( r<s )
such that ( p^{2}+q^{2}=r^{2}+s^{2} )
A. For all odd primes ( p<q ) there exists positive nonprimes ( r<s ) such that ( p^{2}+q^{2}=r^{2}+s^{2} )
B. There exists odd primes ( p<q ) such that for all positive non-primes ( r<s, p^{2}+q^{2}=r^{2}+s^{2} )
C. There exists odd primes ( p<q ) such that for all positive non-primes ( r<s, p^{2}+q^{2} neq r^{2}+s^{2} )
D. For all odd primes ( p<q ) and for all positive nonprimes ( r<s, p^{2}+q^{2} neq r^{2}+s^{2} )
11
21976. For any two statements p and q, the negation of the
expression pvpaq) is: [JEEM 2019-9 April
(a) ~PA – – . (b) PA
(c) paq
d) – pv~9
11
220Contrapositive of the statement “if two number are not equal then their square are not equal is ;
A. If the squares of two number are equal, then the number are not equal
B. If the squares of two number are equal, then the number are equal
C. If the square of two number are not equal then number are equal
D. If the square of two number are not, equal, then the number are not equal
11
221Which of the following is NOT equivalent to ( boldsymbol{p} rightarrow boldsymbol{q} ? )
A. ponly if q
B. q is necessary for ( p )
c. q only if ( p )
D. p is sufficient for ( q )
11
222Which statement represents the inverse of the statement “lf it is snowing, then
Skeeter wears a sweater.”?
A. If Skeeter wears a sweater, then it is snowing
B. If Skeeter does not wear a sweater, then it is not snowing.
C. If it is not snowing, then Skeeter does not wear a sweater
D. If it is not snowing, then Skeeter wears a sweater
11
223Given are three positive integers ( a, b )
and ( c . ) Their greatest common divisor is
( D ; ) their least common multiple is ( M ) Then, which two of the following statements are true?
(1) The product ( M D ) cannot be less
than abc
(2) The product ( M D ) cannot be greater
than abc
(3) ( M D ) equals ( a b c ) if and only if ( a, b, c ) are each prime
(4) ( M D ) equals ( a b c ) if and only if ( a, b, c ) are relatively prime in pairs (This means: no two have a common
factor greater than ( 1 . )
( mathbf{A} cdot 1,2 )
в. 1,3
c. 1,4
D. 2,3
E .2,4
11
224State the converse and contrapositive of each of the following statements:
(i) ( p: A ) positive integer is prime only if it has no divisors other than 1 and itself
(ii) ( q: ) I go to a beach whenever it is a
sunny day
(iii) ( r: ) If it is hot outside then you feel
thirsty
11
225Write the converse and contropositive of
If a parallelogram is a square, then it is a rhombus’.
11
226The contrapositive of : “If two triangles
are congruent then they are similar” is
A. If two triangles are similar then they are congruent
B. If two triangles are not congruent then they are not similar
C. If two triangles are not similar then they are not congruent
D. None
11
227Consider the. following compound
statementt
(i) Mumbai is the capital of Rajasthan or Maharashtra,
(ii) ( sqrt{3} ) is a rational number or an
irrational number,
(iii) 125 is a multiple of 7 or 8
(iv) A rectangle is a quadrilateral or a regular hexagon. Which of the above statements is not
true?
A. (i)
B. (ii)
c. (iii)
( D cdot(mid v) )
11
228Which of the following is a statement?
A. Rani is a beautiful girl.
B. Shut the door.
c. Yesterday was Friday.
D. If its raining then there must be cloud in the sky
11
229How many buses are there for Suryapet from Hyderabad?
( mathbf{A} cdot mathbf{7} )
B. 17
c. 12
D. 15
11
23051. Consider the following statements
[2011]
