Eamcet Mathematics - Mathematical Reasoning Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
Which of the following is an open statement?
- (a)
Good morning to all.
- (b)
wish you all the best
- (c)
x is a whole number
- (d)
Where do you live?
An open statement is a sentence which becomes a statement by giving a certain value to the variable in that sentence.
By giving a certain value to x in the sentence 'x is a whole num,ber', we can obtain a statement.So, this is an open statement
Choose disjunction among the following sentences.
- (a)
It is raining and the Sun is shining
- (b)
Ram and Shyam are good friends
- (c)
2 or 3 is a prime number
- (d)
Everyone who lies in India is an Indian
In conjunction, two sentences are connected with 'OR'.
Which among the following is not a conjunction?
- (a)
Gautam and Rahul are good friends
- (b)
The earth is round and the Sun is hot
- (c)
9>4and12>15
- (d)
None of the above
In the sentence,'Gautam and nRahul are good friends', the word 'and' is not a connective.So, this sentence is not a conjunction.
~(p v q)v(~p^q) is logically equivalent to
- (a)
~p
- (b)
p
- (c)
q
- (d)
nq
~(p v q)v(~p^q)\(\equiv \)(~p^~q)v(~p^q)
\(\equiv \)~p^(~q v q)\(\equiv \)~p^T=~P
Choose disjunction among the following sentences.
- (a)
It is raining and the Sun is shining
- (b)
Ram and Shyam are good friends
- (c)
2 or 3 is a prime number
- (d)
Everyone who lies in India is an Indian
In disjunction, two sentences are connected with 'OR'.
Which among the following is not a conjunction?
- (a)
Gautam and Rahul are good friends
- (b)
The earth is round and the Sun is hot
- (c)
9 > 4 and 12 > 15
- (d)
None of the above
In the sentences, 'Gautam and Rahul are good friends', the word 'and' is not a connective. So, this sentence is not a conjunction.
The negation of '12 > 4' is
- (a)
\(12\le 4\)
- (b)
\(13>5\)
- (c)
\(12>3\)
- (d)
\(12\ge 4\)
Negation is a denial of a statement.
So, \(12\le 4\) is correct option.
The contrapositive of \(p\rightarrow(\sim q \rightarrow \sim r)\) is
- (a)
\(\left( \sim q\wedge r \right) \rightarrow \sim p\)
- (b)
\((q\wedge \sim r)\rightarrow \sim p\)
- (c)
\(p\rightarrow \left( \sim r\vee q \right) \)
- (d)
\(p\wedge \left( q\wedge r \right) \)
The contrapositive of p\(\rightarrow \)q is ~q\(\rightarrow \)~p
∴ Contrapositive of p\(\rightarrow \)(~q\(\rightarrow \)~r) is
~(~q\(\rightarrow \)~r)\(\rightarrow \)~p
\([\because \sim (p\rightarrow q)\equiv p\wedge \sim q\\ \therefore \quad \sim (\sim q\rightarrow \sim r)\equiv \sim q\wedge r]\)
\(\equiv \)(~q\(\wedge \)r)\(\rightarrow \)~p
Which of the following is not equivalent to \(p\leftrightarrow q\) ?
- (a)
p if and only if q
- (b)
p is necessary and sufficient for q
- (c)
q if and only if p
- (d)
None of the above
Options (a), (b) and (c) are equivalent to \(p\leftrightarrow q\)
If \(p\rightarrow q\) is false, then which of the following is true?
- (a)
p is true and q is false
- (b)
p is false and q is false
- (c)
p is true and q is true
- (d)
None of the above
\(p\rightarrow q\) is false only in the case, when p is true and q is false.
\(\sim \left( p\vee q \right) \vee \left( \sim p\wedge q \right) \) is logically equivalent to
- (a)
~p
- (b)
p
- (c)
q
- (d)
nq
\(\equiv p\wedge \left( \sim q\vee q \right) \equiv \sim p\wedge T=\sim p\)
\(\quad \quad \equiv p\wedge \left( \sim q\vee q \right) \equiv \sim p\wedge T=\sim p\)
The statement 'If X2 is not even, then X is not even' is converse of the statement
- (a)
if x2 is odd, then x is even
- (b)
if x is not even, then x2 is even
- (c)
if x is even, then X2 is even
- (d)
if X is odd, then X2 is even
Converse of \(p\rightarrow q\quad is\quad q\rightarrow p\) .
