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CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 5 - MCQExams.com
CBSE
Class 11 Engineering Maths
Mathematical Reasoning
Quiz 5
The contrapositive of $$\sim p \rightarrow ( q \rightarrow \sim r)$$ is
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$$(~q \wedge r) \rightarrow ~p$$
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$$(q \rightarrow r) \rightarrow~p$$
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$$(q \vee ~r) \rightarrow ~ p$$
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None of these.
Explanation
The contrapositive of $$\sim p \rightarrow ( q \rightarrow \sim r)$$ is
$$\equiv \sim ( q \rightarrow \sim r) \rightarrow \sim (\sim p )$$
$$\equiv \sim (\sim q \vee \sim r) \rightarrow p \equiv (q \wedge r) \rightarrow p $$
Let $$S$$ be non-empty subset of $$R$$ then consider the following statement
"Every number $$\displaystyle x\: \epsilon \: S $$ is an even number."
Negation of the statement will be
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There is no number $$\displaystyle x\: \epsilon \: S $$ which is even
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There exists a number $$\displaystyle x\: \epsilon \: S $$ which is not even
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There exists a number $$\displaystyle x\: \epsilon \: S $$ which is odd
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($$B$$) and ($$C$$) both
Explanation
The given statement implies $$x$$ in the set $$S$$ is always even. The negation of this means there exists $$x$$ in $$S$$ which is not even, or which is odd.
So, option $$D$$ is correct.
The compound statement, "If you want to top the school, then you do not study hard" is equivalent t
o
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"If you want to top the school, then you need to study hard".
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"If you will not top in the school, then you study hard".
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"If you study hard, then you will not top the school".
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"If you do not study hard, then you will top in the school".
Explanation
An equivalent statement of a compound statement, has the same meaning as the compound statement.
So, for the given compound statement, the equivalent statement is " If you study hard, then you will not top the school. "
Which of the following is equivalent to $$p\Longleftrightarrow q$$?
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$$p\implies q$$
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$$q\implies p$$
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$$(p\implies q)\wedge(q\implies p)$$
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None of these
Explanation
The truth values of $$ p\Longleftrightarrow q $$ are equal to the truth values of $$ (p\Longrightarrow q)\wedge (q\Longrightarrow p) $$
So, they are equivalent.
The property $$p\wedge(q\vee r)\equiv(p\wedge q)\vee(p\wedge r)$$ is called
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associative law
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commutative law
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distributive law
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idempotent law
Explanation
As the symbol AND $$ \wedge $$ is distributed over OR $$ \vee $$ , the given property is also called Distributive property.
Write the negation of the statement "If the switch is on
, then the fan rotates".
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"If the switch is not on, then the fan does not rotate".
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"If the fan does not rotate, then the switch is not on".
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"The switch is not on or the fan rotates".
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"The switch is on and the fan does not rotate".
Explanation
To find the negation of the statement, we find the opposite of the conclusion.
So, negation of the given statement is " If the switch is on, then the fan does not rotate"
The negation of the statement, "I go to school everyday", is
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I never go to school.
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Some days, I do not go to school.
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Not all the days I do not go to school.
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All days I go to school.
Explanation
The negation or the opposite of the statement, " I go to school everyday" will be "Some days, I do not go to school" as there is a possibility of not going to school on some days rather than not going at all.
In the above network, current flows from N to T w
hen
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p closed, q closed, r opened and s opened.
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p dosed, q opened, s closed and r opened.
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q closed, p opened, r opened and s closed.
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p opened, q opened, r closed and s closed.
Explanation
We can notice that current will flow from M to T, if atleast one of the pair of switches $$ q ; s $$ or $$ p ; r $$ are closed.
So, from the given options , the right option is q closed , open , r opened and s closed
Find the converse of the statement, "If ABCD is
square, then it is a rectangle".
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If ABCD is a square, then it is .not a rectangle.
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If ABCD is not a square, then i is a rectangle.
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if ABCD is a rectangle, then it is square.
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If ABCD is not a square, then it is not a rectangle.
Explanation
In the converse of a statement, we reverse the statements such that the meaning does not change.