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest
The negation of the statement “Suman is brilliant and
dishonest if and only if Suman is rich” can be expressed as
(a) ~(Q (P^~R)) (b) ~QH-PAR
(c) ~(PA~R) HQ (d) ~PAQ H-R)
11
231Which is logically equivalent to “If today is Sunday Matt cannot play hockey”?
A. Today is Sunday and Matt can play hockey
B. If Matt plays hockey then today is not Sunday
c. Today is Sunday and Matt cannot play hockey
D. Today is not Sunday if and only if Matt plays hockey
11
232Which of the following statements is the converse of “You cannot skateboard
if you do not have a sense of balance.”?
A. If you cannot skateboard, then you do not have a sense of balance
B. If you do not have a sense of balance, then you cannot skateboard.
c. If you skateboard, then you have a sense of balance
D. None of these
11
233The inverse of the statement ( (p wedge sim )
( boldsymbol{q}) rightarrow boldsymbol{r} ) is
A. ( (p vee sim q) rightarrow sim r )
в. ( (sim p wedge sim q) rightarrow sim r )
c. ( (sim p vee q) rightarrow sim r )
D. None of these.
11
234Which of the following is logically equivalent to ( sim(sim p Rightarrow q) ? )
A ( cdot p wedge q )
В . ( p wedge sim q )
( mathrm{c} cdot sim p wedge q )
D ( . sim p wedge sim q )
11
235The contrapositive of the statement “If it is raining, then I will not come”, is :
A. If I will come, then it is not raining.
B. If I will not come, then it is raining.
c. If 1 will come, then it is raining
D. If I will not come, then it is not raining
11
236Let ( p, q ) and ( r ) be any three logical
statements. Which one of the following is true?
( mathbf{A} cdot sim[p wedge(sim q)] simeq(sim p) wedge q )
В . ( (p vee q) wedge(sim r) equiv(sim p) vee(sim q) vee(sim r) )
c. ( sim[p vee(sim q)] equiv(sim p) wedge q )
D . ( [p wedge(sim q)] equiv(sim p) wedge sim q )
E . ( [p wedge(sim q)] equiv p wedge q )
11
237( 7 neq 10 . ) Choose the option that
expresses the statement using the correct connective.
A ( . sim(7=10) )
В. ( (7 neq 10) )
c. ( sim(-7=10) )
D. ( (-7 neq-10) )
11
238Which of the following is the inverse of the proposition : ‘If a number is a prime then it is odd”?
A. If a number is not a prime then it is odd
B. If a number is not a prime then it is not odd
c. If a number is not odd then it is not prime
D. If a number is odd then it is a prime
11
239The statement ( boldsymbol{p} rightarrow(boldsymbol{q} rightarrow boldsymbol{p}) ) is
equivalent to
A ( cdot p rightarrow(p wedge q) )
в. ( p rightarrow(p leftrightarrow q) )
c. ( p rightarrow(p rightarrow q) )
D. ( p rightarrow(p vee q) )
11
240( sim p wedge q ) is logically equivalent to
( mathbf{A} cdot p rightarrow q )
в. ( q rightarrow p )
( mathbf{c} cdot sim(p rightarrow q) )
( mathbf{D} cdot sim(q rightarrow p) )
11
241Consider
Statement-1 : (p^-9)^(-p^q) is a fallacy.
Statement-2: (p 9) (-9 -p) is a tautology.
[JEE M 2013]
(a) Statement-1 is true; Statement-2 is true;
Statement-2 is a correct explanation for Statement-1.
(b) Statement-1 is true; Statement-2 is true; Statement-2 is
not a correct explanation for Statement-I.
(c) Statement-1 is true; Statement-2 is false.
(d) Statement-1 is false; Statement-2 is true.
58 AL the studente falace nerformed noorly in
11
24261. The statement p
(a) a tautology (b)
eqivalent to p
2. Let A and nu
~9) 1:
[JEE M 2014)
a fallacy
q (d) equivalent to – p 9
elemento
11
243State the following statement is True or
False

