Mathematical Reasoning & Induction · JEE Main previous year questions

For any two statements p and q, the negation of the expression $p\vee (\sim p\wedge q)$ is :

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

If $p\Rightarrow (q\vee r)$ is false, then the truth values of p, q, r are respectively :-

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

If three distinct numbers a, b, c are in G.P. and the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root, then which one of the following statements is correct?

1. A.$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in G.P.
2. B.d, e, f are in A.P
3. C.d, e, f are in G.P
4. D.$\frac{d}{a},\frac{e}{b},\frac{f}{c}$ are in A.P.

Which one of the following statements is not a tautology?

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

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

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

The expression $\sim (\sim p\to q)$ is logically equivalent to :

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

Contrapositive of the statement " If two numbers are not equal, then their squares are not equal." is :

1. A.If the squares of two numbers are equal, then the numbers are not equal
2. B.If the squares of two numbers are equal, then the numbers are equal
3. C.If the squares of two numbers are not equal, then the numbers are equal
4. D.If the squares of two numbers are not equal, then the numbers are not equal

If the Boolean expression $(p\oplus q)\wedge (\sim p\odot q)$ is equivalent to $p\wedge q$, where $\oplus ,\odot \in \{\wedge ,\vee \}$, then the ordered pair $(\oplus ,\odot )$ is :

1. A.$(\vee ,\wedge )$
2. B.$(\vee ,\vee )$
3. C.$(\wedge ,\vee )$
4. D.$(\wedge ,\wedge )$

Consider the set of all lines $px+qy+r=0$ such that $3p+2q+4r=0$. Which one of the following statements is true?

1. A.The lines are not concurrent
2. B.The lines are concurrent at the point $\left(\frac{3}{4},\frac{1}{2}\right)$
3. C.The lines are all parallel
4. D.Each line passes through the origin

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

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

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

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

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

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

Contrapositive of the statement ‘If two numbers are not equal, then their squares are not equal’, is :

1. A.If the squares of two numbers are equal, then the numbers are equal.
2. B.If the squares of two numbers are equal, then the numbers are not equal.
3. C.If the squares of two numbers are not equal, then the numbers are not equal.
4. D.If the squares of two numbers are not equal, then the numbers are equal.

The proposition $(\sim p)\vee (p\wedge \sim q)$ is equivalent to :

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

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

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

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

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

The statement $\sim (p\leftrightarrow \sim q)$ is :

1. A.equivalent to $\sim p\leftrightarrow q$
2. B.a tautology
3. C.a fallacy
4. D.equivalent to $p\leftrightarrow q$

Let $p$ and $q$ denote the following statements $p$: The sun is shining $q$: I shall play tennis in the afternoon The negation of the statement "If the sun is shining then I shall play tennis in the afternoon", is

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

The logically equivalent proposition of $p\iff q$ is

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

Statement 1: If the points $(1,2,2),(2,1,2)$ and $(2,2,z)$ and $(1,1,1)$ are coplanar, then $z=2$. Statement 2: If the 4 points $P,Q,R$ and $S$ are coplanar, then the volume of the tetrahedron $PQRS$ is 0.

1. A.Statement 1 is false, Statement 2 is true.
2. B.Statement 1 is true, Statement 2 is false.
3. C.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
4. D.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.

Let $X$ and $Y$ are two events such that $P(X\cup Y)=P(X\cap Y)$. Statement 1: $P(X\cap Y^{\prime })=P(X^{\prime }\cap Y)=0$ Statement 2: $P(X)+P(Y)=2P(X\cap Y)$

1. A.Statement 1 is false, Statement 2 is true.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
3. C.Statement 1 is true, Statement 2 is false.
4. D.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.

Let $f(x)=\sin x,g(x)=x$. Statement 1: $f(x)\le g(x)$ for $x$ in $(0,\infty )$ Statement 2: $f(x)\le 1$ for $x$ in $(0,\infty )$ but $g(x)\to \infty$ as $x\to \infty$.

1. A.Statement 1 is true, Statement 2 is false.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
3. C.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
4. D.Statement 1 is false, Statement 2 is true.

The Statement that is TRUE among the following is

1. A.The contrapositive of $3x+2=8\Rightarrow x=2$ is $x\ne 2\Rightarrow 3x+2\ne 8$.
2. B.The converse of $\tan x=0\Rightarrow x=0$ is $x\ne 0\Rightarrow \tan x=0$.
3. C.$p\Rightarrow q$ is equivalent to $p\vee \sim q$.
4. D.$p\vee q$ and $p\wedge q$ have the same truth table.

Let $s_1=\sum _{j=1}^{10}j(j-1){}^{10}{C}_j$, $s_2=\sum _{j=1}^{10}j\cdot {}^{10}{C}_j$ and $s_3=\sum _{j=1}^{10}{j}^2\cdot {}^{10}{C}_j$. Statement-1 : $S_3=55\times 2^9$. Statement-2 : $S_1=90\times 2^8$ and $S_2=10\times 2^8$.

1. A.Statement - 1 is true, Statement- 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
2. B.Statement - 1 is true, Statement-2 is false.
3. C.Statement - 1 is false, Statement-2 is true.
4. D.Statement - 1 is true, Statement-2 is true; Statement - 2 is a correct explanation for Statement - 1.

Let S be a non-empty subset of R. Consider the following statement: $P$ : There is a rational number $x\in S$ such that $x>0$. Which of the following statements is the negation of the statement P?

1. A.There is no rational number $x\in S$ such that $x\le 0$
2. B.Every rational number $x\in S$ satisfies $x\le 0$
3. C.$x\in S$ and $x\le 0\Rightarrow x$ is not rational
4. D.There is a rational number $x\in S$ such that $x\le 0$