If \(p\rightarrow \left( \sim p\vee q \right) \) is false, then the truth value of p and q are respectively
- (a)
F,F
- (b)
T,T
- (c)
T,F
- (d)
F,T
\(p\rightarrow \left( \sim p\vee q \right) \equiv F\Rightarrow p\equiv T\quad and\quad \sim p\vee q\equiv F\)
P is true and both ~p and q are flase.
So, P is true and q is false.
The negation of the statement "Plants take in CO2 and give out O2' is
- (a)
Plants do not take in CO2 and do not give out O2
- (b)
Plants do not take in CO2 or do not give out O2
- (c)
Plants take in CO2 and do not give out O2
- (d)
Plants take in CO2 or do not give out O2
\(\sim \left( p\wedge q \right) =\sim p\vee \sim q\)
\(\sim r\wedge s\) is equivalent to
- (a)
\(r\rightarrow s\)
- (b)
∼(s\(\rightarrow \)r)
- (c)
s\(\rightarrow \)r
- (d)
∼(r\(\rightarrow \)s)
\(\sim r\wedge s\) is equivalent to ∼(s\(\rightarrow \)r)
The dual of the statement \([(r\vee s)\wedge \sim s)\vee (\sim s)]\)
- (a)
\([(r\wedge s)\vee \sim s)]\wedge (\sim s)\)
- (b)
\([(r\wedge s)\wedge \sim r)]\vee (\sim s)\)
- (c)
\([(r\vee s)\wedge \sim r)]\vee (\sim s)\)
- (d)
\([(r\wedge s)\wedge \sim s)]\vee (\sim s)\)
Dual of a statement can be obtained by replacing \('\wedge '\quad by\quad '\vee ','\vee '\quad by\quad '\wedge ',\) T by F and F by T.
Let \(S:[(r\vee s)\wedge \sim s]\vee (\sim s)\)
So, dual of S is \(S:[(r\wedge s)\vee \sim s]\wedge (\sim s)\)
Among the following statements, which is a tautology?
- (a)
\(p\wedge (p\vee q)\)
- (b)
\(p\vee (p\wedge q)\)
- (c)
\([p\wedge (p\rightarrow q)]\rightarrow q\)
- (d)
\(q\rightarrow [p\wedge (p\rightarrow q)]\)
\(p\wedge (p\vee q)\quad is\quad F,\quad when\quad p\equiv F\)
\(p\vee (p\wedge q)\quad is\quad F,\quad when\quad p\equiv F,\quad q\equiv F\)
\(and\quad q\rightarrow [p\wedge (p\rightarrow q)]\quad is\quad F,\quad when\quad p\equiv F,\quad q\equiv T\)
\(So,\quad for\quad [p\wedge (p\rightarrow q)]\rightarrow q\equiv [p\wedge (\sim p\vee q)]\rightarrow q\)
\(\equiv [\{ p\wedge (\sim p)\} \vee (p\wedge q)]\rightarrow q\)
Which of the following is always true?
- (a)
\((p\Rightarrow q)\equiv \sim q\Rightarrow \sim p\)
- (b)
\(\sim (p\vee q)\equiv \sim p\wedge \sim q\)
- (c)
\(p\Rightarrow q\equiv p\wedge q\)
- (d)
\(\sim (p\vee q)\equiv \sim p\wedge \sim q\)
\(\quad p\Rightarrow q\equiv \sim p\vee q\)
\(\therefore \sim (p\Rightarrow q)\equiv p\wedge \sim q\)
\(Also, \sim (p\vee q)\equiv \sim p\wedge \sim q\)
Which among the following is false?
- (a)
\(\sqrt { 6 } \) is a rational number or an irrational number
- (b)
A square is a quadrilateral or a 6 sided polygon
- (c)
Mumbai is the capital of Kolkata or karnataka
- (d)
The school is closed, if there is a holiday or Sunday
In first compound statement, first statement is false and second is true. So, the first compound statement is true.
In second compound statement, one of the component statement is true, so it is true.
In third compound statement, both of the component statement, both the component statements are false, so it is false.
In fourth compound statement, both the component statements are true, so it is true.
