Mathematical Reasoning MCQ Questions & Answers in Algebra | Maths
Learn Mathematical Reasoning MCQ questions & answers in Algebra are available for students perparing for IIT-JEE and engineering Enternace exam.
51.
Negation of the proposition : If we control population growth, we prosper
A
If we do not control population growth, we prosper
B
If we control population growth, we do not prosper
C
We control population but we do not prosper
D
We do not control population, but we prosper
Answer :
We control population but we do not prosper
$$p :$$ we control population, $$q :$$ we prosper
$$\therefore $$ we have $${p \Rightarrow q}$$
Its negation is $$ \sim \left( {p \Rightarrow q} \right){\text{ i}}{\text{.e}}{\text{., }}p \wedge \sim q$$
i.e., we control population but we do not prosper.
52.
For integers $$m$$ and $$n,$$ both greater than 1, consider the following three statements :
$$P : m$$ divides $$n$$
$$Q : m$$ divides $$n^2$$
$$R : m$$ is prime, then
A
$$Q \wedge R \to P$$
B
$$P \wedge Q \to R$$
C
$$Q \to R$$
D
$$Q \to P$$
Answer :
$$Q \wedge R \to P$$
$$\left( B \right)\frac{8}{4} = 2,\frac{{64}}{4} = 16;$$ but 4 is not prime.
Hence $$P \wedge Q \to R,{\text{false}}$$
$$\left( C \right)\frac{{{{\left( 6 \right)}^2}}}{{12}} = \frac{{36}}{{12}} = 3;$$ but 12 is not prime
Hence $$Q \to R,{\text{false}}$$
$$\left( D \right)\frac{{{{\left( 4 \right)}^2}}}{8} = \frac{{16}}{8} = 2;\frac{4}{8}$$ is not an integer
Hence $$Q \to P,{\text{false}}$$
53.
The statement “If $$2^2 = 5$$ then I get first class” is logically equivalent to
A
$$2^2 = 5$$ and I do not get first class
B
$$2^2 = 5$$ or I do not get first class
C
$${2^2} \ne 5$$ or I get first class
D
None of these
Answer :
$${2^2} \ne 5$$ or I get first class
Let $$p$$ and $$q$$ be two proposition given by $$p : 2^2 = 5, q : 1$$ get first class
Here give statement is $$p \to q$$
So contrapositive of $$p \to q{\text{ is }} \sim q \to \, \sim p$$
i.e., if I do not get first class then $${2^2} \ne 5.$$
54.
Which of the following is not a statement ?
A
Please do me a favour
B
2 is an even integer
C
$$2 + 1 = 3$$
D
The number 17 is prime
Answer :
Please do me a favour
"Please do me a favour" is not a statement.
55.
If $$p : 4$$ is an even prime number, $$q : 6$$ is a divisor of 12 and $$r :$$ the HCF of 4 and 6 is 2, then which one of the following is true ?
Given that
$$p : 4$$ is an even prime number.
$$q : 6$$ is a divisor of 12.
and $$r :\,$$ the HCF of 4 and 6 is 2.
$$\therefore \,\, \sim p \vee \left( {q \wedge r} \right){\text{ is true}}{\text{.}}$$
56.
Let $$p$$ be the proposition : Mathematics is a interesting and let $$q$$ be the propositions that Mathematics is difficult, then the symbol $${p \wedge q}$$ meeans
A
Mathematics is interesting ipllies that Mathematics is difficult
B
Mathematics is interesting implies and is implied by Mathematics is difficult
C
Mathematics is interesting and Mathematics is difficult
D
Mathematics is interesting or Mathematics is difficult
Answer :
Mathematics is interesting and Mathematics is difficult
$${p \wedge q}$$ means Mathematics is interesting and Mathematics is difficult.
57.
Which of the following is always true ?
A
$$\left( { \sim p \Rightarrow q} \right) = \,\, \sim q \Rightarrow \, \sim p$$
B
$$\left( { \sim p \vee q} \right) \equiv \vee p \vee \sim q$$
C
$$ \sim \left( {p \Rightarrow q} \right) \equiv p \wedge \sim q$$
D
$$ \sim \left( {p \vee q} \right) \equiv \sim p \wedge \sim q$$
Clearly last column of the above truth table contains only $$F.$$ Hence $$\left( {p \wedge q} \right) \wedge \left( { \sim \left( {p \vee q} \right.} \right)$$ is a contradiction.