So converse of the given statement is " If ABCD is a rectangle, then it is a square"
The counter example of the statement, "All odd
numbers are primes", is
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7
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5
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9
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All the above
Explanation
To prove the statement "All odd numbers are primes" we have to find an odd number which is not prime."
For example, $$ 9 $$ is an odd number which is not prime.
If "All odd numbers are primes and the sum of three angles in a triangle is $$190^\circ$$", then "All odd numbers are primes or the sum of the angles in a triangle is $$190^\circ$$" is a
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tautology
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contradiction
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contingency
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not a statement
Explanation
Both the statements are false as all odd numbers are primes and the sum of three angles of a triangle is not $$ 190^o $$
A tautology is a statement that is always true. And Two false statements make a True statement in a Tautology.
In the above network, current flows from M to N, when
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q closed, r opened and p closed.
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q opened, p opened and r closed.
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q opened, p closed and r closed.
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q closed, p closed and r opened.
Explanation
We can notice that current will flow from M to N, if atleast one of the switches $$ q $$ or $$ p $$ is closed and $$ r $$ is surely closed.
SO, from the given option is , the right option is q opened, p closed, , r closed
Which of the following connectives can be used for
describing a switching network?
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$$\vee$$
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$$\wedge$$
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Both (1) and (2)
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None of these
Explanation
We can use both AND $$ \wedge $$ to denote Series networks and OR $$ \vee $$ to denote Parallel networks to describe a switching network.
"No square of a real number is less than zero" is
equivalent to
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for every real number a, $$a^2$$ is non negative.
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$$\forall a\in R$$, $$a^2\ge0$$.
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either (1) or (2).
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None of these
Explanation
Both the options $$ 1 $$ and $$ 2 $$ are correct, as option $$ 1 $$ has the statement rewritten with the same words as the original statement and option $$ 2 $$ has the statement written in mathematical way.
Find the truth value of the statement, "The sum of
any two odd numbers is an odd number".
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True
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False
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Neither True nor False
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Cannot be determined
Explanation
Let us take two odd numbers to check the statement.
$$ 5 + 3 = 8 $$
$$ 7 + 5 = 12 $$
We notice that sum of two odd numbers is an even number.
Hence te given statement is False.
The truth value of the statement, "We celebrate our
Independence day on 15 August", is
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T
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F
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neither T nor F
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Cannot be determined
Explanation
The given statement is True as we do celebrate our Independence day on $$ 15th $$ August.
What is the truth value of the statement $$2\times3=6$$ or $$5+8=10$$?
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True
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False
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Neither True nor False
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Cannot be determined
Explanation
The first statement $$ 2 \times 3 = 6 $$ is True.
But the second statement $$ 5 + 8 = 10 $$ is False.
Since the statements are connected using OR, the result will be True, if anyone of them is True.
Hence, truth value of the given statement is T.
The converse of converse of the statement $$p\implies\sim q$$ is
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$$\sim q\implies p$$
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$$\sim p\implies q$$
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$$p\implies\sim q$$
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$$\sim q\implies\sim p$$
Explanation
Converse of a converse of a statement will result in the original statement itself.
Hence, the answer is $$ p\Longrightarrow \sim q $$ which is the original statement itself.
Which of the following connectives satisfy commu
tative law?
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$$\wedge$$
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$$\vee$$
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$$\Leftrightarrow$$
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All the above
Explanation
All the three given symbols satisfy the commutative law.
$$ p\wedge q\Leftrightarrow q\wedge p $$
$$p\vee q\Leftrightarrow q\vee p$$
$$(p\Leftrightarrow q)\Leftrightarrow (q\Leftrightarrow p)$$
In which of the following cases, $$p\Leftrightarrow q$$ is true?
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p is true, q is true.
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p is false, q is true.
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p is true, q is false.
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None of these.
Explanation
As p is equivalent to q, the statement is true only when both are true.
Write the compound statement, "If p, then q and if q,
then p" in symbolic form.
