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CBSE Questions for Class 11 Engineering Maths Mathematical Reasoning Quiz 2 - MCQExams.com
CBSE
Class 11 Engineering Maths
Mathematical Reasoning
Quiz 2
State the following statement is True or False
The truth value of "$$4$$ is even and $$8$$ is odd" is True
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True
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False
Explanation
For "and" to be true both conditions must be true "$$4$$ is even" is true but "$$8$$ is odd" is false.
T and F is FALSE.
Which is logically equivalent to "If today is Sunday Matt cannot play hockey"?
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Today is Sunday and Matt can play hockey
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If Matt plays hockey then today is not Sunday
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Today is Sunday and Matt cannot play hockey
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Today is not Sunday if and only if Matt plays hockey
Explanation
Matt can play Hockey on any day other than Sunday.
Hence, $$B$$ is correct.
Which statement is logically equivalent to "If Yoda cannot use a lightsaber then he cannot help Luke win the battle"
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Yoda cannot use a lightsaber and he will help Luke win the battle
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If Yoda can help Luke win the battle then he can use a lightsaber
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Yoda can use a lightsaber if and only if he can help Luke win the battle
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Yoda cannot use a lightsaber and will not help Luke win the battle
Explanation
A statement is logically equivalent to its contrapositive To form the contrapositive switch the "If" and "then" sections of the statement AND insert "NOTs" into each section Notice in this situation that inserting "NOTs" turns the thoughts positive (you are negating negative thoughts)
Which of the following statements is the contrapositive of the statement, You win the game if you know the rules but are not overconfident.
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If you lose the game then you dont know the rules or you are overconfident.
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A sufficient condition that you win the game is that you know the rules or you are not overconfident.
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If you dont know the rules or are overconfident you lose the game
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If you know the rules and are overconfident then you win the game.
Explanation
Contrapositive is the inverse of the converse of the statement.
It is obtained by first interchanging the hypothesis and conclusion and then adding "not" to both
In this case, converse is "If you win the game, then you know the rules but are not overconfident."
Inverse of this statement gives answer as
A.
Which of the following statements is the inverse
of
"Our pond floods whenever there is a thunderstorm."?
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If there is a thunderstorm, then our pond floods.
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If we do not get a thunderstorm, then our pond does not flood.
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If our pond does not flood, then we did not get a thunderstorm.
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None of these
Explanation
The given statement can be written as ''If there is a thunderstorm, then our pond floods
".
Inverse of a given statement is "
If our pond does not flood, then we did not get a thunderstorm.
".
"
If our pond does not flood, then we did not get a thunderstorm.
".
Hence, option C is correct.
$$\sim \left[ \left( p\wedge q \right) \rightarrow \left( \sim p\vee q \right) \right]$$ is
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Tautology
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Contradiction
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Neither (A) nor (B)
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Either (A) or (B)
Explanation
It is clear from the table that
$$\sim \left[\left( p\wedge q \right) \rightarrow \left( \sim p\vee q \right)\right] $$
is a contradiction.
Which of the following statements is the converse
of
"You cannot skateboard if you do not have a sense of balance."?
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If you cannot skateboard, then you do not have a sense of balance.
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If you do not have a sense of balance, then you cannot skateboard.
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If you skateboard, then you have a sense of balance.
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None of these
Explanation
Converse of "If $$P$$, then $$Q$$" is "If $$Q$$, then $$P$$".
Now, the given statement
"You cannot skateboard if you do not have a sense of balance." can be re-written as "If
you do not have a sense of balance, then you cannot skateboard."
The converse of this statement is option $$B$$.
Which of the following statements is the inverse
of
"If it rains, then I do not go fishing."?
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If I go fishing, then it does not rain.
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If I do not go fishing, then it rains.
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If it does not rain, then I go fishing.
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None of these
Explanation
Therefore, the reason comes after the assertion.
Then, the inverse of the expression "if it rains, then I do not go fishing " is:
If I go fishing, then it does not rain
The negative of the statement "he is rich and happy" is given by
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He is not rich and not happy
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He is not rich or not happy
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He is rich and happy
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He is not rich and happy
Explanation
The negation of the given statement is "he is not rich or not happy".
The inverse of the statement
"If it is raining then the grass is wet"
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"If it is not raining then the grass is not wet".
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"If it is raining then the grass is not wet".
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"If it is not raining then the grass is wet".
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None of these
Explanation
$$p\rightarrow{q}$$
Inverse is $$\sim{p}\rightarrow\sim{q}$$
$$\Rightarrow$$ Inverse of :
It it is raining then the grass is wet.
is:
If it is not raining then the grass is not wet.