The negation of the statement 'If I become a teacher, then I will open a school', is
- (a)
The negation of the statement 'If I become a teacher, then I will open a school', is
- (b)
Neither I will become a teacher nor I will open a school
- (c)
I will not become a teacher or I will open a school
- (d)
I will become a teacher and I will not open a school
Negation of a statement is denial of the statement.
Let p :I become a teacher, q : I will open a school.
The statement is \(p\rightarrow q\equiv (\sim p)\vee q\)
So, negation is \(\sim [(\sim p)\vee q]=p\wedge (\sim q).\)
Which of the following sentences is not a statement?
- (a)
8 is less than 6
- (b)
Every set is a finite set
- (c)
How far is Chennai from here?
- (d)
The sum of two positive numbers is positive
(a) This sentence is false because 8 is greater than 6. Hence it is a statement.
(b) This sentence is also false since there are sets which are not finite. Hence it is a statement.
(c) This sentence is not a statement because it is a question.
(d) This sentence is true since sum of two positive numbers is always positive. Hence it is a statement.
Which of the following is a logical statement?
- (a)
Are you going to Chennai?
- (b)
Shyam is a handsome boy
- (c)
Alas! I were a king
- (d)
Sum of two irrational numbers is always irrational
The statement given in (d) is a logical statement as we can decide logically whether it is true or not. Infact, this statement is false as sum of two irrational numbers may not be irrational.
Which of the following is not a logical statement?
- (a)
3≤3
- (b)
Every square is a rhombus
- (c)
Switch on the fan
- (d)
Sum of the angles of a triangle is always 1800
Which of the following is a statement?
- (a)
Please help me
- (b)
Hurrah! India has won the match
- (c)
Good night to all
- (d)
17 is a prime number
Option (d) is true. So it is a statement. (a) is a request, (b) is exclamatory and (c) is a wish. So none of (a), (b), (c) is a statement.
Which one of the following is not a statement?
- (a)
It is not that the sky is blue
- (b)
Is the sky blue?
- (c)
The sky is dark in the night
- (d)
The sky is not blue in the night
Which of the following is a statement?
- (a)
Roses are black
- (b)
Mind your own business
- (c)
Be punctual
- (d)
Do not tell lies
Which one of the following statements is not a false statement?
- (a)
p: Each radius of a circle is a chord of the circle
- (b)
q: Cricle is a particular case of an ellipse
- (c)
r: \(\sqrt{13}\) is a rational number
- (d)
s: The centre of a circle bisects each chord of the circle
(a) No radius of a circle is a chord. So, p is false.
(b) We know that equation of an ellipse is given by \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \)=1if we take a = b, then we get x2+y2=a2
which satisfies all the conditions of circle .
∴ Circle is the particular case of an ellipse. So, q is not false.
(c) \(\sqrt{13}\) is an irrational number. So, r is false.
(d) The centre of a circle bisects only those chords which are diameters. So, s is false.
Which of the following statement is true after writing negation of the given statements?
- (a)
Australia is a continent
- (b)
There does not exist a quadrilateral which has all its sides equal
- (c)
Every natural number is greater than 0
- (d)
The sum 00 and 4 is 7
(a) The negation of the given statement is Australia is not a continent.
We know that this statement is false.
(b) The negation of the statement is There exists a quadrilateral which has all its sides equal. This statement is true because we know that square is a quadrilateral such that its four sides are equal.
(c) The negation of the statement is
There exists a natural number which is not greater than 0,This is a false statement.
(d) The negation of the statement is
The sum of 3 and 4 is not equal to 7.
This statement is false.
Which of the following compound statements is true after writing the component statements of each compound statement?
- (a)
A line is straight and extends indefinitely in both directions
- (b)
0 is greater than every positive integer and less than every negative integer
- (c)
All living things have two legs and two eyes
- (d)
42 is divisible by 4 and 5
(a) The component statements are
p : A line is straight.
q :A line extends indefinitely in both directions.
Both these statements are true. Therefore, the compound statement is true.
(b) The component statements are
p : 0 is greater than every positive integer.
q : 0 is less than every negative integer.
Both statements are false. Therefore, the compound statement is false.
(c) The component statements are
p : All living things have two legs.
q : All living things have two eyes.
Both these statements are false. Therefore, the compound statement is false.
(d) The component statements are
p : 42 is divisible by 4.
q : 42 is divisible by 5.
Both the statements are false. So compound statement is false.
Consider the following compound statements. Which of the following statements is not true?
