Mathematical Reasoning & Induction · JEE Main previous year questions

For the function $f(x)=e^{\sin |x|}-|x|$, $x\in \mathbf{R}$, consider the following statements : Statement I : $f$ is differentiable for all $x\in \mathbf{R}$. Statement II : $f$ is increasing in $\left(-\pi ,-\frac{\pi }{2}\right)$. In the light of the above statements, choose the correct answer from the options given below :

1. A.Both Statement I and Statement II are true
2. B.Both Statement I and Statement II are false
3. C.Statement I is true but Statement II is false
4. D.Statement I is false but Statement II is true

Let $\mathrm{A}=\left[\begin{array}{cc}1& 2\\ 1& \alpha \end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}3& 3\\ \beta & 2\end{array}\right]$. If ${\mathrm{A}}^2-4\mathrm{A}+\mathrm{I}=\mathrm{O}$ and ${\mathrm{B}}^2-5\mathrm{B}-6\mathrm{I}=\mathrm{O}$, then among the two statements : (S1): $[(\mathrm{B}-\mathrm{A})(\mathrm{B}+\mathrm{A}){]}^{\mathrm{T}}=\left[\begin{array}{cc}13& 15\\ 7& 10\end{array}\right]$ and (S2): $\det (\mathrm{adj}(\mathrm{A}+\mathrm{B}))=-5$,

1. A.only (S1) is correct
2. B.only (S2) is correct
3. C.both (S1) and (S2) are correct
4. D.both (S1) and (S2) are wrong

Among the statements : I: If $\left|\begin{array}{ccc}1& \cos \alpha & \cos \beta \\ \cos \alpha & 1& \cos \gamma \\ \cos \beta & \cos \gamma & 1\end{array}\right|=\left|\begin{array}{ccc}0& \cos \alpha & \cos \beta \\ \cos \alpha & 0& \cos \gamma \\ \cos \beta & \cos \gamma & 0\end{array}\right|$, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =\frac{3}{2}$, and II: If $\left|\begin{array}{ccc}{x}^2+x& x+1& x-2\\ 2x^2+3x-1& 3x& 3x-3\\ x^2+2x+3& 2x-1& 2x-1\end{array}\right|=\mathrm{p}x+\mathrm{q}$, then ${\mathrm{p}}^2=196{\mathrm{q}}^2$,

1. A.both are true
2. B.both are false
3. C.only I is true
4. D.only II is true

For the function $f(x)=(\cos x)-x+1,x\in \mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0,\pi ]$. (S2) $f(x)$ is decreasing in $[0,\frac{\pi }{2}]$ and increasing in $[\frac{\pi }{2},\pi ]$.

1. A.Both (S1) and (S2) are incorrect.
2. B.Only (S1) is correct.
3. C.Only (S2) is correct.
4. D.Both (S1) and (S2) are correct.

For $0<c<b<a$, let $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ and $\alpha \ne 1$ be one of its roots. Then, among the two statements (I) If $\alpha \in (-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$ (II) If $\alpha \in (0,1)$, then $b$ may be the geometric mean of $a$ and $c$

1. A.only (II) is true
2. B.Both (I) and (II) are true
3. C.only (I) is true
4. D.Neither (I) nor (II) is true

Consider the system of linear equations $x+y+z=4\mu ,x+2y+2\lambda z=10\mu ,x+3y+4{\lambda }^{2}z={\mu }^2+15$ where $\lambda ,\mu \in \mathbf{R}$. Which one of the following statements is NOT correct ?

1. A.The system has unique solution if $\lambda \ne \frac{1}{2}$ and $\mu \ne 1,15$
2. B.The system has infinite number of solutions if $\lambda =\frac{1}{2}$ and $\mu =15$
3. C.The system is consistent if $\lambda \ne \frac{1}{2}$
4. D.The system is inconsistent if $\lambda =\frac{1}{2}$ and $\mu \ne 1$

Consider the function $f:\left[\frac{1}{2},1\right]\to \mathbb{R}$ defined by $f(x)=4\sqrt{2}{x}^3-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point. (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos \frac{\pi }{12}$. Then

1. A.Both (I) and (II) are correct.
2. B.Only (I) is correct.
3. C.Both (I) and (II) are incorrect.
4. D.Only (II) is correct.

The negation of the statement $((A\wedge (B\vee C))\Rightarrow (A\vee B))\Rightarrow A$ is

1. A.equivalent to $B\vee \sim C$
2. B.equivalent to $\sim A$
3. C.equivalent to $\sim C$
4. D.a fallacy

The statement $(p\wedge (\sim q))\vee ((\sim p)\wedge q)\vee ((\sim p)\wedge (\sim q))$ is equivalent to _________.

