Mathematical Reasoning & Induction · JEE Main previous year questions

Let g : N $\to$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all $n\ge 0$. Then which of the following statements is true?

1. A.There exists an onto function f : N $\to$ N such that fog = f
2. B.There exists a one-one function f : N $\to$ N such that fog = f
3. C.gogog = g
4. D.There exists a function f : N $\to$ N such that gof = f

The Boolean expression $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to :

1. A.$\sim q$
2. B.q
3. C.p
4. D.$\sim p$

Consider the statement "The match will be played only if the weather is good and ground is not wet". Select the correct negation from the following :

1. A.The match will not be played and weather is not good and ground is wet.
2. B.If the match will not be played, then either weather is not good or ground is wet.
3. C.The match will be played and weather is not good or ground is wet.
4. D.The match will not be played or weather is good and ground is not wet.

Which of the following Boolean expressions is not a tautology?

1. A.$(p\Rightarrow q)\vee (\sim q\Rightarrow p)$
2. B.$(q\Rightarrow p)\vee (\sim q\Rightarrow p)$
3. C.$(p\Rightarrow \sim q)\vee (\sim q\Rightarrow p)$
4. D.$(\sim p\Rightarrow q)\vee (\sim q\Rightarrow p)$

The Boolean expression $(p\wedge \sim q)\Rightarrow (q\vee \sim p)$ is equivalent to :

1. A.$q\Rightarrow p$
2. B.$p\Rightarrow q$
3. C.$\sim q\Rightarrow p$
4. D.$p\Rightarrow \sim q$

If P and Q are two statements, then which of the following compound statements is a tautology?

1. A.$((\mathrm{P}\Rightarrow \mathrm{Q})\wedge \sim \mathrm{Q})\Rightarrow (\mathrm{P}\wedge \mathrm{Q})$
2. B.$((\mathrm{P}\Rightarrow \mathrm{Q})\wedge \sim \mathrm{Q})\Rightarrow \mathrm{Q}$
3. C.$((\mathrm{P}\Rightarrow \mathrm{Q})\wedge \sim \mathrm{Q})\Rightarrow \mathrm{P}$
4. D.$((\mathrm{P}\Rightarrow \mathrm{Q})\wedge \sim \mathrm{Q})\Rightarrow \sim \mathrm{P}$

Which of the following Boolean expression is a tautology?

1. A.$(p\wedge q)\vee (p\to q)$
2. B.$(p\wedge q)\vee (p\vee q)$
3. C.$(p\wedge q)\to (p\to q)$
4. D.$(p\wedge q)\wedge (p\to q)$

The contrapositive of the statement "If you will work, you will earn money" is :

1. A.If you will not earn money, you will not work
2. B.If you will earn money, you will work
3. C.You will earn money, if you will not work
4. D.To earn money, you need to work

The negation of the Boolean expression $p\vee (\sim p\wedge q)$ is equivalent to :

1. A.$p\wedge \sim q$
2. B.$\sim p\vee \sim q$
3. C.$\sim p\wedge q$
4. D.$\sim p\wedge \sim q$

Consider the statement : “For an integer n, if $n^3-1$ is even, then n is odd.” The contrapositive statement of this statement is :

1. A.For an integer n, if n is even, then $n^3-1$ is even.
2. B.For an integer n, if $n^3-1$ is not even, then n is not odd.
3. C.For an integer n, if n is odd, then $n^3-1$ is even.
4. D.For an integer n, if n is even, then $n^3-1$ is odd.

Let p, q, r be three statements such that the truth value of $(p\wedge q)\to (\sim q\vee r)$ is F. Then the truth values of p, q, r are respectively :

1. A.T, F, T
2. B.F, T, F
3. C.T, T, T
4. D.T, T, F

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is :

1. A.If I will catch the train, then I reach the station in time.
2. B.If I do not reach the station in time, then I will not catch the train.
3. C.If I will not catch the train, then I do not reach the station in time.
4. D.If I do not reach the station in time, then I will catch the train.

Which of the following is a tautology ?