Which of the following statements is
logically equivalent
to
"The solution is easy if you read the question carefully."?
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If you do not read the question carefully, the solution is hard.
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If the solution is easy, then you read the question carefully.
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If the solution is hard, then you did not read the question carefully.
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None of these
Explanation
Given Statement:
The solution is easy if you read the question carefully.
Logically equivalent statement:
If you do not read the question carefully, the solution is hard.
If p : A man is happy
q : A man is rich
Then, the statement, "If a man is not happy, then he is not rich" is written as
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$$\sim p\rightarrow \sim q$$
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$$\sim q\rightarrow p$$
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$$\sim q\rightarrow \sim p$$
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$$q\rightarrow \sim p$$
Explanation
Given, p : A man is happy
and q : A man is rich
'If a man is not happy, then he is not rich' is written as $$\sim p \rightarrow \sim q$$.
The inverse of the statement " If a person is mean, then they are a fighter " is-
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If a person is not mean, then they are not a fighter.
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If a person is not mean, then they are a fighter.
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If a person is mean, then they are not a fighter.
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None of the above
Explanation
A person is not mean is the negation of the hypothesis that a person is mean and they are not a fighter is the negation of the conclusion they are a fighter.
$$A$$
Negation of the statement "Every natural number is an integer".
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All natural numbers are whole numbers.
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Every natural number is not an integer.
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Every natural number is not a real number.
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none of the above
Explanation
Negation of "Every Natural number is an integer." is "Every Natural number is NOT an integer."
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Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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Assertion is correct but Reason is incorrect
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Assertion is incorrect but Reason is correct
Explanation
R.E.F image
$$ ar(ABC) = BC\times AD $$
$$ ar(PQR) = PR\times QS $$
As $$ \boxed{QS = AD} $$& $$ \boxed{PR = BC} $$
$$ \Rightarrow $$ area are equal.
The inverse of the statement "If a number is divisible by 4 then it is also divisible by 2" is-
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If a number is divisible by 4, then it is always divisible by 2.
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If a number is not divisible by 4, then it is divisible by 2.
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If a number is not divisible by 4, then it is not divisible by 2.
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None of the above
Explanation
a number is not divisible by 4 is the negation of the hypothesis
a number is divisible by 4
,and it is not divisible by 2 is the negation of the conclusion
it is also divisible by 2
Inverse of the statement "If the triangle with side lengths $$a, b, c$$ is a right triangle, then $$a^2 + b^2 = c^2$$.
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If $$a^2+b^2\neq c^2$$ then the triangle with side lengths $$a, b, c$$ is not a right triangle.
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If $$a^2+b^2 = c^2$$ then the triangle with side lengths $$a, b, c$$ is a right triangle.
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If the triangle with side lengths $$a,b,c$$ is a not right triangle, then $$a^2 + b^2 \neq c^2$$.
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None of the above.
Explanation
Given,
If the triangle with side lengths $$a, b, c$$ is a right triangle, then $$a^2+b^2=c^2$$
This is of the form $$p\rightarrow{q}$$
$$p$$: If the triangle with side lengths $$a, b, c$$ is a right triagle.
$$q: a^2+b^2=c^2$$
Inverse of $$p\rightarrow{q}$$ is $$\sim{p}\rightarrow\sim{q}$$
$$\Rightarrow$$ If the triangle with side lengths $$a, b, c $$ is not a right triangle, then $$a^2+b^2=c^2$$.
Which of the following is a statement?
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Rani is a beautiful girl.
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Shut the door.
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Yesterday was Friday..
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If its raining then there must be cloud in the sky.
Explanation
If its raining then there must be cloud in the sky.
This a statement.
A statement is a closed sentence. It can also be a mathematical identity.
A statement should have a complete meaning independently.
Write the inverse of the statement-
If you do not drink your milk, you will not be strong.
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If you are strong, then you drink your milk,
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If you do not drink your milk, then you are strong.
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If you drink your milk, then you are strong.
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None of the above
Explanation
If you drink your milk is negation of the hypothesis if you donot drink your milk you are strong is negation of the conclusion you will not be strong.
$$C$$
Inverse of a statement can be explained as
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Negating both the hypothesis and conclusion of a conditional statement.
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Antecedent is the negation of the original antecedent and whose consequent is the negation of the original consequent.
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both are correct
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none is correct
Explanation
Inverse of a statement means converting positive statement into a negative one.
In mathematical terms, by applying negation .
Since, option A and B means the same. Hence, option C is the answer.
State whether true or false.
The sum of the interior angles of a quadrilateral is $$350^o$$.
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True
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False
Explanation
False
The sum of interior angles of quadrilateral is $$360^{0}$$.
