MCQ Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 3

The converse of

$p\Rightarrow q$ is

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$p\Rightarrow q$
0%
$q\Rightarrow p$
0%
$-p\Rightarrow -q$
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$-q \Rightarrow -p$

Explanation

Suppose conditional statement of the form "If

$p$ then

$q$ is given.The converse is "If

$p$ then

$q$ .

$p \implies q$ is

$q \implies p$ is converse

$P\rightarrow (q\rightarrow r)$ is logically equivalent to

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$(q\vee q)\rightarrow \sim r$
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$(p\wedge q)\rightarrow \sim r$
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$(p\vee q)\rightarrow r$
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$(p\wedge q)\rightarrow r$

Explanation

$P \rightarrow (\sim q\rightarrow r)$

$= p \rightarrow (\sim q\vee r)$

$\sim p\vee (\sim q\vee r)$

$(\sim p \vee \sim q) \vee r$

$\sim (p\wedge q) \vee r$

$(p\wedge q) \rightarrow r$

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The negation of the statement

$(p\rightarrow q)\wedge r$ is

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$p\wedge \sim q\vee \sim r$
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$(\sim p\wedge q)\wedge (\sim r)$
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$(p\wedge \sim q)\wedge (r)$
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$(p\wedge \sim q)\wedge (\sim r)$

Explanation

$p\rightarrow q$

$=\sim p\vee q---(1)$

$\sim [(p\rightarrow q)\wedge r]$

$\sim [\sim p\vee q\wedge r]$ ...[form(i)]

$\sim (\sim p\vee q)\vee \sim r$ ...[Demorgans low]

$\sim (\sim p)\wedge \sim q\vee q\sim r$

$av(b\wedge c)$

$p\wedge \sim q\vee \sim r$

$=(a\vee b)\vee (a\vee c)]$

1227061_1302288_ans_6e29cc5b0f284b9282e76c6bedf3ac80.jpg

$p\vee q$ is true, when

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either $p$ of $q$ are true
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$p$ is true and $q$ is false
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$p$ is false and $q$ is true
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All of these

Explanation

We draw the table q disjunctive between two variables

$p\quad \quad q\quad \quad pVq\\ T\quad \quad T\quad \quad T\\ T\quad \quad F\quad \quad T\\ F\quad \quad T\quad \quad T\\ F\quad \quad F\quad \quad F$

The disjunction is true when either

$p$ is true,

$q$ is true or both

$p$ and

$q$ is true.

Contrapositive of the statement ''if two number are not equal then their square are not equal is ;

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If the squares of two number are equal , then the number are not equal
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If the squares of two number are equal , then the number are equal
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If the square of two number are not equal then number are equal
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If the square of two number are not, equal , then the number are not equal

Explanation

Contrapositive of

$p \rightarrow q$ is

$\sim q \rightarrow \sim p$

Hence, contrapositive of

$\text{if two number are not equal then their square are not equal is}$

$\text{If the squares of two number are equal, then the number are equal}$ .

State whether the following statement is True or False.
The sum of three odd numbers is even.

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True
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False

Explanation

Odd number are

$1,3,5,7,9,11,13,15$

Taking any three odd numbers and adding them

$1+3+5$

$5+7+9=21$

$11+9+7=27$

The sum of the three odd number is always odd

So, the statement is false.

State whether the following statement is True or False.
The product of two even numbers is always even.

0%
True
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False

Explanation

True

The product of two even numbers is even. The product of two even numbers is even. Let m and n be any integers so that

$2m$ and

$2k$ are two even numbers. The product is

$2m(2k) = 2(2mk)$ , which is even.

State whether the following statement is True or False.
All even numbers are composite numbers.

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True
0%
False

Explanation

False

$2$ is a prime number.

A prime number is a whole number that only has two factors which are itself and one. A composite number has factors in addition to one and itself. The numbers 0 and 1 are neither prime nor composite. All even numbers are divisible by two and so all even numbers greater than two are composite numbers.

State whether the following statement is True or False.
If an even number is divided by

$2$ , the quotient is always odd.

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True
0%
False

Explanation

[False]

If an even number is divided by

$2$ , the quotient is always odd.

That is

$12÷2=6$ which is an even number

State whether the following statement is True or False.

$2$ is the only even prime number.

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True
0%
False

State whether the statement

$p:$ If

$x$ is a real number such hat

$x^{3}+x=0$ , then

$x$ is

$0$ is true/false

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True
0%
False

State whether the statements given are True or False

Two rational with different numerators can never be equal

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True
0%
False

Explanation

Let

$\dfrac{2}{3}$ and

$\dfrac{4}{6}$ be two rational numbers.
Now,

$\Rightarrow \dfrac{4}{6}=\dfrac{4÷2}{6÷2}$

$=\dfrac{2}{3}$

Therefore, two rational numbers with different numerators can be equal.