- (a)
Murnbai is the capital of Rajasthan or Maharashtra
- (b)
\(\sqrt{3}\) is a rational number or an irrational number
- (c)
125 is a multiple of 7 or 8
- (d)
A rectangle is a quadrilateral or a regular hexagon
(a) The component statements are
p : Mumbai is the capital of Rajasthan.
q : Mumbai is the capital of Maharashtra.
We note that p is false and q is true, so the compound statement is true.
(b) The component statements are
p: \(\sqrt{3}\) is a rational number.
q: \(\sqrt{3}\) is an irrational number.
We note that p is false and q is true, so compound statement is true.
(c) The component statements are
p: 125 is a multiple of 7.
q: 125 is a multiple of 8.
We note that p and q both are false statements, so compound statement is false.
(d) The component statements are
p : A rectangle is a quadrilateral.
q : A rectangle is a regular hexagon.
We note that p is true and q is false. So compound statement is true.
"It is raining and weather is cold." The negation of the statement is
- (a)
It is not raining and weather is cold
- (b)
It is raining or weather is not cold
- (c)
It is not raining or weather is not cold
- (d)
It is not raining and weather is not cold
Each of the following statement is represented symbolically. Which one is not correct?
p: It is raining; q: It is pleasant.
- (a)
It is raining and pleasant p v q
- (b)
It is not raining still it is pleasant (- p) \(\wedge \) q
- (c)
It is neither raining nor pleasant (~ p) \(\wedge \) (~ q)
- (d)
If it rains then, it will be pleasant p⟶q
Let p : 7 is not greater than 4.
q : Paris is in France.
be two statements. Then ~ (p v q) is the statement
- (a)
7 is greater than 4 or Paris is not in France
- (b)
7 is greater than 4 and Paris is not in France
- (c)
7 is greater than 4 and Paris is in France
- (d)
7 is not greater than 4 or Paris is not in France
p:7 is not greater than 4
q: Paris is in France
∴ p v q: 7 is not greater than 4 or Paris is in France.
So, ~ (p v q): 7 is greater than 4 and Paris is not in France.
Write the contrapositive of the following statement:
If a number is divisible by 9, then it is divisible by 3
- (a)
If a number is not divisible by 9, then it is visible by 3
- (b)
If a number is not divisible by 3, then it is not
divisible by 9 - (c)
If a number is divisible by 3, then it is not divisible by 9.
- (d)
If a number is divisible by 9, then it is not divisible by 3
Write the converse of the following statement.
If a number n is even, then n2 is even
- (a)
If a number n2 is odd, then n is not even
- (b)
If a number n2 is even, then n is odd
- (c)
If a number n2 is even, then n is even
- (d)
If a number n is odd, then n2 is odd
Write the converse of the following statement.
If you do all the exercises in the book, you get an A-grade in the class.
- (a)
If you get an A-grade in the class, then you have done all the exercises of the book
- (b)
If you did not get an A-grade in the class, then you have not done all the exercises of the book
- (c)
If you have done all the exercises of the book, then you get an A-grade.
- (d)
You will get an A-grade if you will complete all exercises of the book
Write the converse of the following statement.
If two integers a and b are such that a > b, then a - b is always a positive integer.
- (a)
If two integers a and b are such that a < b, then a - b is always positive integer
- (b)
If a < b, then a - b is always a negative integer
- (c)
If two integers a and b are such that a > b, then a - b is not a positive integer
- (d)
If two integers a and b are such that a - b is always a positive integer, then a > b
The contrapositive and converse of the statement : "If the two lines are parallel, then they do not interest in the same plane" is
- (a)
If the two straight lines intersect in a plane, the the lines are not parallel.
If the two lines do not intersect in the same plane, then the two lines are parallel - (b)
If the two straight lines do not intersect in a plane, then the lines are parallel.
If the two lines do not intersect in the same plane, then the two lines are parallel - (c)
If the two straight lines intersect in a plane, then the lines are not parallel.
If the two lines do not intersect in the same plane, then the two lines are not parallel. - (d)
None of these
Contrapositive statement: If two straight lines intersect in a plane, then the lines are not parallel.
Converse statement: If the two lines do not intersect in the same plane, then the two lines are parallel.