1. A.$(\sim p)\wedge (p\vee q)\to q$
2. B.$(q\to p)\vee \sim (p\to q)$
3. C.$(p\to q)\wedge (q\to p)$
4. D.$(\sim q)\vee (p\wedge q)\to q$

Negation of the statement : $\sqrt{5}$ is an integer or 5 is irrational is :

1. A.$\sqrt{5}$ is not an integer and 5 is not irrational.
2. B.$\sqrt{5}$ is irrational or 5 is an integer.
3. C.$\sqrt{5}$ is an integer and 5 is irrational.
4. D.$\sqrt{5}$ is not an integer or 5 is not irrational.

If $p\to (p\wedge \sim q)$ is false, then the truth values of p and q are respectively :

1. A.T, T
2. B.T, F
3. C.F, T
4. D.F, F

Which one of the following is a tautology?

1. A.$P\wedge (P\vee Q)$
2. B.$P\vee (P\wedge Q)$
3. C.$Q\to (P\wedge (P\to Q))$
4. D.$(P\wedge (P\to Q))\to Q$

Which of the following statements is a tautology?

1. A.$\sim (p\wedge \sim q)\to p\vee q$
2. B.$\sim (p\vee \sim q)\to p\vee q$
3. C.$\sim (p\vee \sim q)\to p\wedge q$
4. D.$p\vee (\sim q)\to p\wedge q$

The logical statement $(p\Rightarrow q)\wedge (q\Rightarrow \sim p)$ is equivalent to :

1. A.q
2. B.$\sim p$
3. C.p
4. D.$\sim q$

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha {)}^k+(\beta {)}^k$, $k\ge 1$, then which one of the following statements is not true?

1. A.$(p_1+p_2+p_3+p_4+p_5)=26$
2. B.$p_5=11$
3. C.$p_3=p_5-p_4$
4. D.$p_5=p_2\cdot p_3$

Let A, B, C and D be four non-empty sets. The contrapositive statement of "If $A\subseteq B$ and $B\subseteq D$, then $A\subseteq C$" is :

1. A.If $A⊈C$, then $A⊈B$ or $B⊈D$
2. B.If $A⊈C$, then $A⊈B$ and $B\subseteq D$
3. C.If $A\subseteq C$, then $B\subset A$ or $D\subset B$
4. D.If $A⊈C$, then $A\subseteq B$ and $B\subseteq D$

If the truth value of the statement $p\to (\sim q\vee r)$ is false (F), then the truth values of the statements p, q, r are respectively :

1. A.T, F, T
2. B.F, T, T
3. C.T, T, F
4. D.T, F, F

Let A, B and C be sets such that $\varphi \ne \text{A}\cap \text{B}\subseteq \text{C}$. Then which of the following statements is not true ?

1. A.If $(\text{A}-\text{B})\subseteq \text{C}$, then $\text{A}\subseteq \text{C}$
2. B.$\text{B}\cap \text{C}\ne \varphi$
3. C.$(\text{C}\cup \text{A})\cap (\text{C}\cup \text{B})=\text{C}$
4. D.If $(\text{A}-\text{C})\subseteq \text{B}$, then $\text{A}\subseteq \text{B}$

Let $f(x)=x^2$ , $x\in \mathrm{R}$. For any $\mathrm{A}\subseteq \mathrm{R}$, define $g(\mathrm{A})=\{x\in \mathrm{R}:f(x)\in \mathrm{A}\}$. If S = [0, 4], then which one of the following statements is not true ?

1. A.$g(f(\mathrm{S}))\ne \mathrm{S}$
2. B.$f(g(\mathrm{S}))=\mathrm{S}$
3. C.$f(g(\mathrm{S}))\ne f(\mathrm{S})$
4. D.$g(f(\mathrm{S}))=g(\mathrm{S})$

Which one of the following Boolean expressions is a tautology?

1. A.$(p\vee q)\wedge (\sim p\vee \sim q)$
2. B.$(p\vee q)\vee (p\vee \sim q)$
3. C.$(p\wedge q)\vee (p\wedge \sim q)$
4. D.$(p\vee q)\wedge (p\vee \sim q)$

The negation of the Boolean expression $\sim s\vee (\sim r\wedge s)$ is equivalent to :

1. A.$\sim s\wedge \sim r$
2. B.r
3. C.$s\vee r$
4. D.$s\wedge r$