Select and write the correct answer from the given alternative of the following question:
If

$p \wedge q$ is false and

$p \wedge q$ is true, then __________ is not true.

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$p \vee q$
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$p \leftrightarrow q$
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$\sim p\, \vee \, \sim q$
0%
$q \vee \, \sim p$

Explanation

$p \wedge q$ is false and

$p \vee q$ is true, then

$p \leftrightarrow q$ is not true.

Select and write the correct answer from the given alternative of the following question:
Inverse of statement pattern

$(p \vee q) \rightarrow (p \wedge q)$ is ________.

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$(p \wedge q) \rightarrow (p \vee q)$
0%
$\sim (p \vee q) \rightarrow (p \wedge q)$
0%
$(\sim p \wedge \, \sim q) \rightarrow (\sim p \vee \, \sim q)$
0%
$(\sim p \vee \, \sim q) \rightarrow (\sim p \, \wedge \, \sim q)$

Explanation

Inverse of statement pattern

$(p \vee q) \rightarrow (p \wedge q)$ is

$(\sim p \wedge \, \sim q) \rightarrow (\sim p \vee \, \sim q)$

Select and write the correct answer from the given alternative of the following question:
If

$A = \left \{1 , 2 , 3 , 4 , 5 \right \}$ then which of the following is not true ?

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$\exists \, x \, \in \, A$ such that $x + 3 = 8$
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$\exists \, x \, \in \, A$ such that $x + 2 < 9$
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$\forall \, x \, \in \, A , x + 6 \geq 9$
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$\exists \, x \, \in \, A$ such that $x + 6 < 10$

Explanation

$\forall \, x \, \in \, A , x + 6 \geq 9$ as,

$\because 1+6 \ngeq 9$ ,

$2+6 \ngeq 9$

Select and write the correct answer from the given alternative of the following question:
The negation of inverse of

$\sim p \rightarrow q$ is ___________.

0%
$q \wedge p$
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$\sim p \wedge\, \sim q$
0%
$p \wedge q$
0%
$\sim q \rightarrow \, \sim p$

Explanation

The negation of inverse of

$\sim p \rightarrow q$ is

$q \wedge p$

Select and write the correct answer from the given alternative of the following question:
If

$p \wedge q$ is F ,

$p \rightarrow q$ is F then the truth values of

$p$ and

$q$ are ___________.

0%
T , T
0%
T , F
0%
F , T
0%
F , F

Explanation

$p \wedge q$ is F ,

$p \rightarrow q$ is F then the truth values of

$p$ and

$q$ are T , F.

Select and write the correct answer from the given alternative of the following question:
The negation of

$p \wedge (q \rightarrow r)$ is __________.

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$\sim p \wedge (\sim q \rightarrow \, \sim r)$
0%
$p \vee (\sim q \vee r)$
0%
$\sim p \wedge (\sim q \rightarrow \, \sim r)$
0%
$\sim p \vee (\sim q \wedge \sim r)$

Explanation

The negation of

$p \wedge (q \rightarrow r)$ is

$\sim p \vee (\sim q \wedge \sim r)$

$p\wedge (\sim p) \Rightarrow p$ is

0%
a tautology
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a contradiction
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neither tautology nor contradiction
0%
none of these

Explanation

A proposition is said to be a

$contradiction$ if its truth value is

$F$ for any assignment of truth values to its components.

From the above truth table, we can clearly see that the answer is

$B$ .

$p$ : He is hard working.

$q$ : He is intelligent.
Then

$\sim q\Rightarrow\sim p$ , represents

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If he is hard working, then he is not intelligent.
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If he is not hard working, then he is intelligent.
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If he is not intelligent, then he is not had working.
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If he is not intelligent, then he is hard working.

The contrapositive of "if in a triangle

$ABC, AB=AC$ , then

$\angle B=\angle C$ ", is

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If in a triangle $ABC, \angle B=\angle C$ , then $AB=AC$ .
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If in a triangle $ABC, AB\neq AC$ , then $\angle B\neq\angle C$ .
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If in a triangle $ABC, \angle B\neq\angle C$ , then $AB\neq AC$ .
0%
If in a triangle $ABC, \angle B\neq\angle C$ , then $AB=AC$ .

Explanation

Take

$p:AB=AC$

and

$q:\angle B=\angle C$

So the given statement is symbolically represented as

$p\rightarrow q$

Now by definition, Contrapositive of a conditional statement

$p\rightarrow q$ is

$\sim q\rightarrow \sim p$

$\sim q:\angle B\neq \angle C$

and

$\sim p:AB\neq AC$

Thus

$\sim q\rightarrow \sim p$ is given by

If in a

$\triangle ABC, \angle B\neq \angle C$ then

$AB\neq AC$ .

$p:$ He is hard working.