Then statement that sum of
interior angles of quadrilateral is $$350^{0}$$ is false.
State whether the following statements are true or false. Give reasons for your answers.
For any real number $$x, x^2 \geq 0$$.
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True
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False
Explanation
The square of any real number is always positive.
Let $$x$$ be a positive real number. We can see $$x^2\ge 0$$ as $$x \times x\ge 0$$
If $$x$$ is negative real number, then, $$(-x)(-x)=xx=x^2 \ge 0$$
So, the statement is aways TRUE.
State true or false.
A rhombus is a parallelogram.
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True
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False
Explanation
If the opposite side of a rhombus is equal then it is the parallelogram.
Then statement that a rhombus is parallelogram is true.
State whether the following statement is true or false.
The sum of two even numbers is even.
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True
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False
Explanation
The sum of two even numbers is always even then the statement is true.
For example: $$2x+4x=6x$$ and $$4+8=12$$
If a compound statement is made up of three simple statements, then the number of rows in the truth table is
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$$8$$
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$$6$$
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$$4$$
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$$2$$
Explanation
Fact: If a compound statement is made up of $$n$$ simple statements, then number of rows in the truth table will be $$2^n$$
Here $$n=3$$ is given, so number of rows will be $$=2^3=8$$
If truth values of $$p$$ be $$F$$ and $$q$$ be $$T$$. Then, truth value of $$\sim (\sim p\vee q)$$ is
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$$T$$
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$$F$$
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Either $$T$$ or $$F$$
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Neither $$T$$ not $$F$$
Explanation
$$p$$
$$q$$
$$\sim p$$
$$\sim p\vee q$$
$$\sim (\sim p\vee q)$$
$$F$$
$$T$$
$$T$$
$$T$$
$$F$$
$$\therefore$$ Truth value of $$\sim (\sim p\vee q)$$ is F.
The converse of the contrapositive of the conditional $$ p \rightarrow \sim q $$ is :
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$$ p \rightarrow q $$
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$$ \sim p \rightarrow \sim q $$
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$$ \sim q \rightarrow p $$
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$$ \sim p \rightarrow q $$
Explanation
The contrapositive of $$ p \rightarrow \sim q $$ is
$$ \sim (\sim q) \rightarrow \sim p $$ or $$ q \rightarrow \sim p $$
Also, converse of $$ q \rightarrow \sim p $$ is $$ \sim p \rightarrow q$$.
Hence, converse of the contrapositive of the conditional $$ p \rightarrow \sim q $$ is
$$ \sim p \rightarrow q$$
Let $$p, q$$ and $$r$$ be any three logical statements. Which one of the following is true?
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$$\sim [p \wedge (\sim q)] \equiv (\sim p )\wedge q$$
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$$\sim (p \vee q) \wedge (\sim r) \equiv (\sim p) \vee (\sim q) \vee (\sim r)$$
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$$\sim [p \vee (\sim q)] \equiv (\sim p) \wedge q$$
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$$\sim [p \wedge (\sim q)] \equiv (\sim p) \wedge \sim q$$
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$$\sim [p \wedge (\sim q)] \equiv p \wedge q$$
Explanation
$$\sim [p\wedge (\sim q)]\equiv (\sim p)\vee \sim (\sim q)\equiv (\sim p)\vee q$$
So option $$A$$ is not correct
$$\sim (p\vee q)\wedge (\sim r)\equiv (\sim p)\wedge (\sim q)\wedge (\sim r)$$
So option $$B$$ is also not correct.
$$\sim [p\vee (\sim q)]\equiv (\sim p)\wedge \sim (\sim q)\equiv (\sim p)\wedge q$$
So option $$C$$ is correct.
"If we control population growth, then we prosper". Negative of this proposition is:
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If we do not control population growth, we prosper
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If we control population, we do not prosper
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we control population and we do not prosper
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If we don't control population, we do not prosper
Explanation
Given,
$$\text{"if we control population growth, then we prosper."}$$
Statement contain if......then proposition.
Let $$P=\text{we control population growth}$$ and
$$Q=\text{we prosper}$$
statement=$$P\rightarrow{Q}$$
$$P\rightarrow{Q}$$ is equivalent to "$$\sim P \vee Q$$"
Therefore,
Negation of
$$\sim P \vee Q$$
is "
$$P \wedge \sim Q$$"
Now,
Negation of Statement:-
$$\text{"we control population growth and we do not prosper."}$$
Which one of the following statement is a tautology?
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$$p\rightarrow (p\rightarrow q)$$
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$$(p \vee p)\rightarrow q$$
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$$p \vee (p\rightarrow q)$$
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$$p \vee (q\rightarrow p)$$
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