If each of the statements p ⟶ ~ q, ~ r ⟶ q and p is true, then
- (a)
r is false
- (b)
r is true
- (c)
q is true
- (d)
None of these
Since p is true and also p ⟶ ~q is true, therefore, ~ q is true. (∵ True statement cannot imply a
false statement).
⇒ ~ q is false
⇒ Also ~ r⟶ q is true, therefore, ~ r is false.
⇒ ~ r is true.
Which of the following is not true?
- (a)
~(P ∧ q) = (~p) v (~q)
- (b)
~ (p v q) = (~ p) ∧ (~ q)
- (c)
p ⟶ q = ~ p v q
- (d)
~ (p v q) = ~ p v ~ q
We know that ~ (p v q) = ~ P ∧ ~ q
Which of the following is different from the others?
- (a)
~(p↔️q)
- (b)
~p ↔️ q
- (c)
p ↔️ ~q
- (d)
None of these
We know that
~ (p ↔️ q) = ~ p ↔️ q = p ↔️ q
So, all the three statements are logically equivalent.
Which of the following is true?
- (a)
~(p ∧ q) = ~P ∧ ~q
- (b)
~(p v q) = ~p ∧ ~q
- (c)
~(p⟶q) = pv ~q
- (d)
None of these
We have
~ (p ∧ q) = ~ p v ~ q and ~ (p v q) = ~ p ∧ ~ q
Note that - (p ⟶ q) = ~ (- p v q) = p ∧ ~ q.
Which of the following is not logically equivalent?
- (a)
~(p⟶q) and ((p ∧ ~q)
- (b)
~ (p v q) and (~ p) ∧ (~ q)
- (c)
p ⟶ q and (~p) ∧ q
- (d)
~[pv(~q)] and (~p) ∧ q
p ⟶ q ≡(~p) v q
Make the component statements of the following compound statements and state whether the compound statement is true or false.
(i) 57 is divisible by 2 or 3.
(ii) 24 is a multiple of 4 and 6.
(iii)Allliving things have two eyes and two legs.
(iv) 2 is an even number and a prime number.
- (a)
(i) (ii) (iii) (iv) True True True False - (b)
(i) (ii) (iii) (iv) True False False True - (c)
(i) (ii) (iii) (iv) True True False True - (d)
(i) (ii) (iii) (iv) False True False True
Its component statements are:
p : 57 is divisible by 2. [False]
q: 57 is divisible by 3. [True]
So, it is a true statement.
(ii) Its component statements are
p : 24 is multiple of 4. [True]
q : 24 is multiple of 6. [True]
So, it is a true statement.
(iii) Its component statements are
p : All living things have two eyes [False]
q: All living things have two legs. [False]
So, it is a false statement.
(iv) Its component statement are
p:2 is an even number [True]
q:2 is a prime number [True]
So, it is a true statement.
The connective in the statement
"2 + 7 > 9 or 2 + 7 < 9" is
- (a)
and
- (b)
or
- (c)
>
- (d)
<
The negation of the statement "A circle is an ellipse" is
- (a)
An ellipse is a circle
- (b)
An ellipse is not a circle
- (c)
A circle is not an ellipse.
- (d)
A circle is an ellipse
The negation of the statement "7 is greater than 8" is
- (a)
7 is equal to 8
- (b)
7 is not greater than 8
- (c)
8 is less than 7
- (d)
None of these
The statement
"If x2 is not even, then x is not even" is converse of the statement
- (a)
If x2 is odd, then x is even
- (b)
If x is not even, then x2 is not even.
- (c)
If x is even, then x2 is even
- (d)
If x is odd, then x2 is even
Which of the following is not a negation of "A natural number is greater than zero"?
- (a)
A natural number is not greater than zero
- (b)
It is false that a natural number is greater than zero
- (c)
It is false that a natural number is not greater than zero
- (d)
None of these
Statement-I: The contrapositive of the statement "If x is prime number, then x is odd" is "If a number x is not odd, then x is not a prime number."
Statement-II: The converse of the statement "If x is prime number, then x is odd, then x is not a prime number".
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement-I
- (b)
If both Statement-I and Statement-II are true and Statement-II is not the correct explanation of Statement-I
- (c)
If Statement-I is true but Statement-II is false
- (d)
If Statement-I is true but Statement-II is true
Contrapositive of Statement-I: If a number x is not odd, then x is not a prime number.
Converse of Statement-II: If x is a prime number.