$q:$ He will win.
The symbolic form of "If he will not win then he is not hard working", is

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$p\Rightarrow q$
0%
$(\sim p)\Rightarrow (\sim q)$
0%
$(\sim q)\Rightarrow (\sim p)$
0%
$(\sim q)\Rightarrow p$

Explanation

Given

$p:$ He is hard working

and

$q:$ He will win

we get

$\sim p:$ He is not hard working

and

$\sim q:$ He will not win

Now the given statement in the question is "If he will not win then he is not hard working"

For this conditional statement, the symbolic form is

$\left( \sim q \right) \Rightarrow \left( \sim p \right)$

The converse of: "If two triangles are congruent then they are similar" is

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If two triangles are similar then they are congruent.
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If two triangles are not congruent then they are not similar.
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If two triangles are not similar then they are not congruent.
0%
None

The contrapositive of

$p\Rightarrow q$ , is

0%
$p\Rightarrow q$
0%
$q\Rightarrow p$
0%
$\sim p\Rightarrow \sim q$
0%
$\sim q\Rightarrow \sim p$

Explanation

The contrapositive of "If

$p$ then

$q$ " is "If not

$q$ then not

$p$ " which is represented by

$\sim q\Rightarrow \sim p$ .

The inverse of "If

$x$ has courage, then

$x$ will win", is

0%
If $x$ will win, then $x$ has courage.
0%
If $x$ has no courage, then $x$ will not win.
0%
If $x$ will not win, then $x$ has no courage.
0%
If $x$ will not win, then $x$ has courage.

Explanation

Take

$p:x$ has courage

and

$q:x$ will win

So the given conjugation is

$p\Rightarrow q$

Now we need to find Inverse of this.

Be definition, Inverse of

$p\Rightarrow q$ is

$\sim p\longrightarrow \sim q$

$\sim p:x$ has no courage and

$\sim q:x$ will not win

Thus the inverse

$\sim p\longrightarrow \sim q$ is symbolic form of "If

$x$ has no courage, then

$x$ will not win.

$p:$ He is hard working.

$q:$ He will win.
The symbolic form of "He is hard working then he will win", is

0%
$p\vee q$
0%
$p\wedge q$
0%
$p \Rightarrow q$
0%
$q \Rightarrow p$

Explanation

Given

$p:$ He is hard working

and

$q:$ He will win

He is hard working then he will win is a conditional statement.

$p$ happens then

$q$ will happen.

The symbolic form of this condition is

$p\Rightarrow q$

The converse of "If in a triangle

$ABC, AB=AC$ , then

$\angle B=\angle C$ ", is

0%
lf in a triangle $ABC, \angle B=\angle C$ , then $AB=AC$ .
0%
lf in a triangle $ABC, AB\neq AC$ , then $\angle B\neq\angle C$ .
0%
lf in a triangle $ABC, \angle B\neq\angle C$ , then $AB\neq AC$ .
0%
lf in a triangle $ABC, \angle B\neq\angle C$ , then $AB=AC$ .

Explanation

Take

$p:AB=AC$

and

$q:\angle B=\angle C$

So the given statement is symbolically represented as

$p\rightarrow q$

Now by definition, Converse of a conditional statement

$p\rightarrow q$ is

$q\rightarrow p$

$q\rightarrow p$ is given by

"If in a triangle

$ABC, \angle B=\angle C,$ then

$AB=AC.$ "

The converse of

$p\Rightarrow q$ , is

0%
$p\Rightarrow q$
0%
$q\Rightarrow p$
0%
$\sim p\Rightarrow \sim q$
0%
$\sim q\Rightarrow \sim p$

Explanation

Given conditional statement is "If p then q".
The converse is "If q then p."
Symbolically, the converse of

$p\Rightarrow q$ is

$q\Rightarrow p$ .
A conditional statement is not logically equivalent to its converse.
Hence, option B.

The inverse proposition of

$(p \wedge \sim q)\Rightarrow r$ , is

0%
$\sim r \Rightarrow \sim p\vee q$
0%
$\sim p\vee q \Rightarrow \sim r$
0%
$r \Rightarrow p\wedge \sim q$
0%
none of these.

Explanation

Inverse proportion of

$p \rightarrow q$ is

$\sim p \rightarrow \sim q$

$\therefore$ inverse proportion of

$p \land \sim q \rightarrow r$ is

$\sim (p\ \wedge \sim q ) \rightarrow \sim r$

$\implies \sim p \lor q \rightarrow \sim r$

Which of the following is the inverse of the proposition "If a number is prime, then it is odd"?

0%
If a number is not prime, then it is odd.
0%
If a number is not a prime, then it is not odd.
0%
If a number is not odd, then it is not a prime.
0%
If a number is not odd, then it is a prime.

Explanation

p : A number is prime
q : It is odd
We have

$p\rightarrow q$
The inverse of

$p\rightarrow q$ is

$\sim p\rightarrow\sim q$
i.e. If a number is not a prime then it is not odd.

CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 3 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 3 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

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