If Susan does not like spinach then the truth value of the statement “Susan
likes ice cream and she like spinach” is
True
A. True
B. False

11
244Write the converse and contrapositive of
the statement “lf the two lines are
parallel then they do not intersects in the same plane”.
11
245Let ( S ) be non-empty subset of ( boldsymbol{R} ) then consider the following statement “Every number ( x in S ) is an even
number.”

Negation of the statement will be
A. There is no number ( x in S ) which is even
B. There exists a number ( x in S ) which is not even
C. There exists a number ( x in S ) which is odd
D. ( (B) ) and ( (C) ) both

11
246State whether the following sentences are always true, false or ambiguous. Justify your answer. The earth is the only planet where life exist.11
247( sim[(sim p) wedge q] ) is logically equivalent to
A ( . sim(p vee q) )
В ( . sim[p wedge(sim q)] )
C ( . p wedge(sim q) )
D. ( p vee(sim q) )
E ( .(sim p) vee(sim q) )
11
248Statement ( 1: sim(mathbf{p} leftrightarrow sim mathbf{q}) ) is equivalent
to ( mathbf{p} leftrightarrow mathbf{q} )
Statement ( 2: sim(mathbf{p} leftrightarrow sim q) ) is a
tautology
A. Both Statement 1 and Statement 2 are true and
Statement 2 is a correct explanation for statement 1
B. Both Statement 1and Statement 2 are true and Statement 2 is not a correct explanation for Statement
1
C. Statement 1 is true but statement 2 is false
D. Statement 1 is false but Statement 2 is true
11
249The contrapositive of ( (boldsymbol{p} vee boldsymbol{q}) Rightarrow boldsymbol{r} ) is
A ( cdot r Rightarrow(p vee q) )
в. ( r Rightarrow(p vee q) )
c. ( r Rightarrow(sim p wedge sim q) )
D. ( p Rightarrow(q vee r) )
11
250The inverse of the propositions ( (p wedge sim )
( boldsymbol{q}) rightarrow boldsymbol{r} ) is
( mathbf{A} cdot(sim r) rightarrow(sim p) vee q )
в. ( (sim p) vee q rightarrow(sim p) )
c. ( r rightarrow p vee(sim q) )
D. ( (sim p) wedge(sim q) rightarrow r )
11
251If statements ( p, q, r ) have truth values ( T )
( F, T ) respectively then which of the following statement is true
A ( cdot(p rightarrow q) wedge r )
в. ( (p rightarrow q) vee sim r )
C ( .(p wedge q) vee(q wedge r) )
D. ( (p rightarrow q) rightarrow r )
11
252Write the negation of the statement ” ( sqrt{7} ) is irrational”11
253If statement ( boldsymbol{p} rightarrow(boldsymbol{q} vee boldsymbol{r}) ) is true then
the truth values of statements ( p, q, r )
respectively
( mathbf{A} cdot mathbf{T}, mathbf{F}, mathbf{T} )
в. F, Т, F
( mathrm{c} cdot mathrm{F}, mathrm{F}, mathrm{F} )
D. all of these
11
254State whether the following statements are true or false. Give reasons for your
For any real number ( boldsymbol{x}, boldsymbol{x}^{2} geq mathbf{0} )
A. True
B. False
11
255Which of the following is the inverse of the proposition “If a number is prime, then it is odd”?
A. If a number is not prime, then it is odd.
B. If a number is not a prime, then it is not odd.
c. If a number is not odd, then it is not a prime.
D. If a number is not odd, then it is a prime.
11
256Logically equivalent statement to ( boldsymbol{p} leftrightarrow boldsymbol{q} )
is
( mathbf{A} cdot(p rightarrow q) wedge(q rightarrow p) )
( mathbf{B} cdot(p wedge q) vee(q rightarrow p) )
( mathbf{c} cdot(p wedge q) rightarrow(q vee p) )
D. none of these
11
257Which of the following is not a tautology?
A. ( p rightarrow(p vee q) )
в. ( (p wedge q) rightarrow p )
c. ( (p vee q) rightarrow(p wedge(sim q)) )
D. ( (p vee sim p) )
11
258“If Tom buys a red skateboard then
Which statement below is logically equivalent?
A. If Amanda does not buy green in-line skates then Tom does not buy a red skateboard
B. If Tom does not buy a red skateboard then Amanda does not buy green in-line skates
C. If Amanda buys green in-line skates then Tom buys a red skateboard
D. If Tom buys a red skateboard then Amanda does not buy green in-line skates
11
259( P rightarrow(q rightarrow r) ) is logically equivalent to
A ( cdot(q vee q) rightarrow sim r )
в. ( (p wedge q) rightarrow sim r )
c. ( (p vee q) rightarrow r )
D. ( (p wedge q) rightarrow r )
11
260Which of the following is not proposition?
A. 3 is prime.
B. ( sqrt{2} ) is irrational.
c. Mathematics is interesting
D. 5 is an even integer
11
261The negative of the statement “he is rich and happy” is given by
A. He is not rich and not happy
B. He is not rich or not happy
c. He is rich and happy
D. He is not rich and happy
11
262The converse of ( boldsymbol{p} rightarrow(boldsymbol{q} rightarrow boldsymbol{r}) ) is
( mathbf{A} cdot(q wedge sim r) vee p )
в. ( (sim q vee r) vee p )
( c cdot(q wedge sim r) wedge sim p )
( p )
D. ( (q wedge sim r) wedge p )
11
263State whether the ” Or” used in the
(i) Sun rises or Moon sets
(ii) To apply for a driving licence you should have a ration card or a passport
(iii) All integers are positive or negative
11
264The contrapositive of the statement “If you believe in yourself and are honest
then you will get sucess” is
A. If you do not believe yourself and are dishonest then you will not get success.
B. If you do not believe yourself and are dishonest then you will get success
c. If you get success then you are honest and you also believe in yourself.
D. If you will not get success then you don’t not believe in yourself or are not honest
11
265Consider the statements
(i)Two plus three is five.
(ii) Every square is a rectangle.
(iii) Sun rises in the east.
(iv) The earth is not a star.
Which of the above statements have
truth value (T) ?
A. (i) and (ii)
B. (ii) and (iii)
c. (iii) and (iv)
D. All of these
11
266Determine the contrapositive of the following statement:
Only If Max studies will he pass the test.
11

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