Mathematical Reasoning JEE Main Mathematics: Authentic Shift PYQs (2021-2026) with 2.5-Min Shortcut Tricks & Trap Defense

Correct Answer: (b)

Explanation: Inverse of $p \rightarrow(p \rightarrow q)$ is $\sim p \rightarrow \sim(p \rightarrow q)$ Contrapositive of inverse of $p \rightarrow(p \rightarrow q)$ is $\begin{aligned} & \sim[\sim(p \rightarrow q)] \rightarrow \sim(\sim p) \\ & \equiv(p \rightarrow q) \rightarrow p \\ & \equiv(\sim p \vee q) \rightarrow p \end{aligned}$

Correct Answer: (b)

Explanation: To determine the value of $ k $ such that $ P(n) = 2^{2n} - 1 $ is divisible by $ k $ for all $ n \in \mathbb{N} $, we start by examining a specific case: Consider $ n = 1 $: $ P(1) = 2^{2 \times 1} - 1 = 2^2 - 1 = 4 - 1 = 3 $ Since $ P(1) = 3 $, it follows that $ P(n) $ is divisible by $ 3 $. Therefore, the value of $ k $ is $ 3 $.

Correct Answer: (c)

Explanation: As $(p \leftrightarrow \sim q) \rightarrow(\sim p \wedge q)$ is false, we get $\mathrm{p} \leftrightarrow \sim \mathrm{q} \equiv \mathrm{T}$ and $\sim \mathrm{p} \wedge \mathrm{q} \equiv \mathrm{F}$ $\sim p \wedge q \equiv F$ $\Rightarrow \sim p \equiv F$ and $q \equiv F$ $\Rightarrow p \equiv T$ and $q \equiv F$

Correct Answer: (a)

Explanation: $\begin{aligned} & (p \rightarrow q) \wedge(q \rightarrow \sim p) \\ & \equiv(\sim p \vee q) \wedge(\sim q \vee \sim p) \\ & =(\sim p) \vee(q \wedge \sim q) \\ & \equiv(\sim p) \vee F \equiv \sim p\end{aligned}$

Correct Answer: (c)

Explanation: ---