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$$(p\wedge q)\wedge(q\wedge p)$$
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$$(p\implies q)\vee(q\implies p)$$
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$$(q\implies p)\wedge(p\implies q)$$
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$$(p\wedge q)\vee(q\wedge p)$$
Explanation
Then is denoted by the symbol $$ \Longrightarrow $$ and AND is denoted by the symbol $$ \wedge $$
So, the given statement can be denoted in symbolic form as $$ (q\Longrightarrow p)\wedge (p\Longrightarrow q) $$
When does the truth value of the statement $$(p\vee r)\Leftrightarrow (q\vee r)$$ become true?
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p is true, q is true.
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p is false, q is false.
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p is true, r is true.
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Both (1) and (3)
Explanation
As the LHS and RHS of the statements are equivalent, the statement will be true, when both $$ p $$ and $$ q $$ are either True or False.
Find the negation of the statement, "Some odd numbers
are not prime".
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Some odd numbers are primes.
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There is an odd number which is not a prime.
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All odd numbers are primes.
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Not all odd numbers are primes.
Explanation
To find the negation of the statement, we find the opposite of the conclusion.
So, negation of the given statement is " All odd numbers are primes"
when does the current flow from A to B?
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p is open, q is open, r is closed.
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p is closed, q is open, r is closed.
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p is closed, q is closed, r is open.
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p is open, q is closed, r is closed.
Explanation
The network can be described by the statement of $$p\vee(q\vee r)$$
p
q
r
$$q\vee r$$
$$p\vee(q\vee r)$$
T
T
T
T
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
F
F
F
T
F
T
F
T
T
F
F
T
T
T
F
F
F
F
F
So, the current flows from A to B in the following cases
(i) p is closed, q is closed, r is closed
(ii) p is closed, q is closed, r is open
(iii) p is closed, q is open, r is closed
(iv) p is open, q is closed, r is closed
(v) p is closed, q is open, r is open
(vi) p is open, q is closed, r is open
(vii) p is open, q is open, r is closed
He is smart; He is intelligent.
Write the conjunction.
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He is smart and he is intelligent.
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He is smart maybe he is intelligent.
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He is intelligent because
He is smart
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He is not smart and he is not intelligent.
Explanation
The conjunction is both the statements together, that is " He is smart and he is intelligent"
"If natural numbers are whole numbers, then rational numbers are integers" or "If rational numbers are integers, then natural numbers are whole numbers"
is
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a tautology
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a contradiction
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a contingency
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not a statement
Explanation
A tautology is a statement that is always true.
Here, both the statements of the compound statement are False. Hence the compound statement is Tue as a combination of Two False is True.
Hence, the given statement is a tautology.
Find the inverse of the statement, "If $$\triangle{ABC}$$ is equilat
eral, then it is isosceles".
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If $$\triangle{ABC}$$ is isosceles, then it is equilateral.
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If $$\triangle{ABC}$$ is not equilateral, then it is isosceles.
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If $$\triangle{ABC}$$ is not equilateral, then it is not isosceles.
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If $$\triangle{ABC}$$ is not isosceles, then it is not equilateral.
Explanation
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
So, inverse of the given statement is "If $$ \triangle ABC $$ is not equilateral , then it is not isosceles"
In the above network, current flows from T to M,
when
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p closed, q closed and r opened.
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p closed, q opened and r closed.
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p opened; q closed and r closed.
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All the above
Explanation
For current to flow from $$ T $$ to $$ M $$, the line from T to M should be completely connected.
This can happen when both $$ p, q $$ are closed irrespective of $$ r $$ open or close.
And can happen when $$ r $$ is closed, and either of $$ p , q $$ is closed or open.
So All of the given options are answers.
The compound statement, "If you won the race; then you did not run faster than others" is equivalent t
o
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"If you won the race, then you ran faster than others".
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"If you ran faster than others, then you won the race".
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"If you did not win the race, then you did not run faster than others".
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"If you ran faster than others, then you did not win the race".
Explanation
Two equivalent statements are those which have the same meaning.
From the given options, we can see that the compound statement, "If you ran faster than others, then you did not win the race" hss the same meaning as the given statement.
Which of the following is negation of the statement
"All birds can fly".
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"Some birds cannot fly".
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"All the birds cannot fly".
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"There is at least one bird which can fly".
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All the above
Explanation
To find the negation of the statement, we find the opposite of the conclusion.
So, negation of the given statement is " Some birds cannot fly"
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