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

Among the two statements (S1) : $(p\Rightarrow q)\wedge (p\wedge (\sim q))$ is a contradiction and (S2) : $(p\wedge q)\vee ((\sim p)\wedge q)\vee (p\wedge (\sim q))\vee ((\sim p)\wedge (\sim q))$ is a tautology

1. A.both are false.
2. B.only (S1) is true.
3. C.both are true.
4. D.only (S2) is true.

Let $f:[2,4]\to \mathbb{R}$ be a differentiable function such that $(x{\log }_{e}x)f^{\prime }(x)+({\log }_{e}x)f(x)+f(x)\ge 1,x\in [2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{4}$. Consider the following two statements : (A) : $f(x)\le 1$, for all $x\in [2,4]$ (B) : $f(x)\ge \frac{1}{8}$, for all $x\in [2,4]$ Then,

1. A.Neither statement (A) nor statement (B) is true
2. B.Only statement (A) is true
3. C.Only statement (B) is true
4. D.Both the statements (A) and (B) are true

The number of ordered triplets of the truth values of $p,q$ and $r$ such that the truth value of the statement $(p\vee q)\wedge (p\vee r)\Rightarrow (q\vee r)$ is True, is equal to _________.

For $a\in \mathbb{C}$, let $\mathrm{A}=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})>\mathrm{Im}(\bar{a}+z)\}$ and $\mathrm{B}=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})<\mathrm{Im}(\bar{a}+z)\}$. Then among the two statements : (S1) : If $\mathrm{Re}(a),\mathrm{Im}(a)>0$, then the set A contains all the real numbers (S2) : If $\mathrm{Re}(a),\mathrm{Im}(a)<0$, then the set B contains all the real numbers,

1. A.both are false
2. B.only (S1) is true
3. C.only (S2) is true
4. D.both are true

The converse of $((\sim p)\wedge q)\Rightarrow r$ is

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

The negation of the statement $(p\vee q)\wedge (q\vee (\sim r))$ is :

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

The statement $\sim [p\vee (\sim (p\wedge q))]$ is equivalent to :

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

Negation of $(p\Rightarrow q)\Rightarrow (q\Rightarrow p)$ is :

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

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

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

Statement $(\text{P}\Rightarrow \text{Q})\wedge (\text{R}\Rightarrow \text{Q})$ is logically equivalent to :

1. A.$(\text{P}\Rightarrow \text{R})\wedge (\text{Q}\Rightarrow \text{R})$
2. B.$(\text{P}\Rightarrow \text{R})\vee (\text{Q}\Rightarrow \text{R})$
3. C.$(\text{P}\wedge \text{R})\Rightarrow \text{Q}$
4. D.$(\text{P}\vee \text{R})\Rightarrow \text{Q}$

Among the statements : (S1) : $2023^{2022}-1999^{2022}$ is divisible by 8 (S2) : $13(13{)}^n-12n-13$ is divisible by 144 for infinitely many $n\in \mathbb{N}$

1. A.both (S1) and (S2) are incorrect
2. B.only (S1) is correct
3. C.only (S2) is correct
4. D.both (S1) and (S2) are correct

Among the statements (S1) : $(p\Rightarrow q)\vee ((\sim p)\wedge q)$ is a tautology (S2) : $(q\Rightarrow p)\Rightarrow ((\sim p)\wedge q)$ is a contradiction

1. A.neither (S1) and (S2) is True
2. B.only (S2) is True
3. C.both (S1) and (S2) are True
4. D.only (S1) is True

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

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

(S1) $(p\Rightarrow q)\vee (p\wedge (\sim q))$ is a tautology (S2) $((\sim p)\Rightarrow (\sim q))\wedge ((\sim p)\vee q)$ is a contradiction. Then

1. A.only (S2) is correct
2. B.both (S1) and (S2) are correct
3. C.only (S1) is correct
4. D.both (S1) and (S2) are wrong

The number of values of $r\in \{p,q,\sim p,\sim q\}$ for which $((p\wedge q)\Rightarrow (r\vee q))\wedge ((p\wedge r)\Rightarrow q)$ is a tautology, is :

1. A.2
2. B.1
3. C.4
4. D.3

Among the statements : (S1) $((p\vee q)\Rightarrow r)\iff (p\Rightarrow r)$ (S2) $((p\vee q)\Rightarrow r)\iff ((p\Rightarrow r)\vee (q\Rightarrow r))$

1. A.only (S1) is a tautology
2. B.neither (S1) nor (S2) is a tautology
3. C.both (S1) and (S2) are tautologies
4. D.only (S2) is a tautology